in 2011 , physics world honored aephraim steinberg and colleagues, from the university of toronto , with the top one position in its yearly breakthrough ranking . so farthe quest for fundamental particles involves large energy and length scales from giga to teraelectronvolts ( e.g. , 7 tev in p - p collisions ) , from tenths of meters ( e.g. , the lhc detector has about 14.6 m of diameter and 21.6 m of total length ) to kilometers ( e.g. , the lhc ring has about 27 km length ) . in contrast, steinberg s team developed a very interesting top - table experiment at much smaller scales about 1.3 ev ( 943 nm ) and a few meters , designed to shed some light on another fundamental aspect of the quantum theory : elucidating how quantum particles travel ( on average ) through physical space . with the aid of the so - called `` weak measurement '' technique, consisting of weakly perturbing the quantum system at some time during its evolution and previous to its definite measurement , this team has become `` the first to track the average path of single photons passing through a young s double slit experiment something that steinberg says physicists had been `` brainwashed '' into thinking is impossible . ''but , why is this experiment relevant at all ?the mathematical formulation of the quantum theory is neat and accurate , and its applications have proven to be very powerful an important `` bite '' of the gross internal product in industrialized countries relies on quantum mechanics , including developments and applications in technological , energetic or biomedical areas . from the electron to the higgs , the success of quantum mechanics is indisputable .however , what do we really know about quantum systems ?the famous bohr - einstein debates ended in the 1930s with the orthodox or copenhagian view of quantum systems , which has healthfully survived to date : the quantum world is essentially probabilistic and hence it does not make any sense asking questions intended for going beyond probabilities .actually , these probabilities are such that if we try to determine accurate ( probabilistic ) information about one of the variables ( ) from a pair of ( classical ) canonically conjugate variables , we will be unable to obtain any relevant information about the other ( ) , and vice versa .this is the essence of bohr s complementarity principle , which formally translates into the well - known heisenberg uncertainty relation , where denotes the dispersion in the measurement of either or .  at a more pictorial level, complementarity is often regarded to the alleged wave - corpuscle duality exhibited by quantum systems : an electron behaves as a wave when it passes through a pinhole , and as a corpuscle when it is acted by a highly energetic photon ( e.g. , rays in compton scattering ) .hence , in young s two - slit experiment , if the electron passes without ever being disturbed during the transit , we will observe a nice fringed pattern : the electron distributes within some regions , while avoids some others .all about the electron distribution is known , but nothing about which slit the electron passed through or , in other words , which momentum the electron carried . trying to determine the latter would eventually lead to fringe erasure because of the impossibility to measure both at the same time . according to von neumann, is a manifestation of a non - unitary evolution process known as the `` collapse '' of the wave function .that is , while the system is unperturbed , it displays a unitary ( probability preserving ) evolution in compliance with schrdinger s equation ; once a measurement is performed , such unitarity is broken and the system state randomly `` collapses '' onto any of the pointer states of the measurement device with the probability prescribed by the system wave function . when this argumentation was proposed , it found the strong opposition of einstein and others , who thought that there should be something else , a set of internal or _ hidden_ variables that would determine the outcome , thus removing any trace of randomness . for those physicists seeking for an objective or realistic description of quantum systems , the wave function could not be complete because it was not able to specifically determine all possible information about the system .it was a hard and thorny way to disprove von neumann s statement , according to which quantum mechanics does not admit hidden variables .however , in 1952 david bohm proposed a counterexample to this theorem , showing that an account of the individual evolution of the system is still compatible with the quantum theory by simply assuming that the wave function acts like a field `` guiding '' the system ( a similar idea was proposed about 25 years earlier by louis de broglie ) .as a consequence , there is no need for a `` collapse '' postulate nor even the presence of an external observer .this does not mean that the mathematical structure itself of quantum mechanics has to be changed , but only that there is still much more room for thinking quantum phenomena in different alternative ways .bohm s suggestion remained almost forgotten until the 1960s , when it called the attention of john bell. while working at cern , he decided to go back to von neumann s theorem and re - examine it with the purpose of systematizing which properties should be satisfied by a quantum hidden - variable theory in order to be valid .in so doing , bell noticed that what makes quantum mechanics so special is a feature lacking in classical mechanics , namely nonlocality , i.e. , the fact that any local disturbance in a quantum system immediately affects the whole system .the most striking example where this property manifests is in quantum entanglement : the correlations between two spatially separate systems that interacted in the far past are very important , since they can be used to transfer quantum information between two distant places without incurring in surperluminal signaling .bell s contribution not only started the revolution of quantum technologies , leading to the development of the quantum information theory , quantum computing , quantum cryptography , quantum teleportation , etc . , but indirectly he also motivated a reconsideration of bohm s approach . nowadays , for some people this is just an alternative quantum theory ( this being based on the ontology generated from it ) ; for others it is only an alternative formulation of the quantum theory . if this is simply regarded as a matter of taste , and we decide to remain at a more pragmatic level , what really matters is the fact that bohmian mechanics has open an alternative pathway to understand quantum systems , justifying steinberg s statement that our `` brainwashed '' view of quantum phenomena may be changed by experiments like the one that he and his colleagues performed just on an optical bench .during the early days of quantum mechanics the weight of statistical mechanics and thermodynamics was too strong boltzman s shadow was too long .it is not strange , therefore , that the copenhagen interpretation became widely accepted ( aided by the neopositivist thought - streams of those years ) , healthfully surviving up to now .however , at present quantum experiments can be performed in a , by far , more refined way than in the first decades of the xxth century , thus limiting the importance of statistics . more specifically , the system evolution can be monitored in real time ( e.g. , molecular configurations , entanglement dynamics , electron ionization , surface diffusion , etc . ) , in contrast with traditional spectroscopic measurements based on the energy domain .furthermore , since the 1970s we also have interesting interference experiments corroborating that , even if we know absolutely nothing about how each ( quantum ) particle behaves individually , at least we know that it reaches the detector as a single , localized event , and not as an extended wave . from the former experiments with electrons to the latter with large molecular complexes, has been confirmed that , as time proceeds and the accumulation of ( randomly distributed ) particle arrivals on a distant detector becomes larger , an incipient interference fringed pattern starts emerging from a seemingly disordered distribution of single , ( time ) uncorrelated detections , in clear correspondence with the outcome expected from schrdinger s equation .statistics thus plays an important role in accounting for the general picture , but says nothing about how each individual detection takes place.  chapter 2 of feynman s renowned _ lectures on physics _ starts as `` in this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form . we choose to examine a phenomenon which is impossible , _ absolutely _ impossible , to explain in any classical way , and which has in it the heart of quantum mechanics . in reality, it contains the _ only _ mystery .we can not make the mystery go away by `` explaining '' how it works .we will just tell you how it works . in telling you how it works we will have told you about the basic peculiarities of all quantum mechanics . ''effectively , the two - slit experiment summarizes the essence ( `` mystery '' ) of quantum mechanics .however , like dirac , feynman also thought that each individual particle self - interfered , apparently being unaware of pozzi s and tonomura s experiments on the two - slit experiment .his scientific authority reinforced the copenhagian viewpoint , but should things have been different if feynman would have watched tonomura s movie of his two - slit experiment with electrons ?well , although evidently we have no answer to this question , perhaps we could say that , for someone who succeeded in introducing the concept of trajectory into quantum mechanics ( in spite of bohr s disapproval ) , things would have been , at least , a bit different .  in any case, it is important to keep in mind three very simple ideas :  * appealing to occam s razor , there is no reason to think that the wave function collapses to a local point upon detection , as formerly stated by von neumann. * experiments corroborate a statistical origin of interference patterns ( and , in general , any quantum trait ) , consisting of a large accumulation of single , localized events. * even if all particles are generated at the same source ( where they can interact ) , once they are released experiments also confirm that each particle arrival is independent of all other previous or subsequent ones ( see more recent experiment in ) . that is , one particle is totally unaware of what others do .accordingly , one can not avoid thinking whether single - event descriptions are affordable in quantum mechanics , even being aware that the evolution of a single particle can not be tracked ( at least , not with the current technology ) without irreversibly perturbing it . in some sense , it would still be possible to work at the level of classical fluids : we know nothing about the individual motion of the fluid s constituents , but still its collective dynamical properties could be determined by means of a streamline analysis .   for simplicity ,let us consider the nonrelativistic schrdinger equation , which describes how the wave amplitude associated with a physical system of mass evolves in time through a given configuration space accounting , for instance , for the system position .statistical information about the possible outcomes that can be expected at any place and time are determined from the probability density .it is at this point where bohm felt the need to introduce the concept of hidden variable , just as a way to test the validity of the assumption that `` the most complete possible specification of an individual system is in terms of a wave function that determines only probable results of actual measurement processes . ''these hidden variables in principle would `` determine the precise behavior of an individual system , but which are in practice averaged over in measurements of the types that can now be carried out . ''the evolution of these hidden variables , which he identifies with individual realizations ( trajectories ) of the system , arises after recasting schrdinger s equation as a set of two real equations of motion , one for the probability density and another one for the phase of the wave function ( when the latter is expressed in polar form , ) .this gives rise to the continuity equation and a hamilton - jacobi - like equation , from which bohm postulates that the system trajectories evolve according to the equation of motion where is a local velocity field . within this scenario ,quantum systems thus consist of a wave field ( ) and a particle guided by this field , which follows a trajectory in configuration space .this is essentially what is nowadays known as bohmian mechanics, although the more philosophical ( ontological ) aspects are or are not seriously considered depending on the author .leaving aside metaphysical connotations , from a pragmatist s viewpoint , bohmian mechanics is basically a fluid - dynamical description for quantum systems .it is worth stressing that the same ideas were already proposed ( although not in the context of hidden variables ) by de broglie and madelung, and contemporarily to bohm by takabayasi. furthermore , it should be clear that this formulation is totally equivalent and is at the same level as any other of the more traditional ones ( schrdinger s , heisenberg s , dirac s , moyal - wigner , etc . ) .each formulation emphasizes a different aspect of the quantum theory ; bohmian mechanics stresses the fact that quantum systems can be associated with a quantum fluid ( not in vain schrdinger s equation is just a diffusion equation with an imaginary diffusion constant , ) .actually , descriptions like the bohmian one in terms of streamlines are rather common in the literature, and in recent years it has been possible to recreate bohmian - like systems ( in fact , debroglian ones ) by means of classical fluid - dynamical experiments.  rather than hidden variables , bohmian mechanics constitutes a valuable tool that allows us to determine `` hidden '' quantum properties , i.e. , properties that are not evident within other formulations of the quantum mechanics . for example , based on the bohmian non - crossing property, i.e. , that bohmian trajectories can not pass through the same point of the configuration space at the same time , we find that quantum flows do not mix in configuration space , as steinberg s experiment shows, and accordingly one can properly specify tubes along which the probability remains constant at any time. also , based on the non - crossing property , one notices that the physical implications of the superposition principle go further beyond from its simple mathematical application due to the phase dynamics involved , which immediately modifies the velocity map throughout the whole configuration space. in any case , it is worth stressing that bohmian trajectories only provide us with hydrodynamical information of the quantum fluid or wave field itself , but not of what is in there behind it that is , how the average ensemble evolves , but not how a real individual particle moves. a simple example of bohmian trajectories illustrating the renowned young two - slit experiment is displayed in fig .[ fig1 ] , where the background contour - plot represents the nonlocal velocity field pervading the whole configuration space .direction , produce two gaussian wave packets , which eventually superimpose and generate in the long time ( fraunhofer ) regime a series of interference fringes ( d , e , f , g ) .black solid lines represent ensembles of bohmian trajectories leaving each slit .the background contour - plot corresponds to the velocity field as time proceeds ( time increases along the horizontal axis ) : from blue to red , increasing value of the velocity from negative to positive , respectively .note that the non - crossing appears as a consequence of the sudden change of sign undergone by the velocity field along the system symmetry line ( ).,width=377 ]let us now revisit the well - known delayed - choice experiment proposed by wheeler in the late 1970s a paradigm of the mystery entailed by quantum mechanics to illustrate how bohmian mechanics removes the paradoxical aspects introduced by this experiment .the main idea behind this experiment is very simple : to reveal the puzzling dual wave - corpuscle nature of quantum systems within the same experiment . to this end, wheeler considers an optical mach - zehnder interferometer with a movable second beam splitter .moreover , the experiment is performed in such a way that at each time there is only one photon inside the interferometer . for a visual representation of the interferometer configurations described below , see fig .[ fig2 ] ( do not confuse here the bohmian trajectories displayed with usual optical geometric paths ) .when the photon enters the interferometer , the first ( fixed ) beam splitter ( bs1 ) , oriented at 45 degrees with respect to the photon incidence direction , may produce direct transmission towards a mirror m1 with a 50% of probability , or a perpendicular deflection ( reflection ) towards a mirror m2 . in either case ,when the photon reaches the mirrors , it gets deflected 90 with respect to the corresponding photon incidence direction .eventually , in an open configuration ( see fig . [ fig2](a ) ) , i.e. , without the second beam splitter bs2 , an arrival can be registered by a detector d1 if the photon followed the transmitted path ( let us denote it by p1 ) , or by d2 if otherwise it followed the reflected path ( p2 ) .this is a typical corpuscular scenario , describable in terms of classical optics ( the photon follows geometric rays ) . on the other hand , in a closed configuration ( see fig . [ fig2](b ) ) , when bs2 is introduced at the place where the paths p1 and p2 intersect , the photon will always be detected by d2 . in this casethe photon displays its wave behavior : at bs2 the components of the associated wave interfere destructively in the direction of p1 and constructively along p2 . followingthe bohr - einstein debates, wheeler reformulated one of the main questions coming from them : when does the quantum system make the choice to behave as a wave or as a particle ?instead of considering a double slit , wheeler assumed the above interferometer , but with bs2 movable , so that it could be removed or inserted once the photon had passed through bs1 .the result is very interesting : regardless of when bs2 is put into play , the photon always behaves as it should , that is , just as if it could anticipate what is going to happen in future : the photon makes a delayed choice .although initially conceived as a thought - experiment , this experiment has already been carried out in the laboratory in many different ways , but always confirming this challenging dual behavior.     the weirdness of wheeler s experiment readily dissipates if one starts describing it properly , that is , using quantum mechanics since the very beginning , without any aid of classical corpuscle - based model .a priori the usual schrdinger or heisenberg descriptions might seem of little help , since they essentially stress the role of probabilities .however , if we keep in mind the fact that there is an associated phase dynamics ( and therefore an inherent nonlocal velocity field ) , things change .this is the idea that first bohm and coworkers, and later on hiley and callaghan, tried to convey by analyzing the experiment in terms of bohmian mechanics . because bohmian trajectories , which in essence are elements to make apparent the flow of such probabilities , obey the non - crossing rule ( see sec .[ sec22 ] ) , what happens is that the photon always behave in the same way . if bs2 is absent , because the trajectories coming from p1 and p2 can not cross the symmetry line at 45 , those coming from p1 are reflected in the direction of d2 , and those from p2 in the direction of d1 .that is , it is not that the photon follows p1 or p2 until reaching the corresponding detector , but there is an exchange in the directionality of the associated quantum flow . on the other hand ,if bs2 is introduced , even once the photon is inside the interferometer , the wave recombination process taking place at this beam splitter produces that the two sets of trajectories will eventually go into one detector , namely d2 .this all - the - way wave behavior ( the classical corpuscle notion just disappears , since it is not necessary at all ) is illustrated in fig .[ fig2 ] by means of the numerical simulation of the open ( a ) and closed ( b ) interferometer configurations described above .as it can be seen , there is no choice of the photon at all , but just a modification of the boundary conditions affecting its wave function , which naturally gives rise to different outcomes , regardless of whether bs2 is introduced or removed once the wave function has started its evolution inside the interferometer .this kind of realistic simulations are very important to better understand the physics that is taking place in apparently paradoxical situations , as it has been recently shown in the case of atomic mach - zehnder interferometry, for example , which is typically considered to analyze and discuss fundamental complementarity issues due to its suitability to this purpose. mechanics has proven to be the most successful theory ever devised ( at least , to date ) .this theory not only has addressed the most fundamental physical problems , but its applications constitute an important part of our everyday life ( actually , more sophisticated applications are still to come ) .however , the fact quantum phenomena can not be understood under the same logic as classical ones brings in a puzzling situation , which has left open a tough debate on its interpretation since the 1930s .probably one may think that this debate will never be really closed ( or at least until we will be able to devise a new , more general theory ) and that , in such a case , it might be pointless to continue talking about quantum philosophical issues or trying to further develop the area of the quantum foundations .evidently , this leads to a sort of hopeless situation , with a remarkably close resemblance to plato s myth of the cave : we are enforced to perceive the shadows cast on the wall of the cave ( our reality ) by the real world ( the reality ) , without possibility to ever reaching a true understanding of the physical world .however , we have seen above that , even within such a harsh scenario , there is still room enough to further explore the quantum world from a pragmatist s point of view , just playing around with the quantum rules and its many way to formulate them .one only needs to look back for a while and make a reflection on how the quantum theory has conceptually developed since its early stages , what can be done at present , and which alternative routes can be followed .these are the essential ingredients for new quantum developments and advances .based on recent achievements , the bohmian formulation of the quantum theory seems to be one of these routes , which allows us to understand how quantum systems evolve obeying a non - observable ( i.e. , not directly accessible in the experiment ) phase dynamics .perhaps it will not be possible to determine how a real individual particle evolves without disturbing it , but at least now we have a tool to understand how the flow of many of these identical particles evolves in configuration space .this has helped us to understand that there are no paradoxes in the quantum theory , but only a misconception about how quantum systems behave , anchored in old - fashioned classical prejudices . in thisregard , following nobel laureate anthony leggett, `` if we push quantum mechanics hard enough it will break down and something else will take over something we ca nt envisage at the moment . ''the author thanks roman ryutin and vladimir petrov for their kind invitation to participate in hepft2014 , and anton godizov for his kind assistance and friendship during the event .support from the ministerio de economa y competitividad ( spain ) under project no .fis2011 - 29596-c02 - 01 as well as a `` ramn y cajal '' research fellowship with ref .ryc-2010 - 05768 is acknowledged .n. bohr , discussion with einstein on epistemological problems in atomic physics , in _ albert einstein : philosopher - scientist _a. schilpp , library of living philosophers , vol .7 ( mjf books , new york , 1949 ) , pp .199 - 241 .reprinted in _ quantum theory and measurementj. a. wheeler and w. h. zurek ( princeton university press , princeton , nj , 1983 ) , pp .9 - 48 ( see also commentaries on the issue in pp .3 - 48 ) .p. g. merli , g. f. missiroli and g. pozzi , _ am .j. phys . _* 44 * , 306 ( 1976 ) .see also : g. matteucci , m. pezzi , g. pozzi , g. l. alberghi , f. giorgi , a. gabrielli , n. s. cesari , m. villa , a. zoccoli , s. frabboni and g. c. gazzadi , _ eur . j. phys ._ * 34 * , 511 ( 2013 ) .t. juffman , a. milic , m. mllneritsch , p. asenbaum , a. tsukernik , j. txen , m. mayor , o. cheshnovsky and m. arndt , _ nature nanotech . _* 7 * , 297 ( 2012 ) .a movie on the experiment can be found at the link : https://www.youtube.com/watch?v=vciomqiru7i         e. fort , a. eddi , a. boudaoud , j. moukhtar and y. couder , _ proc .sci . _ * 107 * , 17515 ( 2010 ) ; d. m. harris , j. moukhtar , e. fort , y. couder and j. w. bush , _ phys .e _ * 88 * , 011001r ( 2013 ) .t. hellmuth , h. walther , a. zajonc and w. schleich , _ phys .a _ * 35 * , 2532 ( 1987 ) ; v. jacques , e. wu , f. grosshans , f. treussart , p. grangier , a. aspect and j .- f .roch , _ science _ * 315 * , 966 ( 2007 ) ; j .- s .tang , y .- l .li , x .- y .xu , g .- y .xiang , c .- f . li and g .- c .guo , nature photon .* 6 * , 600 ( 2012 ) .m. s. chapman , t. d. hammond , a. lenef , j. schmiedmayer , r. a. rubenstein , e. smith and d. e. pritchard , _ phys ._ * 75 * , 3783 ( 1995 ) ; a. d. cronin , j. schmiedmayer and d. e. pritchard , _ rev . mod_ * 81 * , 1051 ( 2009 ) .

Abstract: to date , quantum mechanics has proven to be our most successful theoretical model . however , it is still surrounded by a `` mysterious halo '' that can be summarized in a simple but challenging question : why quantum phenomena are not understood under the same logic as classical ones ? although this is an open question ( probably without an answer ) , from a pragmatist s point of view there is still room enough to further explore the quantum world , marveling ourselves with new physical insights . we just need to look back in the historical evolution of the quantum theory and thoroughly reconsider three key issues : ( 1 ) how this has developed since its early stages at a conceptual level , ( 2 ) what kind of experiments can be performed at present in a laboratory , and ( 3 ) what nonstandard conceptual models are available to extract some extra information . this contribution is aimed at providing some answers ( and , perhaps , also raising some issues ) to these questions through one of such models , namely bohmian mechanics , a hydrodynamic formulation of the quantum theory , which is currently trying to open new pathways of understanding . specifically , the chapter constitutes a brief and personal overview on the historic and contextual evolution of this quantum formulation , its physical meaning and interest ( leaving aside metaphysical issues ) , and how it may help to overcome some preconceived paradoxical aspects of the quantum theory .


discontinuous galerkin ( dg ) methods are high - order finite element discretizations which were introduced in the early 1970s for the numerical simulation of the first - order hyperbolic neutron transport equation . in recent years , these methods have become very popular for the solution of nonlinear convection dominated flow problems .these methods allow high - order of accuracy and locality , which make them well suited to parallel computing , -refinement , -multigrid , unstructured meshes , application of boundary conditions , etc .however , the dg method suffers from spurious oscillations in the vicinity of discontinuities that develop in solutions of hyperbolic systems of conservation laws .these oscillations are due to the gibbs phenomenon and may cause the solution to become locally nonphysical leading to robustness issues for the computation .quadrature rules are usually used to compute integrals in the discretization of the equations .properties of the dg method therefore depend on the local structure of the numerical solution at faces and within elements of the mesh .the control of such oscillations at a reasonable cost while keeping accuracy , robustness and stability is essential for the efficiency of the dg method and remains a challenge .strategies have been proposed such as limiters , non - oscillatory reconstructions , -adaptation , shock capturing techniques , etc .the latter methods aim at adding artificial viscosity to spread the structure of the discontinuity so that it can be resolved at the discrete level . for a dg method with polynomials of degree and cells of size ,the resolution scale is thus meaning that the method should resolve the discontinuity inside the element .the behavior of the numerical solution inside the discretization elements is therefore important for understanding the convergence of the dg computations .  in this work ,we focus on the dg method for scalar nonlinear conservation laws which present stationary discontinuous solutions .more precisely , we are interested in the behavior of discrete profiles near shocks .jennings studied the approximation of scalar equations by monotone conservative finite difference schemes and proved the existence and stability of traveling discrete shocks .discrete profiles for scalar equations were also studied in .the analysis was then extended to systems in .bultelle et al . analyzed the linear stability of steady shock profiles obtained for general systems of conservation laws discretized with the godunov method and constructed unstable profiles in the case of the euler equations for gas dynamics .recently , lerat found exact discrete shock solutions for residual - based compact schemes up to seventh - order of accuracy discretizing the inviscid burgers equation .solutions were derived explicitly and were parametrized by the relative position of the shock in the discretization cell . such analysis may help to tune parameters of the numerical method . in the context of dg methods , cockburn andguzmn considered a formally second order approximation of a scalar linear hyperbolic equation with discontinuous initial condition .they gave estimates of the size of extent of oscillations upstream and downstream of the discontinuity based on suitable weights introduced in .recently , their work was extended to arbitrary approximation order in space and a third - order runge - kutta method on non - uniform meshes .the case of -singularities in initial condition and source term of a scalar hyperbolic equation was then investigated in where superconvergence in negative - order norms outside the pollution region was proved and sharp estimates over the whole domain were given .the objective of this work is the theoretical analysis of stationary discrete shock solutions for the dg method discretizing a scalar conservation law with a general convex flux .we will mainly consider the godunov method to evaluate the numerical flux for which explicit solutions can be derived , but we will also focus on other numerical fluxes widely used in the context of dg methods . to this end , we first give general results about the structure of the stationary discrete profiles .these results are local in the sense that they consider profiles that are small perturbations to the exact solution . in the case of the godunov flux ,we establish the mean value of the solution in the cell containing the exact shock position and the complete solution in other cells for an arbitrary approximation order .the analysis also predicts exponential decay of oscillations of the discrete profile on both sides of the shock for a certain class of monotone numerical fluxes .then , the linear stability of these profiles is investigated and eigenvalues are characterized for the godunov flux . as a result , the shock profiles may become unstable at points where the numerical flux is not differentiable .these points contain the situation of a shock at interface which is counter - intuitive since the exact solution is a piecewise constant function over all cells and is included in the function space of the dg method .considering the inviscid burgers equation we derive exact discrete shock profiles for the dg scheme up to fourth - order of accuracy in the spirit of the work of lerat .the results allow to evaluate quantitatively the structure of the solution within elements and predict situations where the numerical solution violates locally an entropy condition at the cell level .the theoretical stability analysis predicts the occurrence of unstable profiles when the exact shock position is close enough to an interface of the mesh .numerical experiments will suggest that many features of the dg method obtained with the godunov flux still hold for other numerical fluxes .in particular , for a given space discretization , the oscillating and unstable characters of the solution depend on the strength and exact position of the shock . as this latter feature is generally unknown, it strongly complicates the analysis for enhancing stability and robustness of the dg method .these results may support approaches based on _ a posteriori _ limitation techniques such as the mood method .artificial viscosity represent another attractive approach providing that the amount of viscosity adapts itself to the regularity of the solution .theoretical results in this work will suggest the addition of artificial viscosity to the highest modes in the dg space spanned by hierarchical basis functions . as an illustration, we apply the spectral vanishing viscosity method with a selective filter in the legendre basis as proposed in .the paper is organized as follows .section [ sec : numer_approach ] presents the model problem and the numerical approach for the discretization . in section[ sec : steady_shock_sol_conv ] , we consider a general convex flux and analyze discrete shock profiles for the dg method .these profiles are explicitly constructed in the case of the inviscid burgers equation in section [ sec : steady_shock_sol_burgers ] .a linear stability analysis is performed in the vicinity of these solutions in section [ sec : ana_stab ] .these results are assessed by several numerical experiments in section [ sec : num_xp ] and a first attempt for stabilizing the numerical scheme is proposed in section [ sec : svm ] . finally , concluding remarks about this work are given in section [ sec : conclusions ] .the discussion in this paper focuses on the discretization of scalar nonlinear hyperbolic equations in one space dimension with a dg method .let be the space domain and consider the following problem  [ eq : model_pb ] _tu + _ xf(u ) & = & 0 , ( 0 , ) , + u(x,0 ) & = & u_0(x ) , .  the physical flux in is assumed to be coercive and strictly convex over the set of admissible states : and for all in .we are particularly interested in steady shock solutions to ( [ eq : model_pb ] ) .such solutions consist in stationary discontinuities between two states   where denotes the shock position and the states satisfy the rankine - hugoniot relation and the lax entropy condition   the above equation may be written in the equivalent form   where is the unique state such that . integrating ( [ eq : model_pb]a ) in space over , one obtains , which induces    the dg method consists in defining a discrete weak formulation of problem ( [ eq : model_pb ] ) .the domain is discretized with a uniform grid with cells  ] and using orthogonality , one obtains the recursion   and hence if and if for the entries of the matrix defined by   note that we know the explicit projection of the derivatives of the legendre polynomials into the legendre basis . indeed , applying the recurrence relation ( [ eq : leg_rec ] )successively one obtains which may be rewritten under the general form where the entries of the matrix are defined by for and for .the semi - discrete form of the dg discretization in space of problem ( [ eq : model_pb ] ) reads : find in such that   together with the initial condition   the discretization of the elementwise explicit residuals in ( [ eq : discr_var_form ] ) reads   where denotes a numerical flux consistent with the physical flux : .we will consider the godunov flux for the theoretical analysis of sections [ sec : steady_shock_sol_burgers ] and [ sec : ana_stab_god ] : \ } , & \mbox{if } u_h^- \leq u_h^+,\\  \max\{f(v):\,v\in[u_h^+,u_h^-]\ } , & \mbox{if } u_h^- > u_h^+ .\end{array } \right.\ ] ]  the godunov flux results from the solution of the riemann problem for ( [ eq : model_pb ] ) with piecewise constant initial data consisting in states and at the left and right of the interface . the values of ( [ eq : godunov_flux ] ) in the set of states are depicted in figure [ fig : solution_pr_godunov_burgers](a ) .comparisons will also be given in the numerical experiments for the local lax - friedrichs ( llf ) flux : where the constant is a stabilization parameter defined by , and the engquist - osher flux ( see figure [ fig : solution_pr_godunov_burgers](b ) ) :   we recall that fluxes ( [ eq : godunov_flux ] ) to ( [ eq : osh_flux ] ) are lipschitz continuous and monotone   where and denote the partial derivatives of with respect to its first and second arguments .numerical experiments tend to show that the effect of the flux on the quality of the approximation decreases as the polynomial degree increases .the analysis in section [ sec : steady_shock_sol_conv ] and numerical experiments in section [ sec : num_xp ] will support this trend and confirm the relevance of the theoretical analysis with the godunov flux in sections [ sec : steady_shock_sol_burgers ] and [ sec : ana_stab ] .   looking for steady - state solutions   of the numerical scheme ( [ eq : discr_var_form ] ) and projecting the discrete scheme into the function basis ,one obtains for all in and :   the local conservation property   is obtained from where denotes the indicator function of .we are interested in discrete solutions of the numerical scheme such that   therefore , we obtain and the elementwise residuals in cells in may be rewritten under the more convenient forms  [ eq : steady_discrete_scheme ] r_j^0(u_h ) & = & ( u_j+^-,u_j+^+ ) - f _ + & = & ( u_j-^-,u_j-^+ ) - f_= 0 , + r_j^k(u_h ) & = & _ _ j f(u_h)d_x_j^k dx - ( 1-(-1)^k)f_= 0 , 1kp .in this section , we present some preliminary results obtained for a convex flux that will be used in the next sections .we assume that the exact shock position is known and we set the cell index containing the shock without loss of generality .cells and will be referred as to supersonic and subsonic cells , respectively .our objective is here to determine the structure of the numerical solution in cells upstream and downstream of the shock position and to provide information about the solution in the shock cell .the analysis will mainly focus on the use of the godunov flux ( [ eq : godunov_flux ] ) . for a steady - state solution to hold ,the conservation property , with a coercive and convex flux , has one of the following two solutions  [ eq : sol_godunov_flux ] u_h^-=u_l & & u_h^+u_r , + u_h^-u_l & & u_h^+= u_r .indeed , equation has two roots or ( see section [ sec : model_eqn ] ) . hence , to solve , it is suficient to consider the solutions of riemann problems with initial conditions either such that and in , or in and such that .consider the first initial condition , then or . since is continuous ,we now consider successive situations when varies .consider first .the following situations ( i ) , ( ii ) , and ( iii ) lead to the following expressions of the godunov flux ( [ eq : godunov_flux ] ) : ( i ) , ( ii ) , and ( iii ) .the other situation , , leads to when , where is defined by ( [ eq : lax_cond_hat - u ] ) , and when .we therefore obtain ( [ eq : sol_godunov_flux]a ) and a similar step applies to infer ( [ eq : sol_godunov_flux]b ) .  in the following, we shall assume a stronger condition on the traces of the numerical solution at left and right faces of the shock cell :  [ eq : sign_condition_conv - flx ] u_-1/2 ^ -=u_l & & u_-1/2^+ > u_r , + u_1/2 ^ - < u_l & & u_1/2^+= u_r .indeed , the godunov flux is no longer differentiable at points with such that ( see figure [ fig : solution_pr_godunov_burgers](a ) ) and the strict inequalities in ( [ eq : sign_condition_conv - flx ] ) ensure its differentiability in our analysis. we will give some comments on the implications of assumption ( [ eq : sign_condition_conv - flx ] ) in section [ sec : comments_trace_ineq ] .we stress that the present results will show that the above assumption is not a restriction on the validity of our analysis .as a consequence of ( [ eq : sign_condition_conv - flx ] ) , the godunov flux ( [ eq : godunov_flux ] ) reduces to a fully upwind flux at interfaces :  [ eq : godunov - flx_interf_jc ] ( u_-1/2 ^ -,u_-1/2^+ ) & = & f(u_-1/2 ^ - ) , + ( u_1/2 ^ -,u_1/2^+ ) & = & f(u_1/2^+ ) .  from ( [ eq : godunov - flx_interf_jc]a ) , the residuals in cell no longer depend on dofs in cell .hence , the dofs in cells now become independent of dofs in cells .likewise , ( [ eq : godunov - flx_interf_jc]b ) implies that dofs in cells are independent of dofs in cells .this is a key result for the theoretical analysis below and follows from the choice of the godunov flux .the first result concerns existence of discrete shock profiles in cells in the supersonic region when using the godunov numerical flux .this result is local in the sense that it is based on the inverse function theorem in each cell and is valid in a neighborhood of the exact solution ( [ eq : exact_sol ] ) .the same analysis is then applied in the subsonic region . in the case of more general monotone numerical fluxes ,we also give conditions that allow an exponential decay of perturbations away from the shock position .[ th : sol_super_zone_conv_flx ] assume that the shock is strictly contained in cell .let , and assume that the discrete solution satisfies ( [ eq : sign_condition_conv - flx]a ) .then in each cell , the constant discrete function in defined by is the unique solution to the numerical scheme ( [ eq : steady_discrete_scheme ] ) over a sufficiently small neighborhood .we first note that , by consistency , trivially satisfies the numerical scheme ( [ eq : steady_discrete_scheme ] ) : , for all and . in other words, is a stationary solution in the supersonic region .let us prove that this is the unique solution in a sufficiently small neighborhood .since ( see figure [ fig : solution_pr_godunov_burgers]a ) , for each interface , there exist neighborhoods of the solution over where the numerical flux in ( [ eq : steady_discrete_scheme]a ) reduces to the upwind flux :   by assumption ( [ eq : sign_condition_conv - flx]a ) , ( [ eq : proof_sup1 ] ) holds also at interface .now let , define and let be the application whose components are the elementwise residuals in ( [ eq : proof_sup1 ] ) and ( [ eq : steady_discrete_scheme]c ) :   setting in , we have by consistency and it is sufficient to prove that the mapping is invertible over a neighborhood of .the partial derivatives of take values  _u_j^l g_j+1/2 ^ 0 & = & f(u_j+1/2 ^ - ) = f(u_l ) , 0lp , + _u_j^l g_j+1/2^k & = & _ _jf(u_h)_j^l d_x_j^k dx = f(u_l)n_k , l , 0<kp,0lp ,  at the point , where the coefficients are defined by ( [ eq : matn_kl ] ) . the application is continuously differentiable over and its jacobian has the following form at the point :   and reduces to because . applying the inverse function theorem , there exists a neighborhood of where the mapping is invertible .we conclude with   a similar result holds for the subsonic region that we formulate below and whose proof uses the same method as for theorem [ th : sol_super_zone_conv_flx ] .[ th : sol_sub_zone_conv_flx ] assume that the shock is strictly contained in cell .let , and assume that the discrete solution satisfies ( [ eq : sign_condition_conv - flx]b ) .then in each cell , the constant discrete function in defined by is the unique solution to the numerical scheme ( [ eq : steady_discrete_scheme ] ) over a sufficiently small neighborhood .  in the case of a general numerical flux ,the upwind property ( [ eq : godunov - flx_interf_jc ] ) is no longer valid with the consequence that the dofs in supersonic and subsonic regions are no longer uncoupled from other dofs and oscillations may appear in cells close to the shock position .however , we now show that , under some assumptions on the numerical flux , these oscillations decay exponentially fast from the shock . this trend will be illustrated in the numerical experiments of section [ sec : num_xp ] with the llf flux and supports the relevance of the choice of the godunov flux for the theoretical analysis .[ th : exp_decay ] suppose is a monotone ( [ eq : monotone_flux ] ) , lipschitz continuous numerical flux consistent with the physical flux and satisfies   assume that there exist and such that the discrete solution satisfies in cells with , and if or if .then , the amplitude of oscillations of the numerical solution around the exact solution decays exponentially fast as .we first consider supersonic cells and proceed by induction . let an interface and assume that the property holds true in cell : for all with . nowdefine the application whose components are the elementwise residuals in ( [ eq : steady_discrete_scheme]a ) , or equivalently ( [ eq : proof_sup1 ] ) , and ( [ eq : steady_discrete_scheme]c ) :   note that also vanishes at by consistency . by assumption of monotonicity of over , there exists a neighborhood of where is continuously differentiable and has partial derivatives  _u_j^l g_j+1/2 ^ 0 & = & _ u^-(u_j+1/2 ^ -,u_j+1/2^+ ) = _ u^-(u_l , u_l ) , 0lp , + _u_j+1^l g_j+1/2 ^ 0 & = & ( -1)^l_u^+(u_j+1/2 ^ -,u_j+1/2^+ ) = ( -1)^l_u^+(u_l , u_l ) , 0lp , + _u_j^l g_j+1/2^k & = & _ _jf(u_h)_j^ld_x_j^k dx = f(u_l)n_k , l , 0<kp,0lp , + _u_j+1^l g_j+1/2^k & = & 0 , 0<kp , 0lp ,  at , where the coefficients are defined by ( [ eq : matn_kl ] ) . in the same way as in proof of theorem [ th : sol_super_zone_conv_flx ], the jacobian becomes   from , monotonicity of and the first inequality in ( [ eq : hyp_exp_decay ] ) . applying the implicit function theorem, there exists a neighborhood about and a continuously differentiable application such that . setting , a taylor development of  reads   we thus obtain that   from ( [ eq : hyp_exp_decay ] ) , the left and right traces of the numerical solution have the same sign .likewise , a taylor development of the applications with about reads   which induces for .the last coefficient is defined through the relation ( [ eq : sign_jump - a ] ) and reads   from ( [ eq : hyp_exp_decay ] ) , we have and the last coefficient of the perturbation , , thus decay exponentially fast as , while other coefficients are always smaller than . therefore , the induction holds true in cell and the proof is complete since it is assumed to be true in cell .the proof is similar for subsonic cells . introducing the application defined by ([ eq : steady_discrete_scheme]b ) and ( [ eq : steady_discrete_scheme]c ) for and assuming small perturbations about in , the implicit function theorem and a taylor development about give for and   note that the assumption ( [ eq : hyp_exp_decay ] ) holds for the llf flux .indeed , we have and with . numerical experiments of section [ sec : num_xp_prof ] will illustrate the results of theorem [ th : exp_decay ] . in ,smyrlis used similar arguments to demonstrate the exponential decay of oscillations in the vicinity of a stationary shock for the lax - wendroff scheme observed in precedent numerical experiments . in this case , monotonicity is lost and oscillations change sign from one cell to another .the proof of theorem [ th : exp_decay ] shows that the oscillations are transmitted from a cell to its neighboring cell by the numerical fluxes and that only the highest dof is affected . in particular , setting one recovers the uniform solution up to first order : .this may motivate limiting techniques of the solution to suppress spurious oscillations and a first attempt in this direction will be shown in section [ sec : svm ] .   in the case of the godunov flux ,the mean value of the numerical solution in the shock cell is explicitly known .this is the object of the following lemma .[ th : shock_sol_dof0 ] let and and use the godunov flux ( [ eq : godunov_flux ] ) , then under the assumptions ( [ eq : sign_condition_conv - flx ] ) the first dof in cell is defined by where denotes the mean value ( [ eq : mean_sol ] ) of the exact solution ( [ eq : exact_sol ] ) in cell , and   is the relative shock position in and has values when the shock is at interfaces .setting into ( [ eq : discr_var_form ] ) and summing up the results over all cells in , one obtains according to ( [ eq : discrete_shock_at_infty ] ) . integrating ( [ eq : model_pb]a ) in space over , one obtains . subtracting this result to the above equation , one obtains from theorems [ th : sol_super_zone_conv_flx ] and [ th : sol_sub_zone_conv_flx ] , which gives from ( [ eq : discr_ci ] ) .the above results with the godunov flux are based on assumption ( [ eq : sign_condition_conv - flx ] ) on the traces of the numerical solution at interfaces of the shock cell . however , from ( [ eq : sol_godunov_flux ] ) we observe that this assumption may be violated when either ( i ) and , or ( ii ) and .our objective is here to show that , up to a shift in the index of the shock cell , , ( [ eq : godunov - flx_interf_jc ] ) still holds .this is a consequence of the following result : for oscillations in cell , situations ( i ) and ( ii ) reduce to  [ eq : sol_violating_hyp_traces ] u_-3/2^+ = u_-1/2 ^ -=u_l , & u_-1/2^+ = u_1/2 ^ -=u_r , + u_-1/2^+ = u_1/2 ^ - = u_l , & u_1/2^+ = u_3/2 ^ - = u_r ,  respectively . to prove these results , consider the local residuals ( [ eq : local_residuals ] ) at steady - state .setting and imposing the conservation property ( [ eq : steady_local_cons ] ) , one obtains   where and from the mean value theorem .the above equation has two solutions : either , or and .now , suppose for instance that case ( i ) holds so that . applying ( [ eq : steady_residual_nrj ] ) with , we obtain either the trivial solution , or with . in the latter case, we have by coercivity and strict convexity of ( see section [ sec : model_eqn ] ) .this latter result can not hold because of the definition of the godunov flux which imposes by ( [ eq : sol_godunov_flux ] ) , so .  then , every solutions such that correspond to situations where the oscillations are in cell with . indeed , by ( [ eq : godunov_flux ] )one would have and since , there exists a neighborhood of the uniform solution in cells where it is the unique solution from theorem [ th : sol_sub_zone_conv_flx ] .hence , by setting the cell containing oscillations according to theorems [ th : sol_super_zone_conv_flx ] and [ th : sol_sub_zone_conv_flx ] , only violates assumption ( [ eq : sign_condition_conv - flx ] ) . using again ( [ eq : steady_residual_nrj ] ), imposes .similar arguments hold for proving that ( [ eq : sol_violating_hyp_traces]b ) corresponds to the only situation of case ( ii ) violating ( [ eq : sign_condition_conv - flx ] ) .  to sum up , it is convenient to introduce an index of faces of the shock cell , . then , the situations violating ( [ eq : sign_condition_conv - flx ] ) reduce to one of the following cases   therefore , the godunov flux is not differentiable at only , : one solution satisfies ( [ eq : godunov - flx_interf_jc ] ) and the analysis of section [ sec : steady_shock_sol_conv ] remains valid , but the other one corresponds to oscillations in a neighboring cell . in the following , we will only consider solutions for which ( [ eq : godunov - flx_interf_jc ] ) holds without loss of generality . among these solutions , the situations violating ( [ eq : sign_condition_conv - flx ] ) will be analyzed in the linear stability analysis of section [ sec : ana_stab ] and numerical experiments of section [ sec : num_xp ] .we are now interested in steady - state solutions of the inviscid burgers equation , , ( [ eq : model_pb ] ) with . according to theorems [ th : sol_super_zone_conv_flx ] and [ th : sol_sub_zone_conv_flx ] ,the uniform numerical solutions in the supersonic region and in the subsonic region hold with the godunov flux .we thus restrict our analysis to the solution of the numerical scheme in the cell containing the shock .the result now depends on the order of the numerical scheme and we give the solutions for in the following theorem .[ th : shock_sol_p0123 ] consider the discrete dg scheme with the godunov flux ( [ eq : godunov_flux ] ) .let the exact position of the shock be in cell and assume that the internal traces satisfy ( [ eq : sign_condition_conv - flx ] ) .then , for , according to the polynomial degree , , and the relative position of the shock in the cell , , defined by ( [ eq : def_sc ] ) , the solution of the discrete scheme ( [ eq : steady_discrete_scheme ] ) reads : with and if : if :  [ eq : shock_sol_p2 ] -1<s_c<- , & & \ {  rcl u_1 & = & 0 , + u_2 & = & ,  .+ - < s_c < , & & \ {  rcl u_1 & = & - , + u_2 & = & -s_c ,  . + < s_c<1 , & & \ {  rcl u_1 & = & 0 , + u_2 & = & - ;  .if :  s_c_1 , & & \ {  rcl u_1 & = & - , + u_2 & = & -7s_c , + u_3 & = & -u_1 ,  .+ s_c_2 , & & \ {  rcl u_1 & = & ^1/2 , + u_2 & = & , + u_3 & = & ,  .[ eq : shock_sol_p3 ] s_c_3 , & & \ {  rcl u_1 & = & ^1/2 , + u_2 & = & , + u_3 & = & ,  .  where , , ; , , and  [ eq : prop_def_deltai ] _ 2 & = & 3_1 - 419s_c^3 + 279s_c , + _ 1 & = & ^.  see appendix [ app : proof_sol_p123 ] .  several conclusions may be inferred from theorem [ th : shock_sol_p0123 ] .first , for a polynomial degree , , a given dg scheme , the solutions are parametrized by the relative shock position in the cell only . in particular ,the numerical solution is independent of the mesh size and the amplitude of the oscillations are proportional to the shock strength .these oscillations occur for and vanish if and only if the shock is at an interface : .then , the coefficients are displayed in figure [ fig : evol_coeffs_p1 - 3](a - c ) and present continuous evolution with .the validity of the assumption ( [ eq : sign_condition_conv - flx ] ) is illustrated in figure [ fig : evol_coeffs_p1 - 3](d ) and is satisfied according to the ranges of solutions in theorem [ th : shock_sol_p0123 ] .we observe that for ( resp . ) , the positions ( resp . ) correspond to situations where or ( see section [ sec : comments_trace_ineq ] ) .it is remarkable that assumption ( [ eq : sign_condition_conv - flx ] ) selects the solutions over the whole range in  ] .results are given for explicit first- and third - order time integration schemes .we are here interested in the slow convergence and even non convergence observed from the numerical experiments in the preceding section and illustrated in figure [ fig : residuals_p2 ] . considering only and approximations thus appears sufficient for our purpose .the results for the godunov flux have already been determined from the stability analysis in section [ sec : ana_stab ] and will be used for comparison .  for the second - order approximation , ,every eigenvalue is contained in the unit disc . in the case of the godunov flux ,the pairs of complex conjugate eigenvalues correspond to modes located in cells and associated to uniform flow , while real eigenvalues correspond to modes in the shock cell as indicated in table [ tab : eigenvalues ] .the spectra obtained for other numerical fluxes look similar even if a scattering of eigenvalues in cells is observed for the llf flux . using a third - order time integration schemeis seen to lower the modulus of eigenvalues as expected .+  +  +  the third - order approximation , , with the forward - euler scheme exhibits unstable modes when the shock position tends to an interface . hereagain , the spectra for every numerical scheme look similar .the complex eigenvalues are concentrated in the right half - plane , while some real eigenvalues are negative and may become unstable . from the stability analysis for the godunov flux in table [ tab : eigenvalues ] , one may distinguish among the former modes between stable eigenvalues associated to the uniform flow and unstable eigenvalues associated to the shock .for other numerical fluxes , one recovers the eigenvalues in cells concentrated at the same locations and .likewise , modes associated to the shock are also very similar .the eigenvalues are concentrated at the unit circle and become unstable as the shock position tends to the interface .we stress that lowering the cfl value ( see eigenspectra for and ) shows that these modes remain unstable as predicted in table [ tab : eigenvalues ] for .figure [ fig : ana_stab_vep_p2 ] displays the structure of associated eigenvectors for two different shock positions corresponding to unstable modes . in section [ sec : ana_stab_god ] , it was shown that these modes only affect the highest dof and result in a quadratic evolution with support in the shock cell .the structure of the modes for the engquist - osher and llf fluxes is very similar and mainly concentrated in the shock cell .these modes are damped with the runge - kutta scheme , but are clustered around the unit circle and remain unstable for large values .these modes are expected to slow - down or prevent the convergence of some computations as observed in figure [ fig : residuals_p2]b and in the numerical experiments in the preceding section .the analytical results in theorem [ th : exp_decay ] and section [ sec : ana_stab_god_burgers ] and numerical experiments in table [ tab : exp_decay_llf ] suggest the possibility of removing oscillations in the supersonic and subsonic regions by damping the higest dof , say with . likewise , the structure of solutions in theorem [ th : shock_sol_p0123 ] and the stability analysis in section [ sec : ana_stab ] ( see , _e.g. _ , table [ tab : eigenvalues_sc_pm1 ] ) support the necessity of acting on a larger range of dofs , say where , whenever the solution becomes discontinuous in cell .these observations motivate the application of the spectral viscosity method to the dg discretization . in this method ,one supplements the explicit residuals ( [ eq : local_residuals ] ) with an artificial viscosity of the form where plays the role of a viscosity coefficient and denotes the spectral viscosity operator with if and otherwise . here, we use a slightly different implementation from . indeed , using ( [ eq : matn_kl ] )we have where the coefficients are defined by . using the matrix defined in ( [ eq : matm_kl ] ) , we get   we propose to apply this method in the context of our numerical experiments by selecting the range of modes where the spectral viscosity is applied according to the local smoothness of the solution : we set when the solution is irregular and otherwise .we apply the shock detection technique from to test the local smoothness of the solution .figure [ fig : svm_effect ] presents the convergence histories and solutions for the llf numerical flux and third and fourth approximation orders .results obtained with the above method and a viscosity coefficient are compared to that obtained without artificial dissipation .we observe the stabilization of the computations by the numerical viscosity .the amplitude of the oscillations are also lowered and are effectively damped in the uniform region where viscosity is applied only to the highest dofs . the technique allows the computations to converge to steady - state solutions and to lower oscillations while keeping accuracy in smooth regions of the flow .discrete shock profiles for a scalar conservation law with a convex flux discretized with a dg method have been analyzed . using the godunov numerical flux , we show existence of stationary profiles that are oscillating for polynomial degree into one discretization cell only. the oscillations may vanish when the shock is located at an interface of the mesh .a linear stability analysis of the shock profiles show however that these latter solutions may be unstable . considering the inviscid burgers equation ,these profiles are constructed analytically for and are shown to be parametrized by the shock strength and its relative position in the cell .the extension of this analysis to other numerical fluxes is also investigated .a theoretical analysis shows that oscillations propagate in neighboring cells but decay exponentially fast from the shock position for some class of monotone numerical fluxes .moreover , numerical experiments indicate that the shock profiles present strong similarities with the profiles obtained with the godunov flux which may be considered as a relevant model for the analysis of the dg method .finally , these results show that , when using a hierarchical functional basis , only the highest dofs are responsible , at first order , of linear instability and propagation of oscillations in neighboring cells . as an application of this property , the spectral vanishing viscosity method is successfully used to stabilize computations and damp oscillations through a selective action on dofs .one main contribution of this work is the analysis of oscillations and unstable character of the dg method for scalar conservation laws .these results may help to design specific stabilization techniques and future investigations will focus on these methods .the author would like to thank prof .alain lerat from arts et mtiers paristech and jean - luc akian from onera for valuable discussions and their constructive comments .   in this appendix, we prove the results of theorem [ th : shock_sol_p0123 ] which gives the solution of the numerical scheme in the cell where the shock is located as a function of the relative shock position , , given by ( [ eq : def_sc ] ) .we note that the solution for the first dof from ( [ eq : shock_sol_dof0 ] ) in lemma [ th : shock_sol_dof0 ] .we also give the following result which holds for the burgers equation and will be used to evaluate the local residuals associated to the equation for the second dof :   where we have used the orthogonality of the function basis ( [ eq : ortho_basis ] ) . according to the assumption of theorem [ th : shock_sol_p0123 ] ,the exact shock position is assumed to satisfy . from theorems[ th : sol_super_zone_conv_flx ] and [ th : sol_sub_zone_conv_flx ] , it follows that the trace at the left interface satisfies and a similar relation holds at the right interface . further assuming ( [ eq : sign_condition_conv - flx ] ) , the numerical fluxes at the left and right interfaces of cell read and .as a consequence , the equation for the first dof always reduces to the trivial relation .as suggested in theorem [ th : shock_sol_p0123 ] , we adopt the following notation for the dofs in the cell : , for all , where according to ( [ eq : shock_sol_dof0 ] ) . finally , we observe that according to ( [ eq : lr_traces ] ) , assumption ( [ eq : sign_condition_conv - flx ] ) may be rewritten as follows    using ( [ eq : intvol_p1 ] ) , the numerical scheme for the second dof reads whose solution reads . the condition ( [ eq : cns_sign_cond]a ) reads and restricts the solution over , while the condition ( [ eq : cns_sign_cond]b ) reads and restricts the solution over .the negative solution is the only one valid over which gives the result ( [ eq : shock_sol_p1 ] ) .the second equation has a trivial solution which gives .the validity of these solutions is imposed by ( [ eq : cns_sign_cond ] ) which reduces to because the left and right traces are identical .therefore , the solution is valid when and the solution is valid when which correspond to solutions ( [ eq : shock_sol_p2]a , c ) .the second solution of ( [ eq : sn_shock_p2]b ) reads and then ( [ eq : sn_shock_p2]a ) leads to which are valid only if .again , conditions ( [ eq : cns_sign_cond ] ) impose the negative solution .we first observe that for a solution to ( [ eq : sn_shock_p3 ] ) , then is also solution .therefore , we only look for solutions with and deduce the other ones by symmetry . then , we also note that the choice imposes through ( [ eq : sn_shock_p3]b ) and then ( [ eq : sn_shock_p3]a , c ) reduce to and which is possible only if and but is excluded by the strict inequalities in ( [ eq : cns_sign_cond ] ) . in the following , we thus consider . equation ( [ eq : sn_shock_p3]b ) induces and subtracting ([eq : sn_shock_p3]a ) and ([eq : sn_shock_p3]c ) , multiplying by , one obtains     taking symmetries into account , the system ( [ eq : sn_shock_p3 ] ) has at most height real solutions . the first root of equation ( [ eq : sol_shock_p3_u2 ] ) leads to the first solution ( [ eq : shock_sol_p3]a ) : if from which we again retain the solution .assumption ( [ eq : cns_sign_cond ] ) requires and which is satisfied if and only if . the three other roots of equation ( [ eq : sn_shock_p3 ] ) read   where , and has been defined in equation ( [ eq : prop_def_deltai]a ) . the last root ( [ eq : sol_shock_p3_u2b]c )is complex . from ( [ eq : prop_def_deltai ] ) , it follows that the first root ( [ eq : sol_shock_p3_u2b]a ) exists only when and .the former condition has only two real roots and is satisfied providing that defined by ^{\tfrac{1}{2}},\ ] ] while the latter condition has one root and requires the additional conditions or . after substituting the former solution ( [ eq : sol_shock_p3_u2b]a ) into ( [ eq : sol_shock_p3_u1 ] ) and ( [ eq : sol_shock_p3_u3 ] ) one obtains the solution ( [ eq : shock_sol_p3]c ) for and when . it may be checked that ( [ eq : cns_sign_cond]b ) is always satisfied over the range \cup[s_1,1) ] . the solutions and with opposite signs can not satisfy those conditions for any value .likewise , the second solution ( [ eq : sol_shock_p3_u2b]b ) holds when , that is when and leads to solution ( [ eq : shock_sol_p3]b ) .this solution satisfies ( [ eq : cns_sign_cond ] ) if .( eds . ) , _ adigma - a european initiative on the development of adaptive higher - order variational methods for aerospace applications _, notes on numerical fluid mechanics and multidisciplinary design , 113 ( 2010 ) , springer verlag .    , _ on a finite element method for solving the neutron transport equation , in mathematical aspects of finite elements in partial differential equations _, de boor ed ., academic press , new york , 1974 , pp .89123 .                ,_ error estimates for the third order explicit runge - kutta discontinuous galerkin method for linear hyperbolic equation in one - dimension with discontinuous initial data _ , numer .math . , 126 ( 2014 ) , pp. 703740 .

Abstract: we present an analysis of stationary discrete shock profiles for a discontinuous galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux . using the godunov method for the numerical flux , we characterize the steady state solutions for arbitrary approximation orders and show that they are oscillatory only in one mesh cell and are parametrized by the shock strength and its relative position in the cell . in the particular case of the inviscid burgers equation , we derive analytical solutions of the numerical scheme and predict their oscillations up to fourth - order of accuracy . moreover , a linear stability analysis shows that these profiles may become unstable at points where the godunov flux is not differentiable . theoretical and numerical investigations show that these results can be extended to other numerical fluxes . in particular , shock profiles are found to vanish exponentially fast from the shock position for some class of monotone numerical fluxes and the oscillatory and unstable characters of their solutions present strong similarities with that of the godunov method .   discontinuous galerkin method , discrete shock profile , scalar conservation laws , convex flux , inviscid burgers equation , linear stability , spectral viscosity  65n30 , 65n12


recent years have seen significant developments in the theory of classification , most notably binary classification , where strong theoretical results are available that quantify rates of convergence and shed light on qualitative aspects of the problem .extensions to multi - class classification have also been explored , and connections to the theory of regression are increasingly well understood , so that overall a satisfactory theory of supervised machine learning has begun to emerge .  in many real - world problems in which labels or responses are available , however , the problem is not merely to classify or predict a real - valued response , but rather to list a set of items in order .the theory of supervised learning can not be considered complete until it also provides a treatment of such _ ranking _ problems .for example , in information retrieval , the goal is to rank a set of documents in order of relevance to a user s search query ; in medicine , the object is often to rank drugs in order of probable curative outcomes for a given disease ; and in recommendation or advertising systems , the aim is to present a set of products in order of a customer s willingness to purchase or consume . in each example, the intention is to order a set of items in accordance with the preferences of an individual or population .while such problems are often converted to classification problems for simplicity ( e.g. , a document is classified as `` relevant '' or not ) , decision makers frequently require the ranks ( e.g. , a search engine must display documents in a particular order on the page ) . despite its ubiquity ,our statistical understanding of ranking falls short of our understanding of classification and regression .our aim here is to characterize the statistical behavior of computationally tractable inference procedures for ranking under natural data - generating mechanisms .we consider a general decision - theoretic formulation of the _ supervised ranking problem _ in which preference data are drawn i.i.d . from an unknown distribution , where each datum consists of a _ query _ , , and a _ preference judgment _, , over a set of candidate items that are available based on the query . the exact nature of the query and preference judgment depend on the ranking context . in the setting of information retrieval , for example, each datum corresponds to a user issuing a natural language query and expressing a preference by selecting or clicking on zero or more of the returned results .the statistical task is to discover a function that provides a query - specific ordering of items that best respects the observed preferences .this query - indexed setting is especially natural for tasks like information retrieval in which a different ranking of webpages is needed for each natural language query .following existing literature , we estimate a _ scoring function _ , where assigns a score to each of candidate items for the query , and the results are ranked according to their scores . throughout the paper ,we adopt a decision - theoretic perspective and assume that given a query - judgment pair , we evaluate the scoring function via a loss . the goal is to choose the minimizing the risk .\ ] ] while minimizing the risk ( [ eqnrisk ] ) directly is in general intractable , researchers in machine learning and information retrieval have developed surrogate loss functions that yield procedures for selecting . unfortunately , as we show , extant procedures fail to solve the ranking problem under reasonable data generating mechanisms . the goal in the remainder of the paper is to explain this failure and to propose a novel solution strategy based on preference aggregation .  let us begin to elucidate the shortcomings of current approaches to ranking .one main problem lies in their unrealistic assumptions about available data .the losses proposed and most commonly used for evaluation in the information retrieval literature have a common form , generally referred to as ( normalized ) discounted cumulative gain ( ( n)dcg ) .the ndcg family requires that the preference judgments associated with the datum be a vector of _ relevance scores _ for the entire set of items ; that is , denotes the real - valued relevance of item to the query . while having complete preference information makes it possible to design procedures that asymptotically minimize ndcg losses ( e.g. , ) , in practice such complete preferences are unrealistic : they are expensive to collect and difficult to trust . in biological applications , evaluating the effects of all drugs involved in a study or all doses on a single subject is infeasible . in web search ,users click on only one or two results : no feedback is available for most items .even when practical and ethical considerations do not preclude collecting complete preference information from participants in a study , a long line of psychological work highlights the inconsistency with which humans assign numerical values to multiple objects ( e.g. , ) .the inherent practical difficulties that arise in using losses based on relevance scores has led other researchers to propose loss functions that are suitable for _ partial preference data _ .such data arise naturally in a number of real - world situations ; for example , a patient s prognosis may improve or deteriorate after administration of treatment , competitions and sporting matches provide paired results , and shoppers at a store purchase one item but not others . moreover, the psychological literature shows that human beings are quite good at performing pairwise distinctions and forming relative judgments ( see , e.g. , and references therein ) .more formally , let denote the vector of predicted scores for each item associated with query .if a preference indicates that item is preferred to then the natural associated loss is the zero - one loss .minimizing such a loss is well known to be computationally intractable ; nonetheless , the classification literature has shown that it is possible to design convex fisher - consistent surrogate losses for the 01 loss in classification settings and has linked fisher consistency to consistency . by reduction to classification ,similar consistency results are possible in certain bipartite or binary ranking scenarios .one might therefore hope to make use of these surrogate losses in the ranking setting to obtain similar guarantees .unfortunately , however , this hope is not borne out ; as we illustrate in section [ secconsistency ] , it is generally computationally intractable to minimize any fisher - consistent loss for ranking , and even in favorable low - noise cases , convex surrogates that yield fisher consistency for binary classification fail to be fisher - consistent for ranking .  we find ourselves at an impasse : existing methods based on practical data - collection strategies do not yield a satisfactory theory , and those methods that do have theoretical justification are not practical . our approach tothis difficulty is to take a new approach to supervised ranking problems in which partial preference data are aggregated before being used for estimation .the point of departure for this approach is the notion of _ rank aggregation _( e.g. , ) , which has a long history in voting , social choice theory and statistics . in section [ secaggregation] , we discuss some of the ways in which partial preference data can be aggregated , and we propose a new family of -statistic - based loss functions that are computationally tractable . sections [ secconsistency ] and [ secu - statistics ] present a theoretical analysis of procedures based on these loss functions , establishing their consistency .we provide a further discussion of practical rank aggregation strategies in section [ secrank - aggregation ] and present experimental results in section [ secexperiments ] .section [ secconclusions ] contains our conclusions , with proofs deferred to appendices .we begin by considering several ways in which partial preference data arise in practice .we then turn to a formal treatment of our aggregation - based strategy for supervised ranking .[ itempaired - comparison ] _ paired comparison data_. data in which an individual judges one item to be preferred over another in the context of a query are common .competitions and sporting matches , where each pairwise comparison may be accompanied by a magnitude such as a difference of scores , naturally generate such data . in practice, a single individual will not provide feedback for all possible pairwise comparisons , and we do not assume transitivity among the observed preferences for an individual .thus , it is natural to model the pairwise preference judgment space as the set of weighted directed graphs on nodes .  _selection data_. a ubiquitous source of partial preference information is the selection behavior of a user presented with a small set of potentially ordered items .for example , in response to a search query , a web search engine presents an ordered list of webpages and records the url a user clicks on , and a store records inventory and tracks the items customers purchase .such selections provide partial information : that a user or customer prefers one item to others presented ._ partial orders_. an individual may also provide preference feedback in terms of a partial ordering over a set of candidates or items . in the context of elections , for example ,each preference judgment specifies a partial order over candidates such that candidate is preferred to candidate whenever .a partial order need not specify a preference between every pair of items .using these examples as motivation , we wish to develop a formal treatment of ranking based on aggregation . to provide intuition forthe framework presented in the remainder of this section , let us consider a simple aggregation strategy appropriate for the case of paired comparison data .let each relevance judgment be a weighted adjacency matrix where the entry expresses a preference for item over whenever this entry is nonzero . in this case ,a natural aggregation strategy is to average all observed adjacency matrices for a fixed query .specifically , for a set of adjacency matrices representing user preferences for a given query , we form the average . as , the average adjacency matrix captures the mean population preferences , and we thereby obtain complete preference information over the items .this averaging of partial preferences is one example of a general class of aggregation strategies that form the basis of our theoretical framework . to formalize this notion ,we modify the loss formulation slightly and hereafter assume that the loss function is a mapping , where is a problem - specific _ structure space_. we further assume the existence of a series of _ structure functions _, , that map sets of preference judgments into .the loss depends on the preference feedback for a given query only via the structure . in the example of the previous paragraph, is the set of adjacency matrices , and .a typical loss for this setting is the pairwise loss  where is a set of scores and is the average adjacency matrix with entries . in section [ secrank - aggregation ] , we provide other examples of structure functions for different data collection mechanisms and losses . hereafter , we abbreviate as whenever the input length is clear from context .  to meaningfully characterize the asymptotics of inference procedures, we make a mild assumption on the limiting behavior of the structure functions .[ assumptionlimit - structures ] fix a query .let the sequence be drawn i.i.d .conditional on , and define the random variables . if denotes the distribution of , there exists a limiting law such that   for example , the averaging structure function satisfies assumption [ assumptionlimit - structures ] so long as < \infty ] , and , for any fixed vector , is continuous in the topology of .  with our assumptions on the asymptotics of the structure function and the loss in place, we now describe the risk functions that guide our design of inference procedures .we begin with the pointwise conditional risk , which maps predicted scores and a measure on to  ] , and we remark that the standard ndcg criterion uses and .the `` precision at '' loss can also be written in the form ( [ eqnndcg ] ) , where ( assuming that ) and for and otherwise , which measures the relevance of the top items given by the vector .this form generalizes standard forms of precision , which assume .  to analyze the consistency of surrogate losses for the ndcg family ( [ eqnndcg ] ) , we first compute the loss and then state a corollary to proposition [ propositionall - consistency ] .observe that for any , since the function is increasing in its argument , minimizing corresponds to choosing any vector whose values obey the same order as the points . in particular , the range of is finite for any since it depends only on the permutation induced by , so we have corollary [ corollaryndcg - consistency ] .[ corollaryndcg - consistency ] define the set a surrogate loss is fisher - consistent for the ndcg family ( [ eqnndcg ] ) if and only if for all ,   corollary [ corollaryndcg - consistency ] recovers the main flavor of the consistency results in the papers of ravikumar et al . and buffoni et al . .the surrogate is consistent if and only if it preserves the order of the integrated terms : any sequence tending to the infimum of must satisfy for large enough .zhang presents several examples of such losses ; as a corollary to his theorem 5 ( also noted by ) , the loss is convex and structure - consistent ( in the sense of definition [ definitionstructure - consistent ] ) whenever is nonincreasing , differentiable and satisfies .the papers contain more examples and a deeper study of ndcg losses . to extend corollary [ corollaryndcg - consistency ] to a uniform result, we note that if for all and is compact , then is 0-coercive over the set , , so we may arbitrarily set a value for . ]whence theorem [ theoremstructure - consistency ] implies that structure consistency coincides with uniform consistency .another family of loss functions is based on a cascade model of user behavior  .these losses model dependency among items or results by assuming that a user scans an ordered list of results from top to bottom and selects the first satisfactory result . here, satisfaction is determined independently at each position .let denote the index of item that assigns to rank .the form of such expected reciprocal rank ( err ) losses is where  ] , and empirically correlates well with user satisfaction in ranking tasks  .  computing the expected conditional risk for general is difficult , butwe can compute it when is a product measure over .indeed , in this case , we have \prod _ { j=1}^{i-1 } \bigl(1 - \e_\limitlaw\bigl[g ( \structure_{\pi_\alpha^{-1}(j)})\bigr ] \bigr).\end{aligned}\ ] ] when one believes that the values represent the a priori relevance of the result , this independence assumption is not unreasonable , and indeed , in section [ secrank - aggregation ] we provide examples in which it holds .regardless , we see that depends only on the permutation , and we can compute the minimizers of the conditional risk for the err family ( [ eqnerr ] ) using the following lemma , with proof provided in the supplementary material .[ lemmaerr - minimizers ] let  ] for  ] , so that the limiting law .since is a product measure over ^\numresults ] is a dag for each query ( recall section [ seclow - noise - inconsistency ] ) , structure consistency is obtained if for each , for each pair of results . as an example , in this setting + \phi(\alpha_i - \alpha_j)\ ] ] is consistent when is nonincreasing , convex , and has derivative .the fisher - consistent loss ( [ eqnpairwise - consistent ] ) is similar to the inconsistent losses ( [ eqnsurr - edge - loss ] ) considered in section [ secconsistency - hard ] , but the coefficients adjoining each summand exhibit a key difference . while the inconsistent losses employ coefficients based solely on the average weight , the consistent loss coefficients are nonlinear functions of the edge weight differences : they are precisely the edge weights of the difference graph introduced section [ seclow - noise - inconsistency ] . since at least one of the two coefficients + ] is always zero , the loss ( [ eqnpairwise - consistent ] ) penalizes misordering either edge or .this contrasts with the inconsistent surrogates of section [ secconsistency - hard ] , which simultaneously associate nonzero convex losses with opposing edges and .note also that our argument for the consistency of the loss ( [ eqnpairwise - consistent ] ) does not require definition [ defrev - triangle ] s low - noise assumption : consistency holds under the weaker condition that , on average , a population s preferences are acyclic .in this section , we describe strategies for solving the convex programs that emerge from our aggregation approach to ranking and demonstrate the empirical utility of our proposed procedures .we begin with a broad description of implementation strategies and end with a presentation of specific experiments .   at first glance, the empirical risk ( [ eqnu - statistic - emprisk ] ) appears difficult to minimize , since the number of terms grows exponentially in the level of aggregation .fortunately , we may leverage techniques from the stochastic optimization literature to minimize the risk ( [ eqnu - statistic - emprisk ] ) in time linear in and independent of .let us consider minimizing a function of the form where is some collection of data , is convex in its first argument , and is a convex regularizing function ( possibly zero ) .duchi and singer , using ideas similar to those of nemirovski et al . , develop a specialized stochastic gradient descent method for minimizing composite objectives of the form ( [ eqnstochastic - objective ] ) .such methods maintain a parameter , which is assumed to live in convex subset of a hilbert space with inner product , and iteratively update as follows . at iteration ,an index  ] to .\label{eqnexperiment - limiting - score}\ ] ] in our setting , we may thus evaluate the asymptotic ndcg risk of a scoring function by computing the asymptotic scores ( [ eqnexperiment - limiting - score ] ) .in addition , corollary [ corollaryndcg - consistency ] shows that if all minimizers of a loss obey the ordering of the values then the loss is fisher - consistent . a well - known example ofsuch a loss is the least - squares loss , where the regression labels are : we compare the least - squares aggregation loss with a pairwise logistic loss natural for the pairwise data generated according to the btl model ( [ eqnlogistic - pairwise - sampling ] ) . specifically , given a data pair with , the logistic surrogate loss is which is equivalent or similar to previous losses used for pairwise data in the ranking literature . for completeness, we also compare with a fisher - consistent surrogate that requires access to complete preference information in the form of the asymptotic structure scores ( [ eqnexperiment - limiting - score ] ) .following ravikumar et al . , we obtain such a surrogate by granting the regression loss ( [ eqnregression - loss ] ) direct access to the asymptotic structure scores .note that such a construction would be infeasible in any true pairwise data setting .having described our sampling procedure , aggregation strategy , and loss functions , we now describe our model .we let denote the feature vector for the result from query , and we model the scoring function for a vector . for the regression loss ( [ eqnregression - loss ] ) , we minimize the -statistic - based empirical risk ( [ eqnu - statistic - emprisk ] ) over a variety of orders , while for the pairwise logistic loss ( [ eqnlogistic - loss ] ) , we minimize the empirical risk over all pairs sampled according to the btl model ( [ eqnlogistic - pairwise - sampling ] ) .we regularize our estimates by adding to the objective minimized , and we use the specialized stochastic method ( [ eqnfobos - update ] ) to minimize the empirical risk .estimated using the logistic pairwise loss ( [ eqnlogistic - loss ] ) and the -statistic empirical risk with chosen to be regression loss ( [ eqnregression - loss ] ) . the horizontal axis of each plot is the order of the aggregation in the -statistic ( [ eqnu - statistic - emprisk ] ) , the vertical axis is the ndcg risk , and each plot corresponds to a different number of samples . ; ; ; . ]estimated using the -statistic empirical risk ( [ eqnu - statistic - emprisk ] ) with chosen as the regression loss ( [ eqnregression - loss ] ) under various choices of the regularization parameter , . ]our goals in the experiments are to understand the behavior of the empirical risk minimizer as the order of the aggregating statistic is varied and to evaluate the extent to which aggregation improves the estimated scoring function .a secondary concern is to verify that the method is insensitive to the amount of regularization performed on .we run each experiment 50 times and report confidence intervals based on those 50 experiments .let denote the estimate of obtained from minimizing the empirical risk ( [ eqnu - statistic - emprisk ] ) with the regression loss ( [ eqnregression - loss ] ) on samples with aggregation order , let denote the estimate of obtained from minimizing the empirical pairwise logistic loss ( [ eqnlogistic - loss ] ) , and let denote the estimate of obtained from minimizing the empirical risk with surrogate loss ( [ eqnregression - loss ] ) using the asymptotic structure scores ( [ eqnexperiment - limiting - score ] ) directly . theneach plot of figure [ figk - orders ] displays the risk as a function of the aggregation order , using and as references .the four plots in the figure correspond to different numbers of data pairs .broadly , the four plots in figure [ figk - orders ] match our theoretical results .consistently across the plots , we see that for small , it appears there is not sufficient aggregation in the regression - loss - based empirical risk , and for such small the pairwise logistic loss is better. however , as the order of aggregation grows , the risk performance of improves .in addition , with larger sample sizes , the difference between the risk of and becomes more pronounced .the second salient feature of the plots is a moderate flattening of the risk and widening of the confidence interval for large values of .this seems consistent with the estimation error guarantees in the theoretical results in lemmas 7 and 10 in the appendices , where the order being large has an evidently detrimental effect .interestingly , however , large values of still yield significant improvements over .for very large , the improved performance of over is a consequence of sampling artifacts and the fact that we use a finite dimensional representation .[ by using sufficiently many dimensions , the estimator attains zero risk by matching the asymptotic scores ( [ eqnexperiment - limiting - score ] ) directly . ]figure [ figlambda - size ] displays the risk for pairs , , and multiple values of the regularization multiplier on .the results , which are consistent across many choices of , suggest that minimization of the aggregated empirical risk ( [ eqnu - statistic - emprisk ] ) is robust to the choice of regularization multiplier .in this paper , we demonstrated both the difficulty and the feasibility of designing consistent , practicable procedures for ranking . by giving necessary and sufficient conditions for the fisher consistency of ranking algorithms, we proved that many natural ranking procedures based on surrogate losses are inconsistent , even in low - noise settings . to address this inconsistencywhile accommodating the incomplete nature of typical ranking data , we proposed a new family of surrogate losses , based on -statistics , that aggregate disparate partial preferences .we showed how our losses can fruitfully leverage any well behaved rank aggregation procedure and demonstrated their empirical benefits over more standard surrogates in a series of ranking experiments .our work thus takes a step toward bringing the consistency literature for ranking in line with that for classification , and we anticipate several directions of further development .first , it would be interesting to formulate low - noise conditions under which faster rates of convergence are possible for ranking risk minimization ( see , e.g. , the work of , which focuses on the minimization of a single pairwise loss ) .additionally , it may be interesting to study structure functions that yield nonpoint distributions as the number of arguments grows to infinity .for example , would scaling the thurstone mosteller least - squares solutions ( [ eqnskew - symmetric - score ] ) by achieve asymptotic normality induce greater robustness in the empirical minimizer of the -statistic risk ( [ eqnu - statistic - emprisk ] ) ?finally , exploring tractable formulations of other supervised learning problems in which label data is naturally incomplete could be fruitful .we thank the anonymous reviewers and the associate editor for their helpful comments and valuable feedback .

Abstract: we consider the predictive problem of supervised ranking , where the task is to rank sets of candidate items returned in response to queries . although there exist statistical procedures that come with guarantees of consistency in this setting , these procedures require that individuals provide a complete ranking of all items , which is rarely feasible in practice . instead , individuals routinely provide partial preference information , such as pairwise comparisons of items , and more practical approaches to ranking have aimed at modeling this partial preference data directly . as we show , however , such an approach raises serious theoretical challenges . indeed , we demonstrate that many commonly used surrogate losses for pairwise comparison data do not yield consistency ; surprisingly , we show inconsistency even in low - noise settings . with these negative results as motivation , we present a new approach to supervised ranking based on aggregation of partial preferences , and we develop -statistic - based empirical risk minimization procedures . we present an asymptotic analysis of these new procedures , showing that they yield consistency results that parallel those available for classification . we complement our theoretical results with an experiment studying the new procedures in a large - scale web - ranking task .  ,