for now more than two centuries , the incompressible navier - stokes equations have withstood the minds of mathematicians and physicists alike : the derivation of the nature of turbulence from the equations , as well as the global existence of smooth solutions is not known to date .the huge mathematical difficulties concerning the latter problem were recognized by its elevation to the status of `` millennium prize problem '' by the clay mathematics institute ( see the official problem description by fefferman , or review articles e.g. ) .a proof of existence of global regular solutions to the navier - stokes equation is believed to entail the development of completely new methods for the analysis of partial differential equations .the absence of mathematical certainty for the navier - stokes equations may seem to leave the physicist in a somewhat embarrassing position : the equation is known and well tested in application , but the existence of solutions is unclear in relevant cases .  yet , the actual impact of a supposed breakdown of solutions for the navier - stokes equations on physics of fluids is smaller than one might expect and appears like a mere technical detail on second thought .singularities in the navier - stokes equation would , if existent , appear on very small scales .obviously , continuum mechanics do not hold on these smallest scales and the breakdown of the model equation would appear in a regime in which the model does not describe reality at any rate .furthermore , the nature of supposed singularities for the navier - stokes equation is proven to be unphysical in nature , as it requires the existence of infinite momentum . without external forcing , from smooth initial conditions and in the presence of friction ,the occurrence of infinite momentum is impossible to justify physically .the impact of singularities is additionally limited by the fact that the space - time dimension of the singular region is proven to be less than or equal to one for the navier - stokes equations .  in the inviscid limit, the situation is quite the opposite .the incompressible euler equations for ideal fluids , appear to be of little physical significance in most applications , since friction is the dominating process on small scales .the ignorance regarding existence of global solutions is even larger for the inviscid case : the notion of weak solutions , which are well established for the navier - stokes equations since leray , is unknown for the three - dimensional euler equations .nevertheless , the formation of finite - time euler singularities significantly concerns our understanding of ( viscid ) fluid dynamics .euler singularities , if existent , would coincide with the development of large gradients in the velocity field .since no friction is limiting the increase in velocity gradients , infinite momentum is not mandatory for a blowup of the euler equations .the inviscid limit is , therefore , not only a mere description of ideal fluids , but explores the possibility of inherent dynamical processes beyond friction that limit the transition to smaller and smaller scales .this has immediate implications on the existence of a cut - off velocity in high reynolds - number navier - stokes flows , leading to the slightly exaggerated question , quoting constantin : `` do we need schrdinger s equations to calculate the flow around a moving car ? or to predict tomorrow s weather ? '' for that reason , the problem of singularities for the euler equations is of far greater importance to the physical understanding of fluids than the analogous problem for the navier - stokes equations .a similar argument is valid for turbulence . todays phenomenological description of turbulence ( e.g. ) , which is built on the basis of the celebrated theory by kolmogorov , contains as a central point that , in the limit of vanishing viscosity , energy dissipation has to stay finite .this behavior could be explained by the formation of finite - time euler singularities , as implied by onsager sconjecture . for three - dimensional incompressible flow , non - conservation of energymight be caused not only by viscosity but by missing regularity in the velocity field .energy dissipation might occur , if the hlder continuity exponent is smaller than for the velocity field .this conjecture was proven in terms of besov spaces . as a consequence ,a mathematical description of turbulence might be possible in terms of weak solutions for the euler equations , if smooth solutions gain enough roughness in finite time .therefore , insight into the formation of finite - time singularities for the euler equations could uncover a mechanism essential for the understanding of viscous turbulence .  the search for finite - time singularities of the euler equations has resulted in extensive literature , with many analytical results being relatively young .especially the advent of scientific computing has given research a new direction : reports of numerical evidence supporting or denying the existence of finite - time singularities for the euler equations are numerous ( see e.g. for a compiled list ) .as a now classical result , the blowup criterion of beale et .al ( bkm ) connects the existence of solutions for the incompressible euler equations in three dimensions to the critical accumulation of vorticity .more recently , geometric analysis of the flow has helped increasing insight into the process of vorticity growth . among these geometric blowup criteria , theorems developed by deng et .al may be seen as the first to be suitable for verification by direct numerical simulations .an approach along this way will be presented in this paper .given the results of analytical considerations and the experience gained from numerical simulation of the euler equations , certain scenarios are known to be possibly compatible with the analytic requirements of a finite - time blowup , namely the global notion of self - similar collapse to a point and the local process of vorticity accumulation by vorticity - strain coupling .it has been tried in the past to construct explicit initial conditions exploiting these scenarios to obtain numerical evidence for or against a finite - time singularity , with surprisingly inconsistent results .the major reason for this ambiguity is the critical dependence on extrapolation , which renders the identification of singular versus near - singular behavior next to impossible by numerical means .the hopes are high that the situation is less vague when considering geometric properties of the flow , as mentioned above .we will present the application of such geometric criteria to numerical data to sharpen the distinction between singular and near - singular flow evolution and identify the processes connected with this behavior .this paper is organized as follows : we first review the relevant geometric blowup - criteria that form the basis of our numeric method .this includes a motivation of how geometric properties such as curvature or spreading of lagrangian vortex line segments are connected to the accumulation of vorticity .we then describe the special class of high symmetry initial conditions used for our numerical experiments , discuss the implications of flow symmetries on the process of vorticity - strain coupling and introduce different vorticity profiles for vortex dodecapole initial conditions . using this setup , details of the its implementation for our massively parallel simulations with up to mesh pointsare given .results are presented concerning the growth of vorticity and strain , the bkm - criterion and the geometric criteria .these findings act as numerical evidence against the formation of a finite - time singularity for this class of initial conditions . a conclusion and outlooksummarize the paper .classical criteria for the development of a finite - time euler singularity have in common that they focus on global features ( such as certain norms of the velocity or the vorticity fields ) or on point - wise eulerian features ( such as ) of the flow .this comes at the disadvantage of neglecting the structures and physical mechanisms of the flow evolution . a strategy to overcome such shortcomingswas established by focusing more on geometrical properties and flow structures , such as vortex tubes or vortex lines .starting with the works of constantin et al . and cordoba and fefferman ,some of these `` geometric '' criteria ( e.g. ) have reached a phase where they allow direct verification of their assumptions with the help of numerical simulations .special focus is placed on the criteria presented by deng , hou and yu , as the assumptions are in close reach for numerical simulations .they examine the lagrangian evolution of vortex line segments and formulate a combined bound on velocity blowup and vortex segment collapse .one of the trivial consequences of the bkm theorem is the fact that no blowup can occur for the two dimensional euler equations . since the vorticity is bounded by the initial conditions for all times , a critical accumulation is impossible .this is a direct consequence of the vorticity pointing out of the plane of motion , therefore having the vortex - stretching term vanish everywhere .this may be interpreted as a motivation to focus on the behavior of the direction of vorticity , in the three - dimensional case . for 2d, is a constant in space and time ( neglecting sign ) . in 3d, the consequences of the regularity of on the growth - rate of vorticity and ultimately of the applicability of bkm can be precisely stated .  for the euler equations , this was introduced by constantin et al . . they state , roughly , that for a smoothly directed vorticity in an -region there may be no blowup in finite time as long as the velocity remains finite in this region .even though this criterion takes into account the local structure of the flow and follows the evolution of vortex lines , the ( global ) bound on the velocity makes this theorem hard to apply in practice .numerical simulations of the euler equations give no evidence for the velocity to be uniformly bounded in time .this restriction on the velocity field is weakened in a similar criterion by cordoba and fefferman .they consider vortex tubes with some properties concerning their regularity and a surrounding region of the flow . from thisit is possible to deduce , with the help of a milder assumption on the surrounding velocity , that the vortex tube can not reach zero thickness in finite time . even though the velocity field is no longer required to be uniformly bounded in time , the notion of `` regular tube '' of length is too restricting , compared to the experiences of numerical simulations .vortex lines of the three - dimensional incompressible euler equations , defined as integral curves of the vorticity direction field , are transported with the flow . as a consequence ,two points and on the same vortex line stay on the same vortex line for all times .furthermore , as a direct implication of the solenoidality of the vorticity vector field , the vorticity flux through a vortex tube is the same for each cross - section .these two arguments may be combined to get a differential notion of the connection between the vorticity at two different points on the same vortex line .a simple consequence of the solenoidality of results in since for a vortex line it holds by definition that , we have , where is the partial derivative in direction of the vortex line . thus , integrating eq .( [ eq : t1_1 ] ) along the vortex line yields paraphrased , this means : the vorticity at two different points on the same vortex line is connected by the amount of converging or diverging of neighboring vortex lines along their interconnecting path .the more violent vortex lines converge around a vortex line , the faster the vorticity grows along that line .such that it is far outside of the critical region of maximum vorticity and monitor the behavior of .right : for the position of maximum vorticity , choose such that . for a point - wise singularity , and must collapse in finite time .[ fig : t1],title="fig : " ]  such that it is far outside of the critical region of maximum vorticity and monitor the behavior of .right : for the position of maximum vorticity , choose such that .for a point - wise singularity , and must collapse in finite time .[ fig : t1],title="fig : " ]  this finding was connected with bkm by deng et al . to formulate a geometric blowup criterion .it is obvious from equation ( [ eq : vorticityonaline ] ) that the maximum vorticity at a given time can be estimated by the vorticity on its vortex line , as long as remains finite . in detailthis means :  * deng - hou - yu theorem 1 : * _ let be a family of points such that for some it holds .assume that for all there is another point on the same vortex line as , such that the direction of vorticity along the vortex line between and is well - defined .if we further assume that _ _ for some absolute constant c , and _ _ then there will be no blowup up to time . _  it is immediately clear how this criterion can be applied to numerical simulations : if the maximum vorticity exhibits fast growth in time for which it is hard to decide whether it is a finite - time blowup compatible with bkm , instead one could monitor the vorticity outside the critical region , but on the same vortex line .if it remains well bounded , and along the vortex line does not scale critically in time , it is safe to deduce a non - critical growth of .this approach is sketched in fig .[ fig : t1 ] ( left ) .however , due to the freedom of the choice of on the critical vortex line , theorem 1 may be employed in a different way to distinguish different scenarios for a finite - time singularity , as depicted in fig .[ fig : t1 ] ( right ) .suppose that the maximum vorticity grows in a way compatible with bkm .now , choose to be the position of maximum vorticity and define via for some constant independent of the time , where denotes the arc - length parameter of the curve from to .in words , choose on the same vortex line as such that the accumulation of tightening of nearby vortex lines is the same for every instance in time .this provides us with the ability to clearly distinguish between to separate cases of supposed blowup :  a.  for every constant , approaches in finite time to collapse to a single point .this would constitute the desired behavior for a point - wise singularity in the origin .b.  if for any constant , and stay well separated in time and do not collapse to a point , the whole vortex - line from to has to blow up in order to maintain critical growth in .this scenario , however unlikely , is not ruled out analytically .  the insight provided by the theoremcould successfully be used as evidence excluding a point - wise singularity for the considered initial conditions as presented below .   .a slightly curved vortex line with approximately parallel neighboring vortex lines ( left ) exhibits small .vortex lines with tightening neighboring vortex lines ( center ) or vortex lines with high curvature , in comparison to their length ( right ) have high .,title="fig:",scaledwidth=30.0% ] .a slightly curved vortex line with approximately parallel neighboring vortex lines ( left ) exhibits small .vortex lines with tightening neighboring vortex lines ( center ) or vortex lines with high curvature , in comparison to their length ( right ) have high .,title="fig:",scaledwidth=30.0% ] .a slightly curved vortex line with approximately parallel neighboring vortex lines ( left ) exhibits small .vortex lines with tightening neighboring vortex lines ( center ) or vortex lines with high curvature , in comparison to their length ( right ) have high .,title="fig:",scaledwidth=30.0% ]  vortex stretching is recognized as the mechanism responsible for the accumulation of vorticity .revisited from a geometric point of view , vortex lines are transported with the flow , yet twist and turn due to vortex stretching . since in the absence of dissipation vortex lines are unable to reconnect , the topological properties of vortex lines are fixed .a complex flow will therefore entangle , stretch and twist vortex lines in a non - trivial way .the geometric equivalent of the vortex stretching term is the increase in length for a lagrangian vortex line .it is possible to quantify this stretching and establish a sound connection to the vorticity dynamics of the flow .this in turn can then be used to reformulate blowup criteria in terms of geometric constraints on lagrangian vortex lines .this section is meant to give an overview over this procedure to understand the implications of the second theorem of .details regarding the statements below are given therein .consider a vortex line segment at time and its lagrangian image .let , be the arc length parameters of at times and .then , a direct implication of the vorticity transport formula describes the evolution of the absolute vorticity at a lagrangian fluid element meaning that the local stretching of the length of a vortex line segment that is transported with the flow is equivalent to the growth of vorticity at the corresponding transported fluid element .this result can be transformed into a bound for the length of a vortex line by the vorticity along this line .denote with the length of the vortex line segment at time and define with the maximum vorticity on the vortex line segment .furthermore , let be the quantity of vortex line convergence and vortex line curvature , and define .then , the relative increase of the length of the vortex line segment in a time interval , , is bounded as   this result , which is a slight modification of a lemma in , may be viewed in its own right : the relative increase in length along a time - interval is bounded by the vorticity increase and a factor .thus , is a dimensionless number , characterizing the geometric `` tameness '' of the vortex line filament .  as depicted in fig .[ fig : t2lambda ] , a vortex line segment has a huge , if its maximum curvature is large , relative to its length ( the segment is `` kinked '' instead of `` curved '' ) , or if the surrounding vortex lines collapse to the considered segment in at least a point ( the surrounding is `` tightening '' instead of `` parallel '' ) .a relatively unbent vortex line segment with approximately parallel neighboring vortex lines possesses a low value of .this quantifies the constricted notion of `` relatively straight '' and `` smoothly directed '' given in in a sharper way .the segment has to be included in the lagrangian evolution of of , but is free to be just a fraction of it .[ fig : t2 ] ]  connecting the stretching process stated above with the lagrangian accumulation of vorticity , |{{\boldsymbol{\omega}}}|\ ; ,  \label{eq : lagrangevorticity}\ ] ] where is the strain tensor , and noting that the _ curvature _ of the vortex line fulfills with being the unit normal vector of the vortex line , the lagrangian evolution of vortex line stretching , by inserting eq .( [ eq : t2stretch ] ) , becomes at this point it becomes obvious how the process of vortex line stretching interacts with the velocity in two distinct ways : the velocity in direction of the vortex line elongates the segment by drawing it out , while a part of the velocity normal to the vortex line increases the segment s length by enlarging its curves .  integrating ( [ eq : lagstretching ] ) along the vortex line ( from to ) and over time ( from to ) results in \mathrm{d}\tau\;,\ ] ] for instead of starting the above reasoning at time , the results are identical for a later time .this result may be understood as an upper bound for vortex line stretching in terms of velocity and vortex line geometry . in conjunction with the connection between length increase and vorticity amplification , given in equation ( [ eq : t2lemma ] ) , one arrives at \;.\ ] ] this is an inequality for the control of growth rate of the vorticity by geometric flow properties . from this estimate , by combining it with bkm to distinguish critical from sub - critical vorticity growth , the central non - blowup criterion of is derived :  * deng - hou - yu theorem 2 : * _ assume there is a family of vortex line segments and , such that for all .we also assume that is monotonically increasing and for some when is sufficiently close to .furthermore , we assume that _  a.  [ itm : t2_2 ] b.  _ for some _ .[ itm : t2_3 ] c.  _ for some _ [ itm : t2_1 ]  _ then there will be no blowup in the 3d incompressible euler flow up to time t. _  here , means there exists a constant such that ( and accordingly for ) .the choice of lagrangian vortex segments is sketched in fig .[ fig : t2 ] .it should be noted that theorem 2 again includes assumptions on the dimensionless number .especially assumption ( [ itm : t2_2 ] ) poses a uniform bound in time for .this translates to words as the process of `` zooming in '' to the location of maximum vorticity in order to keep the considered vortex line segment relatively straight in comparison to its length .the assumed accompanying collapse in length to keep bounded is then linked in its growth rate to the blowup of the velocity components .it is worth mentioning that the above presented criterion , even though it is obviously inspired by the classical geometric criteria , still differs in crucial aspects .the posed assumptions are purely local and restricted to the geometry of a single critical vortex line filament .assumptions on the velocity do not , in contrast to constantin et al . , impose a uniform bound ( which is not observed in simulations ) , but allow for a finite - time blowup of velocity , strictly connected in its growth rate to the geometrical evolution of the filament .the vortex line segment itself is not assumed to be of length ( as in ) or to be contained in an -region ( which , again , was not observed in simulations ) .these aspects in combination render it a promising theorem to be directly tested by numerical simulations .it has been established that a singularity of the euler equations in finite time necessitates rapid accumulation of vorticity .locally , vorticity - strain coupling is identified as the mechanism for nonlinear amplification in finite time .globally , the notion and possibility of self - similar or locally self - similar collapse to a point is introduced .these aspects serve as a basis for the construction of initial conditions suitable for the possible formation of a finite - time singularity .  due to the incompressibility condition , the trace of vanishes and due to the symmetry of its eigenvalues are real and the corresponding eigenvectors are orthogonal .thus , the biggest eigenvalue , fulfills in regions with non - vanishing strain . as a direct consequence of equation ( [ eq : lagrangevorticity ] ) ,if is proportional to the vorticity in direction then the vorticity growth would be compatible with a finite - time singularity according to bkm .this coupling of the strain to the vorticity is crucial .if the strain rate is constant instead , the growth in vorticity is merely exponential .several cases have been suggested in which this mechanism of coupling may take place .it should be noted that , contrary to expectations , the vorticity - strain coupling does not readily appear in nature .one would expect a tendency of the vorticity to align itself to the eigenvector of the strain tensor with the largest eigenvalue all by itself , since the parallel component is amplified , while the orthogonal components are damped or stay nearly constant .nevertheless , for viscid turbulent flows , quite a different behavior is observed both in numerical simulations and experiments : the vorticity is most likely to align to the intermediate eigenvector of the strain tensor and similarly for the euler equation .one can therefore expect that functional vorticity - strain coupling is inherently unstable .the process has to be designed `` artificially '' by choosing suitable initial conditions .  a more precise notion of the process of vorticity alignment in turbulent flows is given by hamlington , schumacher and dahm .they distinguish , evaluating the biot - savart law numerically , between strain induced _ locally _ by the immediate neighborhood and _ globally _ by long - range interaction .for turbulent flows , they observe a most likely alignment of the vorticity to the most positive eigenvector of the global strain .taking into account also the local strain restores the alignment of vorticity to the intermediate eigenvector .contrary to this , for the successful emergence of a finite - time singularity of the euler equations in a point - wise sense , vorticity - strain coupling should be induced by the local strain to support a collapse to a point . a numerical application of this technique from to euler blowup simulations will be presented elsewhere .different initial conditions were introduced and subsequently improved or refined to construct flows with prolonged intervals of vorticity - strain coupling , and it seems natural to search for techniques to artificially keep the coupling existent .one such technique is the introduction of symmetries to the flow .early examples such as the taylor - green vortex or kerr s initial conditions are employing such symmetries .is normal and the strain tensor possesses a parallel eigenvector with corresponding eigenvalue .the curvature in the symmetry plane has to increase in order to support critical vorticity - strain coupling.,scaledwidth=50.0% ]  consider the plane to be a plane of reflectional symmetry , as shown in fig .[ fig : reflectionalsymmetry ] , defined by for the velocity vector field , which leads to in the plane of symmetry .accordingly , the vorticity obeys and in particular or in the plane of symmetry . due to these properties ,the strain tensor has , or it immediately follows that the eigenvector corresponding to the eigenvalue is directed normally to the symmetry plane , and the vorticity vector is aligned to it .note that this is the sole consequence of the reflectional symmetry and is in no way influenced by the flow . a vortex tube normal tothe symmetry plane therefore seems like a natural candidate for critical accumulation of vorticity by means of vorticity - strain coupling : all that is needed is a sufficiently long period of time in which at one point of the symmetry plane .this possibility has been analyzed by pelz : taking into account only the -component , the strain tensor in the plane of symmetry is given by the biot - savart law as equation ( [ eq : symmetrystrain ] ) shows that in the plane of symmetry does not scale with , but does instead depend on and , which are both equal to zero in the plane . yet ,in close proximity to the plane , and may grow , depending on the curvature of the vortex line intersecting the symmetry plane : a taylor expansion around the plane yields : for small up to first order , for . therefore, if the curvature is huge close to the plane of symmetry , with approximately scales like and thus may scale with too .however , for this to happen , we need . as a matter of fact , the dimensionless number plays a similar role as the characteristic geometric number introduced in section [ ssec : vortexlinestretching ] . for to blow up like , the curvature has to increase in a way to counter the shrinking of the length scale .on the other hand , the axial strain stretches the vortex tube in -direction .this counteracts any increase in curvature to a certain degree .more precisely , the lagrangian evolution of the curvature components and were calculated in : the axial strain diminishes both and .these counteracting processes of vortex line geometry are by no means analytically exact , since all long - range interactions have been ignored .nevertheless , they constitute an intrinsic resistance of a single vortex line to `` self - stretch '' in a critical way .the argument may be readily translated to the case of ( perturbed ) anti - parallel vortex tubes : since no other components of vorticity are introduced , still only depends on and which in turn rely on high curvature to scale like close to the plane of symmetry .  one way to counterthis is to induce the axial strain by neighboring tubes instead of relying on a sufficiently large kink .this will be presented in the following section by introducing additional rotational symmetry .one notable high - symmetry flow was introduced by kida and has subsequently been used extensively to probe a possible euler blowup numerically or analytically ( e.g. ) as well as study the onset of turbulence ( e.g. ) .the kida - pelz flow has a three - fold rotational symmetry about the diagonal and a reflectional symmetry about all three cartesian planes .flows with these two properties are termed as invariant under the full octahedral group .the euler ( and navier - stokes ) equations preserve the kida - pelz symmetries . in generalone can write these initial conditions as this means , for a computational domain spanning the interval  ] ) . at some point , depending on the value of , the threshold is reached and the length of the vortex line segment decreases . yet , for all considered cases of , the length does not collapse to a point , but saturates at early times without approaching .this behavior appears to be stable up to the latest time of the simulation .the final length of the vortex line segments is at least for the smallest case of ( ) , which is still well resolved with at least .this result , therefore , is a numerical evidence against a point - wise blowup for the vortex dodecapole class of initial conditions .this is in concordance with the estimate in .monitoring the development of yields , as shown in fig .[ fig : t1results ] ( right ) , a similar growth rate for the accumulation of vorticity at the endpoint as for the beginning of the vortex line segment .this is hardly surprising , since by construction a constant value for directly links the growth rates of to .nevertheless , a numerical verification of this analytic equality may be seen as a confirmation that the observed growth rate of is by no means a numerical artifact in an isolated small area , but is reproduced at points far away from the critical region , which appear to be well - behaved at first view .the possibly critical growth in the perspective of bkm is , thus , confirmed by the global flow .furthermore , since for a large portion of the simulation the distance is approximately constant , this could possibly be seen as an evidence for the existence of a non - vanishing vortex line segment that blows up in every point .the popular scenario of a collapse to a single point , on the other hand , is clearly conflicting the numerical evidence . the discovery of a possibly critical vortex line segment in the vortex dodecapole flow , however , is to be handled with care , since distinguishing between critical and sub - critical blowup of the whole segment is in no way more conclusive than distinguishing between critical and sub - critical growth of .thus , learning from the lesson taught by 25 years of numerically testing bkm , this should not be interpreted as clear evidence in favor of a finite - time singularity .the geometric properties of lagrangian vortex line segments , especially their curvature and the tightening of their surroundings have been established as revealing parameters in understanding the nature of rapid accumulation of vorticity in euler flows and a sound connection to singular behavior is made through theorem 2 .the ambition here is to utilize these geometric properties , monitored in a numerical simulation , as more reliable means of distinguishing between a finite - time singularity and a mere fast accumulation of vorticity .  despite high hopes from an analytical point of view that these considerations will shed light on the nature of vorticity accumulation , numerical results observing geometrical properties of lagrangian vortex filaments are scarcethis is primarily due to the fact that eulerian quantities such as are readily trackable in post - processing , while monitoring the lagrangian evolution requires additional computational effort . on top of that ,the geometry of integral curves at an instance in time , though in principle computable in post - processing , as well as derived quantities such as their convergence and curvature , are quite inaccessible in comparison to simple eulerian criteria .this section is devoted to the presentation of results concerning the assumptions of theorem 2 of for the vortex dodecapole initial conditions .quite similar to the first theorem , there is considerable freedom in the choice of the involved quantities .the strategy we chose in the context of this paper is as follows :  * identify the lagrangian fluid element , which will contain the maximum of vorticity at the latest time of the simulation , . a vortex line segment here will intrinsically be `` comparable '' to the maximum of vorticity ( as in ) at late stages of the simulation .the assumptions concerning the segment are therefore automatically met . in the numerics this procedureis implemented by carrying out a precursory identical simulation with a huge number of tracer particles ( 1 million ) randomly distributed across the domain .particles that accumulate huge amounts of vorticity are selected for the subsequent production run .* for the production run , at each instance in time start a vortex line integration at along the vorticity direction field . monitor the maximum curvature and the maximum vortex line convergence during the integration and calculate . stop the integration , as soon as reaches a fixed , arbitrary constant .this defines . in the numericsthis is again implemented with a third - order runge - kutta integration and bisectioning to obtain the endpoint of . * for this vortex line segment , calculate the length , and the velocity components and . from the collapse of the length the exponent .this in turn provides the critical growth exponent for the velocity variables , . *compare the increase in and to to distinguish between critical and sub - critical growth of velocity .this can be interpreted rather intuitively . by prescribing an arbitrarily fixed ,the vortex line segment is kept relatively geometrically uncritical , as the length - scale is always adjusted accordingly .this process of `` zooming in '' just enough to retain the geometric `` criticalness '' prescribes the rate of collapse to a point , at least in the direction of the vortex line .all that is left to check is whether the velocity growth in the immediate surrounding is fast enough to be compatible with a finite - time singularity .the results of the previous section , concerning a point - wise singularity versus the blowup of a whole vortex segment , already anticipates that the increase in around the critical vortex line is bounded .if the curvature of the vortex line segment remains controllable ( which is to be expected from the pictures ) , then just a mild collapse of occurs .this leaves much room for and to still be distinguishable from a critical growth .   of the critical vortex filament for different lagrangian fluid elements .the length does not decrease as for any , which would be faster than linear .the lagrangian collapse of the vortex segment is decelerating instead .bottom : evolution of the quantities and in time . does not appear to be growing , while , though increasing in time , does not exhibit a finite - time blowup as .,title="fig:",scaledwidth=45.0% ]  of the critical vortex filament for different lagrangian fluid elements .the length does not decrease as for any , which would be faster than linear .the lagrangian collapse of the vortex segment is decelerating instead .bottom : evolution of the quantities and in time . does not appear to be growing , while , though increasing in time , does not exhibit a finite - time blowup as .,title="fig:",scaledwidth=45.0% ]  fig .[ fig : t2results ] shows the results for the vortex dodecapole initial conditions for a fixed constant .different choices of the constant produce identical , rescaled results .the top plot pictures the length of the vortex line segment for the tracer that is arriving at a position of very huge vorticity at late stages of the simulation .the subplot depicts the long - term behavior of the particle entering the critical region , while the final stage of length decrease is magnified .the decrease in length does not agree with a collapse in final time , but instead the shrinkage of the segment decelerates clearly in time .this contradicts a scaling in time proportional to for any , which would be faster than ( or , in the limiting case , equal to ) linear .it should be noted that for the observed collapse in length , the vortex segment curvature is the dominating term in , shadowing the effects of .this may lead to a change of regime in the rate of collapse , if at some point exceeds in quantity .it could furthermore be argued that the limit is hard to exclude , since the drop in length would be virtually instantaneous in time , with a close to constant scaling before . in this limit , the quantities and would have to grow roughly as to still allow formation of a finite - time singularity . quantifies the largest difference in axial velocity along the segment . for an isolated collapsing vortex tube, this quantity can be expected to not increase critically , since the tangential velocity is less likely to rapidly change than the radial velocity . however, this initial explanation does not consider the influence of the mirror tubes . on the other hand may be interpreted as the velocity of the vortex tube s core itself .again , in an isolated setup this velocity is not expected to blow up .[ fig : t2results ] ( bottom ) shows the observed behavior of and in time for the tracer as considered above . stays roughly constant in time , showing no signs of a blowup . , even though increasing in time , does not fit to critical growth , in particular not like in time .thus , the assumptions of theorem 2 are well met .this result therefore poses a strong numerical evidence against a finite - time singularity for the class of vortex dodecapole initial conditions .in this paper we present numerical evidence against the formation of a finite - time singularity for the vortex dodecapole initial condition .we use data obtained from high resolution adaptively refined numerical simulations to test the assumptions presented by geometric blowup criteria .the applied numerical method allows for a clearer insight into the formation of the possible singularity .most notably , it implies numerical techniques to distinguish between a point - wise blowup and the blowup of a whole vortex line segment .furthermore , by tracking curvature and spreading of lagrangian vortex line segments , the distinction between singular and non - singular behavior can be made much more clearly than the usual approach via bkm .  in this paperwe used vortex dodecapole initial conditions with two different vorticity profiles .comparison of the simulation shows that different vorticity profiles yield similar visual and geometrical appearance .this serves as an argument that the obtained results may apply to the whole class of vortex dodecapole flows .monitoring the growth rate of quantifies the well - known vorticity amplification .amplification by more than two orders of magnitude was reached for both vorticity and strain , exceeding by far values achieved by previous simulations .applying this data to bkm would lead to the conclusion that a finite - time singularity at time fits via extrapolation . yet , as the history of euler simulations has shown , statements obtained by extrapolation are to be handled with care .  following the presented argument , a point - wise collapse should coincide with a blowup of at the point of maximum vorticity .as shown , this statement can be broadened : a finite - time singularity must either lead to a blowup of at the point of maximum vorticity , or the whole critical vortex line segment has to blow up . utilizing the geometric information obtained via vortex line integration from the numerical simulation, it is observed that does not grow in a way to be compatible with a point - wise collapse . yet , measuring the growth - rates on the critical vortex line , high rates of amplification are measured far away from the critical region . even though it is hard to distinguish , whether this amplification is critical or sub - critical , this might be interpreted as an evidence for the blowup of the complete vortex line segment , even though it suffers exactly the same vulnerabilities as extrapolation in bkm . a point - wise blowup , on the other hand, cleary contradicts the numerical results up to the time reached .evidence against a blowup of the whole critical vortex line segment is found when looking at the geometric properties of lagrangian vortex line segments .it was shown by theorem 2 in that a blowup of vorticity is directly connected to the interplay between velocity growth and the collapse of vortex line filaments , when maintaining the overall same shape in geometric means ( i.e. the same ) . since curvature and not increase in order to support a finite - time collapse of the segment , velocity components in the vicinity of the vortex line filament would have to increase as to support the blowup hypothesis . up to the time reached ,this critical velocity growth may be excluded by numerical means .this poses a numerical evidence against the formation of a singularity in finite time for vortex dodecapole configurations .we would like to thank j. dreher for his work on the computational framework .this work benefited from support through project of the deutsche forschungsgesellschaft .access to the bluegene / p multiprocessor computer jugene at the forschungszentrum jlich was made available through project hbo35 .t. grafke , h. homann , j. dreher , r. grauer , numercial simulations of possible finite time singularities in the incompressible euler equations : comparison of numerical methods , physica d : nonlinear phenomena 237 ( 2008 ) 19321936 .m. e. brachet , m. meneguzzi , a. vincent , h. politano , p. l. sulem , numerical evidence of smooth self - similar dynamics and possibility of subsequent collapse for three - dimensional ideal flows , phys .fluids 4 ( 12 ) ( 1992 ) 28452854 .m. d. bustamante , m. brachet , on the interplay between the bkm theorem and the analyticity - strip method to investigate numerically the incompressible euler singularity problem , published online , ` http://arxiv.org/abs/1112.1571 ` ( 2011 ) .

Abstract: we present a numerical method of analyzing possibly singular incompressible 3d euler flows using massively parallel high - resolution adaptively refined numerical simulations up to mesh points . geometrical properties of lagrangian vortex line segments are used in combination with analytical non - blowup criteria by deng et al [ commun . pde * 31 * ( 2006 ) ] to reliably distinguish between singular and near - singular flow evolution . we then apply the presented technique to a class of high - symmetry initial conditions and present numerical evidence against the formation of a finite - time singularity in this case .   euler equation , existence , uniqueness and regularity theory , vortex line geometry , adaptive mesh refinement


in this paper , we investigate the problem of mining numerical data .this problem arises in many practical situations , e.g. analysis of gene and transcriptomic data in biology , soil characteristics and land occupation in agronomy , demographic data in economics , temperatures in climate analysis , etc .we introduce an original framework for mining numerical data based on advances in itemset mining and in formal concept analysis ( fca ,  ) , respectively condensed representations of itemsets and pattern structures in fca  .the mining of frequent itemsets in binary data , considering a set of objects and a set of associated attributes or items , is studied for a long time and usually involves the so - called `` pattern flooding '' problem .a way of dealing with pattern flooding is to search for equivalence classes of itemsets , i.e. itemsets shared by the same set of objects ( or having the same image ) . for an equivalence class , there is one maximal itemset , which corresponds to a `` closed set '' , and possibly several minimal elements corresponding to `` generators '' ( or `` key itemsets '' ) . from these elements , families of association rulescan be extracted .these itemsets are also related to fca , where a concept lattice is built from a binary context and where formal concepts are closed sets of objects and attributes .the present work is rooted both in fca and pattern mining with the objective of extracting interval patterns from numerical data .our approach is based on `` pattern structures '' where complex descriptions can be associated with objects . in  introduced closed interval patterns in the context of gene expression data mining .intuitively , an interval pattern is a vector of intervals , each dimension corresponding to a range of values of a given attribute ; it is closed when composed of the smallest intervals characterizing a same set of objects .  in the present paper , we complete and extend this first attempt . considering numerical data , some general characteristics of equivalence classes remain , e.g. one maximal element which is a closed pattern and possibly several generators which are minimal patterns w.r.t . a subsumption relation defined on patterns .we show that directly extracting patterns data from numerical is more efficient using pattern structures than working on binary data with associated scaling procedures .we also provide a semantics to interval patterns in the euclidean space , design and experiment algorithms to extract frequent closed interval patterns and their generators .the problem of mining patterns in numerical data is usually referred as quantitative itemset / association rule mining  .generally , an appropriate discretization splits attribute ranges into intervals maximizing some interest functions , e.g. support , confidence . however, none of these works covers the notion of equivalence classes , closed patterns , and generators , and this is one of the originality of the present paper .the plan of the paper is as follows .firstly , we introduce the problem of mining numerical data and interval patterns .then , we recall the basics of fca and interordinal scaling .we pose a number of questions that we propose to answer using our framework of interval pattern structures dealing with numerical data .we then detail two original algorithms for extracting closed interval patterns and their generators .these algorithms are evaluated in the last section on real - world data .finally , we end the paper in discussing related work and giving perspectives to the present research work . as a complement , an extended version of this paperis given in  , providing algorithms pseudo - code and a longer discussion on the usefulness of interval patterns in classification problems and privacy preserving data - mining .we propose a definition of interval patterns for numerical data .intuitively , each object of a numerical dataset corresponds to a vector of numbers , where each dimension stands for an attribute .accordingly , an interval pattern is a vector of intervals , and each dimension describes the range of a numerical attribute .we only consider finite intervals and that the set of attributes / dimensions is assumed to bed ( canonically ) ordered .a numerical dataset is given by a set of objects , a set of numerical attributes , where the range of is a finite set noted . means that is the value of attribute for object .  .anumerical dataset .[ cols="<,<,<,<",options="header " , ]   in both cases , using binary data is better when the minimal support is high ( e.g. 90% ) . for low supports , a critical issue , our algorithms deliver better execution times .most importantly , the global redundancy effect discards the use of binary data , e.g. only of all fisg are actually fipg in dataset bl .finally , the algorithm _ minintchangeg - t _ outperforms _ minintchangeg - h_. _ minintchangeg - t _ however needs more memory since it stores each closed set of objects as a word in the trie , and to each word the list of associated fipg .it is very interesting to analyse the compression ability of closed interval patterns and generators .for that , we compare in each dataset the number of those patterns w.r.t . to all possible interval patterns .it gives the ratio of closed ( generators ) in the whole search space . in both cases ,ratio varies between and .this means that the volume of useful interval patterns , either closed or generators , is very low w.r.t .the set of all possible interval patterns , justifying our interest in equivalence classes for interval patterns .we discussed the important problem of pattern discovery in numerical data with a new and original formalization of interval patterns .the classical fca / itemset - mining settings are adapted accordingly : from a closure operator naturally rise the notions of equivalence classes , closed and generator patterns , and we designed corresponding algorithms .an appropriate semantics of interval patterns shows from a theoretical ( redundancy ) and practical ( computation times ) points of view that mining equivalent binary data ( encoding all possible intervals ) is not acceptable .this is due to the fact that interval patterns are provided with a stronger partial ordering than is - itemsets ( classical set inclusion ) , hence pattern structures yield significantly less generators w.r.t .their semantics .  dealing with interval patterns has applications in computational geometry , machine learning and data - mining , e.g.  and references therein .it is indeed highly related to the actual problem of ( maximal ) -boxes which corresponds to interval patterns ( generators ) with support . when , it corresponds to largest empty subspaces of the data .our contribution to this field is the characterization of smaller subsets ( closed and generators ) .  in data - mining ,closed patterns and their generators are crucial for extracting valid and informative association rules  , while generators can be preferable to closed patterns following the minimum descriptions length principle for so - called itemset - based classifiers  .how these notions can be shifted to interval patterns is an original perspective of research rising questions concerning missing values , fault - tolerant patterns , and interestingness measures that are critical issues even in classical itemset mining : although the compression ability of closed interval patterns and generators is spectacular , the number of patterns remains too high for large datasets .however , bringing the problem of numerical pattern mining into well known settings in favor of these perspectives of research .e. boros , k. elbassioni , v. gurvich , l. khachiyan , and k. makino .an intersection inequality for discrete distributions and related generation problems . in _ international colloquium on automata , languages and programming ( icalp ) _ , lncs ( 2749 ) , pages 543555 .springer , 2003 .j. li , h. li , l. wong , j. pei , and g. dong .minimum description length principle : generators are preferable to closed patterns . in _ innovative applications of artificial intelligence conference_.aaai press , 2006 .

Abstract: we investigate the problem of mining numerical data with formal concept analysis . the usual way is to use a scaling procedure transforming numerical attributes into binary ones leading either to a loss of information or of efficiency , in particular w.r.t . the volume of extracted patterns . by contrast , we propose to directly work on numerical data in a more precise and efficient way . for that , the notions of closed patterns , generators and equivalent classes are revisited in the numerical context . moreover , two algorithms are proposed and tested in an evaluation involving real - world data , showing the quality of the present approach .


during the last decade , first - principles total - energy methods based on density functional theory ( dft ) combined with the pseudopotential method , have become established as a major tool in the study of condensed matter  .the dft - pseudopotential approach is now widely used for both static and dynamic simulations on an enormous range of condensed - matter problems . however , these methods suffer from a severe drawback in that their computational cost generally increases as the cube of the number of atoms in the system .this unfavorable scaling limits the size of systems that can be studied with current methods and today s computers to a few hundred atoms at most .this scaling appears in spite of the fact that the complexity of the problem increases only linearly with the system size .this observation suggests that the unfavorable scaling of current methods is a consequence of the way in which the electronic structure problem is being addressed .conventional methods rely either on diagonalization of the hamiltonian or orthonormalization of a set of occupied orbitals , both of which are intrinsically operations .it is clear that more efficient methods in which the effort is proportional to the number of atoms must be possible , and in recent years a considerable effort has been devoted to finding such ` linear - scaling ' schemes  .the earliest linear - scaling scheme appears to be the ` divide and conquer ' method of yang  .this obtains the electronic density and hence the total energy by dividing the system into overlapping sub - systems that can be treated independently .the density is calculated for each sub - system with conventional lcao - dft .the hamiltonian for each sub - system , which includes the potential due to the other sub - systems , is diagonalized independently , thus avoiding the need to diagonalize the full hamiltonian .this procedure is repeated until self - consistency is achieved .baroni and giannozzi  also proposed a scheme that directly determines the electron density .they do this by discretizing the kohn - sham hamiltonian on a real - space grid , and then using the recursion method of haydock , heine and kelly  to obtain the diagonal elements of the green s function , from which the electron density can be computed by contour integration . in this caselinear scaling results from the fact that the continued fraction used to evaluate a particular diagonal element of the green s function can be truncated once a certain neighborhood of each point has been explored . this neighborhood is independent of the system size for sufficiently large systems .  more recently , several new schemes that resemble traditional first - principles methods have been reported .galli and parrinello  pointed out that some improvement could be achieved in the scaling of a conventional dft calculation by requiring spatial localization of the electronic orbitals .this localization was achieved by adding certain non - local constraining terms to the hamiltonian , or by using a filtering procedure .the total energy can then be obtained as a functional of the localized orbitals and their _ conjugate _orbitals , but in order to obtain these conjugate orbitals , the overlap matrix has to be inverted .since spatial localization implies sparsity of , this can be achieved in operations , so that some improvement with respect to is obtained .a step further in this direction was made independently by mauri , galli and car  ( hereafter referred to as mgc ) and by ordejn _et al . _they introduced a new functional of the occupied orbitals that possesses the same ground state as the conventional energy functional , but with the added advantage of leading naturally to orthogonal orbitals when minimized . if this new functionalis minimized with respect to orbitals that are constrained to remain localized in chosen regions of space , as suggested by galli and parrinello  , a linear scaling method results . in the original formulation , the number of orbitals entering the new functional is equal to half the number of electrons in the system .this restriction seems to lead to very slow convergence , and to the appearance of spurious local minima in the functional .this problem has been recently overcome by kim , mauri and galli  , by generalizing the functional so that it depends on an arbitrary number of orbitals .the linear - scaling scheme most relevant to the present work is that put forward by li , nunes and vanderbilt  ( hereafter referred to as lnv ) in the context of tight - binding semi - empirical calculations . in this method ,linear scaling is achieved by taking advantage of the real space localization properties of the density matrix , . by introducing a spatial cutoff in , such that is set to zero if , the number of non - zero elements in increases only linearly with the system size .the electronic structure problem is then formulated as a minimization of the total energy with respect to the truncated density matrix , subject to the constraints of idempotency ( ) and correct trace ( , where is the number of electrons ) .the scheme of lnv consists of an algorithm for imposing these constraints that at the same time fulfils the goal of linear scaling .the idempotency of is the most difficult constraint to impose , and this scheme achieves it by expressing in terms of an auxiliary matrix , which we denote in this paper by .this is subjected to a _ purifying _ transformation due to mcweeny  .if is a near - idempotent matrix , i.e. if its eigenvalues lie close to 0 or 1 , this transformation will return as a more nearly idempotent matrix , and thus it is possible to minimize the total energy with respect to while ensuring the near idempotency of . by construction , the method is variational ( i.e. ) , and it has been shown that the convergence of calculated properties with the parameter is fairly rapid  .it is now being widely used in tight - binding simulations of large systems .recently , the idea of working with the density matrix has been applied to dft linear scaling schemes .this has been done independently by hierse and stechel  and by hernndez and gillan  . in both cases ,the density matrix is represented in real space as : where the are a set of localized functions , and is a symmetric matrix .the total energy is expressed in terms of , and minimization is carried out with respect to both the and the .hierse and stechel  use a number of functions equal to the number of occupied orbitals , but this restriction is not present in our scheme .the consequences of this and other differences between the two methods will be addressed later in this paper .previously , only a brief description of our method has been published  . in this paperwe give a detailed description of the method , together with some illustrations of its practical performance and a discussion of its relation to other methods . in section 2 , the method is outlined and its theoretical foundations are discussed .the practical implementation of the method is then described in section 3 .the tests we have performed to probe the practical usefulness of the scheme are presented in section 4 . in section 5 , we assess what has been achieved and we discuss possible future developments , with particular attention to the problems that need to be overcome before the method can be generally applied . some of the mathematical analysis is reported in an appendix .we need to recall briefly the principles of dft  .the total energy of the system of valence electrons and atomic cores is expressed as : where the terms on the right are the kinetic , pseudopotential , hartree and exchange - correlation energies of the electrons , and is the madelung energy of the cores .the first two energies are : where are the kohn - sham ( ks ) orbitals , is the total pseudopotential operator , and is the number of occupied orbitals .the energies and can be written in terms of the electron number density : where for simplicity we assume the local density approximation ( lda ) for , with the exchange - correlation energy per electron .the number density is : the important principle for present purposes is that the true ground - state energy and electron density are obtained by minimizing with respect to the ks orbitals , subject to the constraint that the latter are kept orthonormal .  in the standard formulation of dft , which we have just summarized ,all the occupied orbitals are fully occupied .however , it is frequently convenient , for physical , computational or formal reasons , to generalize the theory so that orbitals can be partially occupied .spatial orbital , rather than containing 2 electrons , may now contain electrons , where the occupation number lies in the range .the number density now becomes : and the kinetic and pseudopotential energies are : the expressions for and in terms of are unchanged .the usual physical reason for making this generalization is that one wishes to treat the electrons at a non - zero temperature , in which case the are fermi - dirac occupation numbers  ; computationally , the generalization is sometimes made in order to get rid of the troublesome discontinuity at the fermi level in metallic systems  .our reason for considering it here is that it will be relevant to the density matrix formulation .we shall assume that if is minimized both with respect to the ( subject to orthonormality ) and with respect to the ( subject to the restriction and the condition that the sum be equal to ) , then we arrive at exactly the ground state that is obtained by the more usual minimization with respect to fully occupied states .another way of putting this is that the energy can not be reduced below the normal ground state by allowing partial occupation .now we turn to the density matrix , which is defined by it follows from this definition that is a hermitian operator whose eigenvalues are all in the interval  ] over grid points .the exchange - correlation potential can also be calculated at each point , and is given as \right\ } \ ; .\end{aligned}\ ] ]  to obtain the hartree energy and potential we use the fast fourier transform ( fft ) method to transform the calculated electronic density into reciprocal space , thus obtaining its fourier components .the hartree energy is then given as where is the volume of the simulation cell .the hartree potential in reciprocal space is : this can be constructed on the reciprocal - space grid , and transformed to obtain the hartree potential in real space .fft is , of course , an operation rather than an operation , but the difference is negligible for present purposes .we restrict ourselves here to local pseudopotentials , so that the value of the total pseudopotential at grid point is formally given by : where is the ionic pseudopotential and is the position of ion . in practice , however , can not be calculated like this , because has a coulomb tail at large , where is the core charge . in order to obtain a linear - scaling algorithm for , we proceed as follows .the ionic pseudopotential is represented as the sum of the coulomb potential due to a gaussian charge distribution and a short - range potential .the total charge in is , and the distribution is given by : where the parameter governs the rate of decay of the gaussian .we therefore have : the part of coming from can now be calculated as a direct sum over ions , as in eq .( [ eq : pseudo_sum ] ) .since can be neglected beyond a certain radius , this part of the calculation scales linearly .the part of coming from the array of gaussians can be treated in exactly the same way as the hartree potential .the pseudopotential energy is then calculated by summation over the real - space grid :    once the contributions to the total energy have been obtained as outlined above , we need to vary both and in order to minimize it .the and are independent variables , and the problem breaks naturally into two separate minimizations that can be carried out in an alternating manner : one with respect to with fixed , and the other with respect to with fixed . indeed , the choice of object function can be different for the two types of variation , and when minimizing with respect to the we find it more convenient to take as our object function , where is the chemical potential and is the electron number .we return to this point below .expressions for the derivatives with respect to and are obtained in appendix a. the partial derivative of with respect to is given by {\alpha \beta } \label{eq : partialab}\end{aligned}\ ] ] where , and is the matrix representation of the ks hamiltonian in the support function representation .it is worth noting that this expression is exactly the same as would be obtained in a non - orthogonal tight - binding formalism . there is , however , one important difference : in self - consistent dft calculations the hamiltonian matrix elements depend on through the electronic density .the partial derivative of the total energy with respect to at grid point is given by \phi_\beta({\bf r}_\ell ) , \label{eq : partialphi}\end{aligned}\ ] ] where is the kohn - sham operator , which is made to act on support function .it is important to notice that because of the spatial localization of the support functions , and the finite range of , all the matrices involved in the calculation of these derivatives are sparse , when the system is large enough .provided this sparsity is exploited in the computational scheme , the method scales linearly with the size of the system .  in the scheme of lnv , it is proposed to work at constant chemical potential , rather than at constant electron number .we prefer to maintain the electron number constant .the variations with respect to and will in general cause the electron number to differ from the correct value , and it is therefore necessary to correct this effect as the minimization proceeds .we achieve this in the following manner : during the minimization with respect to , the current search direction is projected so that it is tangential to the local surface of constant , i.e. perpendicular to at the current position .this ensures that the minimization along this direction will cause only a small change in , and it is expected that at the new minimum will differ only slightly from the required value . in any case , it is possible to return to a position as close as desired to the constant surface by following the local gradient .if the value of the chemical potential is appropriately chosen , this correction step can be carried out without losing the reduction in obtained by performing the line minimization , and this is why we prefer to take as the object function instead of the total energy , when minimizing with respect to .we find that this scheme is capable of maintaining the electron number close to its correct value throughout the minimization , and is also simple to implement .the gradient has elements which , as all other gradients discussed earlier , can be calculated in operations .minimization with respect to will also have the effect of changing the electron number .however , given that the two types of variation are performed alternately , the correction during the minimization is sufficient to counteract this effect .given that variation of causes the electronic density to change , and this in turn implies that the hamiltonian matrix elements change , it would seem necessary to update the hamiltonian at each step of the minimization with respect to .however , we find that this can be avoided by considering fixed during this part of the minimization . strictly speaking ,if is held fixed while is varied , we are not minimizing but rather , where is given by \ ; .\end{aligned}\ ] ] if this minimization were carried out through to convergence , this would be equivalent to diagonalizing in the representation of the current support functions . at convergence, it will be found in general that and are not mutually consistent , and if consistency is required , one needs to update and repeat the minimization , iterating this cycle until consistency was achieved .this is not necessary in practice , because will be updated at the next variation with respect to the support functions .the minimization of has practical advantages in that it avoids the updating of the hamiltonian at each step , and , because of its construction , it is a cubic polynomial in every possible search direction , so it is possible to find the exact location of line minima during its minimization .the minimization with respect to can be carried out by simply moving along the gradient eq .( [ eq : partialphi ] ) ( steepest descents ) or by using this expression to construct mutually conjugate directions ( conjugate gradients ) .in order to test our dft scheme , we have performed calculations on a system of 512 si atoms treated using a local pseudopotential .the purpose of these tests is to find out how the total energy depends on the two spatial cutoff radii : the support - region radius , and the -matrix cutoff radius .the practical usefulness of the scheme , and the size of system for which linear - scaling behavior is attained depend on the rate of convergence of to its exact value as and are increased . here , ` exact ' refers only to the absence of errors due to the truncation of ; other sources of inexactness , such as the use of a discrete grid and a local pseudopotential , are of no concern here .the system treated is a periodically repeating cell containing 512 atoms of diamond - structure si having the experimental lattice parameter ( 5.43 ) .the local pseudopotential is the one constructed by appelbaum and hamann  , which is known to give a satisfactory representation of the self - consistent band structure .the lda exchange - correlation energy is calculated using the ceperley - alder formula  .we use a grid spacing of 0.34 , which is similar to the spacing typically used in pseudopotential plane - wave calculations on si , and is sufficient to give reasonable accuracy .the second derivatives of the needed in the calculation of are computed using the second - order formula given in eq .( [ eq : finitedifference ] ) .a support region is centred on every atom , and each such region contains four support functions .one can imagine that these support functions correspond roughly to the single 3 function and the three 3 functions that would be used in a tight - binding description , but we stress that nothing obliges us to work with this number of support functions . in keeping with the tight - binding picture , the initial guess for the support functionsis taken to be a gaussian multiplied by a constant , , or , so that the functions have the symmetry of and states . as an initial guess for the -matrix ,we take the quantity , where is the overlap matrix calculated for the initial support functions .this guess for , which represents the expansion of to first order , is crude , and does not yield the correct value of .this error is corrected by displacing iteratively along the gradient until is within a required tolerance of the correct value .the initial guesses for the and the define the initial hamiltonian and overlap matrices . from this starting point, we make a number of conjugate - gradient line searches to minimize by varying , with the hamiltonian and overlap matrices held fixed .this is followed by a sequence of line searches in which the are varied .we refer to the sequence of moves followed by a sequence of moves as a _cycle_. the entire energy minimization consists of a set of cycles . in practice, we have found that cycles consisting of five moves and two moves work satisfactorily , and that is converged to within  ev / atom after typically 50 - 60 cycles .this would not be an efficient rate of convergence for routine applications , but is more than adequate for present purposes .our test calculations confirm our earlier finding  that for the si perfect crystal is already quite close to its exact value when = 5.0  .we have therefore used this value of to make calculations of as a function of ( see figthe results show that converges very quickly with increasing , and that it is within  0.1 ev of its fully converged value for = 3.05  . in order to show how depends on , we present a series of results at the two region radii = 2.04 and 2.38  ( see fig .these results indicate that there is only a slow variation with and that this variation is almost the same for different values of .this means that it is possible to converge the total energy to satisfactory accuracy with easily manageable spatial cutoffs .it is interesting to know the form of the support functions for the self - consistent ground state .these are shown in fig . 2 for the case = 3.05 , = 5  .the support functions shown here are the first ( initially  gaussian ) and second (  gaussian ) .profiles of the support functions along the [ 100 ] , [ 110 ] and [ 111 ] directions are shown .the support functions are seen to be symmetric with respect to the center of the support region ( ) along the [ 100 ] and [ 110 ] directions . along the [ 111 ] directionthere is a slight asymmetry resulting from the presence of a nearest neighbour ion , which lies at 2.35  from the origin in the positive direction .remarkably , the -like support function seems to be almost perfectly spherically symmetric , except near the peak at  . it is encouraging to see that the support functions go rather smoothly to zero at the region boundary , and this confirms that the boundary is having little effect on the results .we have tried to do three things in this work : to develop the basic formalism needed to underpin dft - pseudopotential methods ; to implement one such method and identify the main technical issues in doing so ; and to present the results of tests on a simple but important system , which allow us to gauge the usefulness of the method .we have shown that a rather general class of dft - pseudopotential methods can be based on a formulation of dft in terms of the density matrix , and that this formulation is equivalent to commonly used versions of dft that operate with fractional occupation numbers . from this viewpoint ,the key challenge is to ensure that the eigenvalues of the variable density matrix lie between 0 and 1 , and we have seen that the method of lnv  gives a way of doing this . the implementation of the basic ideas has been achieved by performing all calculations in real space , with the dft integrals approximated by sums on a grid except for the use of fft to treat the hartree term .an alternative here would be to work with atomic - like basis functions , but we note that the use of a grid preserves an important link with conventional plane - wave methods , as will be analysed in more detail elsewhere .our test results on perfect - crystal si show that the total energy converges rapidly as the real - space cutoffs are increased , and that it is straightforward to achieve a precision comparable with that of normal plane - wave calculations .an important question for any method is the system size at which it starts to beat a standard method a plane - wave method in the present case .this will clearly depend strongly on the system , but even for si it is too soon to answer it on the basis of practical calculations .the cross - over point depends on the prefactor in the linear scaling , and this is strongly affected by the efficiency of the coding .all we have attempted to do here is to address the problem of achieving behavior .the question of the prefactor is a separate matter , which will need separate investigation .it should be clear that there is much more to do before the present methods can be routinely applied to real problems .we have deliberately not discussed in detail the problems of doing calculations on a real - space grid .such problems have been discussed outside the linear - scaling context in several recent papers  , and it should be possible to apply the advances reported there to dft calculations .in particular , curvilinear grids  for the treatment of strongly attractive pseudopotentials are likely to be very important for calculations . we have also not discussed here the calculation of forces on the atoms , the problems that may arise when the boundaries of support regions cross grid points , and the general question of translational invariance within grid - based techniques .we have noted already that our method is related to other recently proposed methods . as shown in sec .2 , the mauri - galli - car scheme is obtained from ours by taking an alternative polynomial expression for the density matrix in terms of the auxiliary matrix .it would clearly be interesting to repeat the calculations done here using this approach . in a sense , this is what has already been done by hierse and stechel  , except that instead of performing calculations on a grid , they use a minimal atomic basis set .the hydrocarbon systems used by them for test purposes are also rather different from the si crystal we have studied .  finally , we note that our linear - scaling scheme is intended for calculations on very large systems , and this means that parallel implementation will play a key role .the test calculations we have presented were , in fact , performed on a massively parallel machine , and the parallel - coding techniques we have developed will be described in a separate paper .the work of cmg is supported by the high performance computing initiative ( hpci ) under grant gr / k41649 , and the work of eh by epsrc grant gr / j01967 . the major calculations were done on the cray t3d at edinburgh parallel computing centre using an allocation of time from the hpci .code development and subsidiary analysis were made using local hardware funded by epsrc grant gr / j36266 . we gratefully acknowledge useful discussions with d. vanderbilt .we derive here expressions for the derivatives and .   in dft , the total energy eq .( [ eq : totalenergy ] ) has two types of contributions : those that can be written as the trace of some operator acting on the density matrix , as is the case for the kinetic and pseudopotential energies ( see eq .( [ eq : traces ] ) ) , and those that depend only on the diagonal elements of , i.e. the electron density , namely the hartree and exchange - correlation energies .the madelung term , , does not depend on either or , so it will make no contribution to the variation in total energy as these are changed . denoting by the kinetic or pseudopotential contribution to the energy , we have : c_{\delta \gamma } \ ; , \label{eq : energycont}\end{aligned}\ ] ] where for the kinetic energy , and for the pseudopotential where is in general a non - local pseudopotential operator .clearly , does not depend on for either operator , so this term does not change as is varied .it is thus easy to see that , \label{eq : ecab}\end{aligned}\ ] ] where is the matrix representation of the corresponding operator ( kinetic energy or pseudopotential ) in the basis of the support functions .  for the hartree and exchange - correlation contributions , denoted by ,we have that the electron density is simply , so that : \phi_\delta({\bf r}).\end{aligned}\ ] ] in the case of the hartree contribution , we have where is the hartree potential , while in the case of the exchange - correlation contribution we have = \mu_{\rm xc}({\bf r}),\end{aligned}\ ] ] where is the exchange - correlation potential . if we take to represent either or as the case may be , we see that expression ( [ eq : evab ] ) reduces to ,\end{aligned}\ ] ] where by comparing this expression with eq .( [ eq : ecab ] ) , it is easy to see that the partial derivative of the total energy with respect to can be written more compactly as , \label{eq : etotalab}\end{aligned}\ ] ] where is the sum of the corresponding matrix elements of the kinetic , pseudopotential , hartree and exchange - correlation operators , i.e. the matrix representation of the kohn - sham hamiltonian in the basis of the support functions .recall that in practice , we do not vary but rather vary with respect to .however , it is trivial to obtain from eq .( [ eq : etotalab ] ) by simply substituting the matrix elements of by those of . once this is done , eq .( [ eq : etotalab ] ) corresponds to eq .( [ eq : partialab ] ) .  [[ functional - derivative - of - e_rm - tot - with - respect - to - phi_alpha ] ] functional derivative of with respect to ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  according to eq .( [ eq : separable ] ) , the density matrix , and hence the total energy , can be regarded as a function of the quantities and .when is varied , therefore varies firstly because of its direct dependence on , and secondly because of the implicit dependence of the matrix on through its dependence on the overlap matrix elements ( see eq . ( [ eq : k_of_l ] ) ) ; we call these two types of variation type 1 and type 2 .  to see how variations of type 1 behave , consider first the kinetic and pseudopotential energies. the type 1 variation of either of these is given by : where represents the kinetic energy or the pseudopotential operator .the variation of the integral gives the last equality follows from the fact that is a hermitian operator .the type 1 variation of can therefore be expressed as : where represents the action of the operator on evaluated at the point .  now consider the type 2 variation of due to the variation of the overlap matrix elements .the variation of is and the type 2 variation of is then obtained by applying this expression to eq .( [ eq : energycont ] ) . after a little manipulation ,one obtains : here , is the matrix whose elements are : combining eqs .( [ eq : conphi1 ] ) and ( [ eq : conphi2 ] ) , we obtain the following expression for the total variations of the kinetic and pseudopotential energies : \phi_\beta({\bf r } ) .\label{eq : de_dphi_kps}\end{aligned}\ ] ]  for the remaining terms ( hartree and exchange - correlation ) , variation in the energy results from variation in the electron density .thus we need to calculate \phi_\gamma({\bf r}').\end{aligned}\ ] ] again , we will have variations coming directly from the change in and variations coming indirectly from changes in the overlap matrix elements .the total variation of will be \nonumber \\ & & + \ ; 2 \sum_{\beta \gamma}\phi_\beta({\bf r}')\phi_\gamma({\bf r } ' ) \frac{\delta}{\delta \phi_\alpha({\bf r } ) } [ 3(lsl)_{\beta \gamma } - 2 ( lslsl)_{\beta \gamma } ] \ ; .\end{aligned}\ ] ] substituting this expression into where the quantity represents the hartree or exchange - correlation potential , we find , after some manipulation : \phi_\beta({\bf r } ) \ ; .\end{aligned}\ ] ] combining this expression for the hartree and exchange - correlation derivatives with eq .( [ eq : de_dphi_kps ] ) for the kinetic and pseudopotential derivatives , we find : \phi_\beta({\bf r } ) \ ; , \label{eq : de_dphi}\end{aligned}\ ] ] where is the kohn - sham operator , and is its matrix representation in the basis of support functions .note that in the practical grid - based calculations , the derivative we actually want is , which describes the variation of with respect to change of at the grid point .the formula for is identical to eq .( [ eq : de_dphi ] ) except that we need to multiply by the volume per grid point .see e.g. m.c .payne , m.p .teter , d.c .allan , t.a .arias and j.d .joannopoulos , rev .* 64 * , 1045 ( 1993 ) ; m.j .gillan , in _ computer simulation in materials science _ , edited by m. meyer and v. pontikis , kluwer , dordrecht ( 1991 ) ; g. galli and a. pasquarello , in _ computer simulation in chemical physics _ , edited by m.p .allen and d.j .tildesley , kluwer , dordrecht ( 1993 ) .

Abstract: a recently proposed linear - scaling scheme for density - functional pseudopotential calculations is described in detail . the method is based on a formulation of density functional theory in which the ground state energy is determined by minimization with respect to the density matrix , subject to the condition that the eigenvalues of the latter lie in the range [ 0,1 ] . linear - scaling behavior is achieved by requiring that the density matrix should vanish when the separation of its arguments exceeds a chosen cutoff . the limitation on the eigenvalue range is imposed by the method of li , nunes and vanderbilt . the scheme is implemented by calculating all terms in the energy on a uniform real - space grid , and minimization is performed using the conjugate - gradient method . tests on a 512-atom si system show that the total energy converges rapidly as the range of the density matrix is increased . a discussion of the relation between the present method and other linear - scaling methods is given , and some problems that still require solution are indicated .


in this paper , we investigate the following infinite horizon optimal control problem : given , and a convex compact subset of , solve for any  subject to the dynamical constraint here and are continuous functions . given and  ] , we have since is -lipschitz continuous w.r.t uniformly on , by gronwall inequality there exists a constant , such that .\ ] ] due to the continuity of and , we deduce that and are uniformly bounded in  ] , , it is deduced by the same arguments as in lemma [ lemoct ] that there exists {\rightarrow}{\mathbb{r}}^d ] , in this framework , we investigate the sparsity properties for the optimal controls which can be derived from the optimality condition .in general , the first order necessary optimality conditions for the infinite horizon problem are the following ( ( * ? ? ?* remark iii.2.55 ) ) .[ optimality ] assume that * ( h1)-(h2 ) * hold and suppose in addition that are with respect to the first variable .given , let be a locally optimal control for problem with the initial point and corresponding optimal trajectory .then there exists an adjoint state satisfying :  a.  for almost all , the following equality holds in the carathodory sense ^t { \varphi}(s)-e^{-\lambda s}\frac{\partial}{\partial y}\ell(\bar y(s),\bar u(s)).\ ] ] b.  c.  for almost all and all ,   the optimality condition shows that for almost all , a local optimal control maximizes where will be of importance throughout this section .the cases and will be treated separately .denote by the euclidean basis of .the first sparsity result that can be obtained for is the following .  given , and ,let be a locally optimal control , be the corresponding optimal trajectory and be the adjoint state .then the following holds in the almost everywhere sense : if then   the arguments are carried out for an arbitrary , and to simplify the notation the dependence of on is not indicated .  by lemma [ optimality ], maximizes the following function at first consider the case when for . for any , if there exists some such that , then due to the fact that we have and consequently for , we set then the maximization of is transformed to where is the polygon defined as for any , we have the constraints satisfied by imply that thus , is strongly convex in .it is also strongly convex in since is continuous .thus the maximum of is obtained at the vertices of the polygon . by comparing the values of at each vertex ,we get the following properties for the maximum  the desired result for , for , is obtained by replacing by .the other cases when have different signs can be treated analogously .condition is always satisfied for .we proceed to give the next sparsity result for the case with -constraints . for any , the following notation is used : .\ ] ]  [ constraintsinf ] given and , assume that is a local optimal control for problem with \ ] ] where .let be the corresponding optimal trajectory and be the adjoint state .then the following holds : for almost all   again the dependence of on is not indicated to simply the notation .  for ,let be defined by it is trivial to see that }\,g_i(v).\ ] ] at first consider the case with .then and hence  ] is either obtained at or . due to the fact that it is deduced that besides , if and , then has a unique solution and .thus , otherwise , for and , we have for  ] satisfies pontryagin s maximum principle for the problem as well . by* corollary , pp.220 ) , } ] is the optimal solution of the problem [ finiteoc ] for a sufficient large by remark [ inftofini ] . if is the optimal control , then it is also optimal for [ finiteoc ] by remark [ inftofini ] .thus on  ] , which is discretized with a mesh parameter .the discount factor is set , and the control weight .we begin by recalling standard results in the control of eikonal dynamics of the form where .we first consider the case of infinite horizon control with a quadratic control penalization , i.e. .results shown in figure [ eikp2 ] illustrate a simple setting where the control constraint is inactive and , as in the linear quadratic setting , the associated value function is quadratic and the optimal feedback is linear .the presented trajectories for the initial condition exhibit asymptotic stabilization , which is the standard result for this case ., -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ]   in this first case related to sparse controls , we set .results presented in figure [ eikp1 ] , illustrate the effect of the control penalization norm on the optimal control problem .the value function loses its quadratic nature , and more importantly , a sparsity region where the optimal control is is created around the origin . as predicted in section 6, the width of this region is equal to .the optimal control is bang - bang ( w.r.t .the euclidean norm constraint ) , or zero ( sparse ) , with the euclidean norm constraint active outside of the sparsity region .we also observe two bands outside the sparsity region where one coordinate is zero while the other makes use of the total control constraint ( directional sparsity ) .all these features are illustrated in the trajectory plot at the end of figure [ eikp1 ] .the structure of the optimal control confirms the results obtained in the third case of theorem 6.3 .  comparing figures 2 and 4we note that in the eikonal case for the limit , the optimal control of the penalized control problem converges to the optimal control of the minimum time problem .this implies a finite time arrival to the origin , which differs from the asymptotic stabilization properties that can be derived for the control penalization .furthermore , it can be verified that this result is independent of the parameter .  .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ]  . row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] . row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ] .row 1 and 2 : value function , -norm of the optimal control , optimal controls and , over the state space ^ 2 ] .row 3 : trajectories for the initial condition ,title="fig:",scaledwidth=49.5% ]we have studied both theoretically and computationally , sparsity properties of infinite horizon , optimal control problems with -penalties ,  ] : where and satisfies the dynamical system .due to lemma [ lemoct ] , there exists an optimal control for .denote by the optimal trajectory corresponding to and .\ ] ] then is optimal for with the final time .the aim is to construct an admissible control defined on the infinite time interval .the corresponding extended state will be denoted by .the construction is described step by step as follows .consider at first the sequence on the time interval  ] .note that each control is an optimal control for a corresponding problem of the form for some integer on the time interval  ] for .analogously to the previous step there exists of such that ,\ \text{as}\ k{\rightarrow}\infty;\ ] ] for satisfying on  ] , and it is denoted again by the same symbol .each control is an optimal control for a corresponding problem of the form for some integer on the time interval  ] , and each is optimal for the problem with .moreover , for all , ,\ \text{as}\ k{\rightarrow}\infty;\ ] ] let us take the diagonal sequence and denote then the following properties hold :  a.  , the control is defined on the time interval $ ] with , is an optimal control for the problem * * of the form , and is optimal for the problem with .b.  , we have \ \text{as}\ k{\rightarrow}\infty,\ ] ] c.  there exists such that   we proceed with proving that is an optimal control for the problem . arguing by contradiction , if is not optimal for , there exists and satisfying such that by the assumption * ( h2 ) * , there exists satisfying such that since is optimal for ( ) , we have for any that then for any and any with , and thus by taking , it holds that we thus obtain , and therefore there exists such that for any  due to the fact that uniformly , there exists such that for all , which implies that since is optimal for * * , note that , and thus together with and , we have finally , by we deduce that which contradicts .hence is an optimal control for , which ends the proof .cannarsa , p. , sinestrari , c. : semiconcave functions , hamilton - jacobi equations , and optimal control .progress in nonlinear differential equations and their applications , 58 .birkhuser boston , inc . ,boston , ma ( 2004 )    garg , d. , patterson , m. , francolin , c. , darby , c. , huntington , g.t . ,hager , w. , rao , a. : direct trajectory optimization and costate estimation of finite - horizon and infinite - horizon optimal control problems using a radau pseudospectral method .49*(2 ) , 335358 ( 2011 )     aseev , s. , kryazhimskii , a. , tarasyev , a. : first order necessary optimality conditions for a class of infinite horizon optimal control problems .iiasa interim rept .ir-01 - 007 ( laxemburg , austria , 2001 )  aubin , j.p . , cellina , a. : differential inclusions , _ grundlehren der mathematischen wissenschaften [ fundamental principles of mathematical sciences ] _ , vol .springer - verlag , berlin ( 1984 ) .set - valued maps and viability theory  bardi , m. , capuzzo - dolcetta , i. : optimal control and viscosity solutions of hamilton - jacobi - bellman equations .systems & control : foundations & applications .birkhuser boston , inc . ,boston , ma ( 1997 ) . with appendices by maurizio falcone and pierpaolo soravia

Abstract: a class of infinite horizon optimal control problems involving -type cost functionals with is discussed . the existence of optimal controls is studied for both the convex case with and the nonconvex case with , and the sparsity structure of the optimal controls promoted by the -type penalties is analyzed . a dynamic programming approach is proposed to numerically approximate the corresponding sparse optimal controllers .