the persistence of a temporal signal is the probability that remains below ( or above ) a given level , for all times  ] has remained below the level .thus , the distribution of the maximum of is simply the derivative with respect to of .the distribution of the extrema has been analytically obtained for the brownian process , but its derivation remains a formidable task for general non - markovian gaussian processes . on this account ,the persistence problem is also related to extreme value statistics , which is a notoriously difficult problem for correlated variables , and which has recently attracted a lot of attention among physicists .hence , the persistence problem has obvious applications in many other applied and experimental sciences , where one has to deal with data analysis of complex statistical signals .for instance , statistical bounds of noisy signals are extremely useful for image processing ( for instance in medical imaging or astrophysics ) , in order to obtain cleaner images by correcting spurious bright or dark pixels . in general, it is important to be able to evaluate the maximum of a correlated temporal or spatial signal originating from experimental noise .the same question can arise when the signal lives in a more abstract space .for instance , in the context of genetic cartography , statistical methods to evaluate the maximum of a complex signal has been exploited to identify putative quantitative trait loci .finally , this same problem arises in econophysics or finance , where the probability for a generally strongly correlated financial signal to remain below or above a certain level is clearly of great interest .  in the present paper, we are interested in the persistence of a non - markovian gaussian process , which can be either stationary or scale - invariant .more specifically , this study focuses on the properties of the distributions of time intervals during which the considered process remains below or above a given level .we shall see that these distributions are simply related to the persistence itself , and contain valuable physical information .we now summarize the content of the present work , and mention briefly our main results . in section [ ii ] , we introduce our main quantities of interest , and among them : _ _ i.__the persistence and which measure the probability to remain above or below the level up to time ; _ _ ii.__the associated distributions of time intervals , , during which the process remains above or below the level . for a general level , we then shortly review the iia calculation introduced in , which leads to analytical approximate expressions for the laplace transform of , , and .in fact , we present a simpler formulation compared to , as well as new additional results , which permit a fast and efficient numerical implementation of the iia results . in section [ iii ] , we first introduce several examples of physically relevant smooth gaussian processes , on which the general results of this paper are numerically tested . within the iia ,we then present the calculation of the persistence exponents , defined , for stationary processes , by , when .we also obtain exact estimates for , for large , including exact bounds for and , and compute the exact asymptotic distributions . for large , takes the form of the wigner distribution , whereas becomes poissonian .all these results are correctly reproduced by the iia , except for the large asymptotics of .finally , in section [ secdist ] , we obtain the exact small time behavior of , , and , for `` very smooth '' processes ( a term to be defined in the next section [ ii ] ) , and find that the iia again reproduces these exact results . for marginally smooth processes , we also obtain exact results , but the iia is not exact in this case , although its results remain qualitatively correct and even quantitatively accurate. many of these results are also obtained by means of simple heuristic arguments , helping to understand them on physical grounds .this study is also illustrated by means of extensive numerical simulations , revealing a very satisfying accuracy of the iia for moderate , in a regime where , contrary to the limit , there are no available exact results for , , and .in this section , we introduce our main physical quantities of interest interval distributions , persistence , sign autocorrelation , constrained densities of crossings and relate their general properties .we then summarize the iia calculation introduced in , and obtain more explicit expressions for the interval distributions and the persistence probability , in the case of a gaussian process .these new results will prove useful in the next sections more specifically devoted to the interval distributions .one considers a stationary non - markovian gaussian process of zero mean and unit variance .its distribution at any time is then and we define its cumulative distribution as ,   due to its gaussian nature , such a stationary process is entirely characterized by its two - point correlation function , it is understood that the process starts from , so that the distribution of the initial condition at is given by eq .( [ gauss1 ] ) .in addition , all derivatives of , when they exists , are random gaussian variables of zero average and second moment ^ 2\right\rangle=(-1)^n f^{(2n)}(0),\ ] ] where the superscript refers to a derivative of order .  the process is assumed to be `` smooth '' , although this constraint will be sometimes relaxed in the present paper . by `` smooth '' ,we mean that the velocity of the process is a continuous function of time .`` very smooth '' processes will have a differentiable velocity .in particular , the most general stationary _ markovian _ gaussian process defined by the equation of motion , does not belong to this class of smooth processes ( is a gaussian -correlated white noise ) . in practice ,a process is smooth if its correlator is twice differentiable at , and is very smooth if the fourth derivative of exists at .the process introduced in eq .( [ markov ] ) has a correlator which has a cusp at , and is thus not even differentiable at .  throughout this paper, we will be interested in the probability that the process remains below or above a certain threshold , for all times in the interval  ] . more precisely , we define and as the persistence , knowing that the process started respectively above and below the level . in other word , is also the probability that the process remains above the level during the considered time interval , and is the probability that remains below the threshold . note that the persistence probes the entire history of the process , and is therefore a _infinite - points _ correlation function : for instance is the probability that the process remains below the level between the times and , _ and _ and , ... _ and _ and . for non - markovian processes , it is thus understandable that this quantity is extremely difficult to treat analytically , and there are very few examples where or can be actually computed . and are intimately related to the interval distributions and by the relations , indeed , if has remained below the level up to time , it belongs to a interval of duration , starting at an initial position uniformly distributed between and , an interpretation which leads to eq .( [ pnew2 ] ) .the above expressions can also be differentiated twice , giving if  , by symmetry of the process under the transformation , one has and . in general , we have the symmetrical relations , so that we will restrain ourselves to the case .the knowledge of and also provides valuable information on the distribution of the extrema of the considered process .the statistical properties of extremal events is an active field of research among mathematicians and physicists . by definition , is also the probability that the maximum of the process in a time interval of duration is less than , and is the probability that the minimum of the process remains bigger than for all times in  ] ( is heaviside s function : , if ; , if ; ) , \ , \theta [ x(0)-m]\rangle,\label{ap}\\ a_<(t)&=&\langle \theta [ m - x(t)]\ , \theta [ m - x(0)]\rangle,\nonumber\\    & = & 2g(m)-1+a_>(t ) , \label{am}\end{aligned}\ ] ] where the last relation is obtained by using .in addition , since the process is invariant under the transformation , one also has , for a gaussian process , these quantities can be explicitly expressed in terms of the correlation function  , for , these integrals can be explicitly performed , giving finally , the time derivative of and can be simply written as   we now introduce and , the mean number of -crossings in the time interval  ] , starting respectively from and .by definition of , one has since is on the same side of as if and only if the number of -crossings in the interval  ] ) by the time scale , which is the mean crossing time interval for ,   we have performed extensive numerical simulations of the processes associated to the correlators , , and , and measured the persistence and the crossing intervals distributions , and in particular , the persistence exponents .let us briefly describe how to generate long trajectories of a stationary gaussian process solely characterized by its two - times correlator . in real time , the most general form of such a process reads where is a -correlated gaussian white noise .the gaussian nature of and the linear form of eq .( [ kernel ] ) ensures that is a gaussian process .moreover , the convolution product of the noise with the kernel ( instead of a general kernel ) ensures stationarity . taking the fourier transform of eq .( [ kernel ] ) , we obtain where , and the noise fourier transform satisfies .the fourier transform of the correlator of is hence which relates the kernel to the correlation function , through their fourier transform , .note that a necessary condition for to be the correlator of a gaussian process is that its fourier transform remains positive for all real frequencies .finally , a trajectory of is obtained after sampling on a frequency mesh , and performing an inverse fast fourier transform of the obtained .the fourier transform of the correlators , , and having simple explicit expressions , this procedure for obtaining long trajectories is extremely efficient .otherwise , one has to tabulate the fourier transform of , before simulating eq .( [ fourier1 ] ) .  in practice ,our numerical results are obtained after averaging trajectories of length ( sometimes ) and with a frequency mesh of typically points spaced by . since for a general , , we obtain typically -crossings , a number which decays rapidly for large . despite the loss of statistics for large positive , we find that can still be measured with great accuracy , since and becomes purely poissonian in this limit ( see next section ) .on the other hand , the error bars for increase rapidly with , due to the occurrence of less -crossings .in addition , the determination of is also plagued by the fact that the exponential asymptotics of and only develops for increasingly large , as increases , which may produce uncontrolled systematic errors in the numerical estimates of ( see next section ) .table [ tab1 ] compares for the correlators ( for ) and , as obtained from numerical simulations and from the iia calculation of eq .( [ thm ] ) .the agreement between the theory and the simulations is excellent for both correlators , and is even improving as increases .in fact , we will show in the next section that the iia becomes exact for , when , and we will obtain an analytic asymptotic expression for .our theoretical and numerical results are also consistent with the numerical bounds computed in for the correlator , for and : it is also clear that the iia provides much better estimates of the persistence exponent than these bounds , which are however exact , although they require a much bigger numerical effort than the iia .table [ tab2 ] compares for the correlators and , as obtained from numerical simulations and from the iia calculation of eq .( [ thp ] ) .the agreement between the theory and the simulations is satisfying , although we will show that the iia ultimately fails in predicting the exact asymptotics of as , although the limiting form of will be given exactly by the iia , for not too large ..[tab1 ] we report the value of as obtained from the iia calculation ( ) and simulations ( ) , for different values of , calculated for the processes associated to the correlators ( ) and ( ) , introduced in eqs .( [ f1],[f2 ] ) . [ cols="^,^,^,^,^,^",options="header " , ]   for the process associated to the correlator ( ) , , compared to the exact value obtained in .the agreement between the iia and numerical estimates of greatly improves as increases , as observed for the two other correlators in table [ tab1 ] , whereas is only fairly reproduced for large ( as already observed in table [ tab2 ] ) , suggesting that the iia somewhat fails to reproduces in the limit , as will be confirmed in the next section .finally , we note that for the three processes considered here ( and all other smooth gaussian processes known to the author ) , one has , with a proportionality constant close to unity .   in the limit , and using and eqs .( [ taugauss],[t2 ] ) , the mean of the intervals are hence , as expected physically , the typical length of the intervals is becoming increasingly large as ( of order ) , whereas the typical length of the is going slowly to zero .   for large , and using eqs .( [ w],[v ] ) , we obtain that moreover , in this limit both functions can be approximated by developing up to second order in , the constraint that is consistently recovered up to leading order , after integrating eq .( [ wap ] ) . moreover , the explicit form of eq .( [ wap ] ) implies that the laplace transform of can be approximated by its value at , provided that finally , using eqs .( [ t2],[pm1 ] ) , we find that the laplace transform of is given by under the conditions of eq .( [ condvalm ] ) .hence , we conclude that for large , one has and , taking the inverse laplace transform of eq .( [ poisson ] ) , that the distribution of intervals is essentially poissonian except in a narrow region of time , corresponding to the conjugate time domain of the condition of eq .( [ condvalm ] ) .the behavior of and , for , will be obtained exactly in section [ secdist ] .the fact that the distribution of the long intervals is becoming poissonian can be physically interpreted . indeed , since when , the process can be considered to be markovian at this time scale .in addition , this also shows that for the intervals , the iia is in fact exact . in the next section , in the process of finding exact bounds for , we will prove the exact result , valid in the opposite limit , where , is the average of over all trajectories for which for all times , and .  on fig .[ fig1 ] , we plot obtained by simulating the process associated to the correlator of eq .( [ f2 ] ) .we also plot the iia result of eq .( [ thm ] ) , which is in perfect agreement with numerical simulations .in addition , we illustrate the rapid convergence of to the exact asymptotics of eq .( [ asthm ] ) . on fig .[ fig2 ] , we plot for , which has already perfectly converged to its asymptotic poissonian form of eq .( [ poissonfin ] ) .   for , and using the asymptotics eq .( [ largem ] ) and the iia expression for of eq .( [ pp1 ] ) , we obtain , in real time , eq . ( [ pps1 ] ) reads which is valid for . using eqs .( [ ap],[taumlarge ] ) , the relation can be explicitly written as when the process crosses the level for the first time at time , after the preceding crossing at time 0 , one has .( [ ppapprox ] ) states that for small times , the probability of having a interval of length is of the same order as the probability of having an -crossing at time , knowing that there is such a crossing at time , which follows a interval ( hence the factor ) . in other words , for short time , a crossing at time is very often the first crossing following the crossing at time .the form of found in eq .( [ wigner ] ) can be obtained from a simple physical argument , which will be our basis for obtaining exact results for the small behavior of in section [ secdist ] .indeed , let us consider a interval of length .for , one can expand the _ smooth _ process in power of up to second order , starting from , remembering that is a gaussian variable of second moment , the probability distribution of the velocity , at a -crossing following a interval , is given by latexmath:[\[\rho(v)=\frac{\left\langle |x'(0)|\delta(x(0)-m)\ , \delta(x'(0)-v)\right\rangle_{x'(0)>0 } } { \left\langle   and are uncorrelated ( as ) , these gaussian averages are easily performed , leading to the exact result in addition , the distribution of , conditioned to the fact that , is a gaussian of mean and mean square deviation , which is of order , and is independent of . hence , for large , one can replace by its average , and the interval length can be obtained by finding the first -crossing of the trajectory of eq .( [ trajec ] ) , hence , for small time , on has finally , using eq .( [ distv ] ) and the asymptotic expression for of eq .( [ taumlarge ] ) , we obtain the distribution of intervals given by eq .( [ wigner ] ) .thus we find that the distribution of intervals is given by the wigner distribution for , a result also obtained in .however , note that the ratio can be arbitrarily large in the limit , or . the probability distribution of eq .( [ wigner ] ) is correctly normalized to unity and has a mean equal to . of course, for , the actual distribution of interval should decay exponentially , with a rate . matching the two asymptotics at , we find that up to a so far unknown multiplicative constant , for large .in addition , the above argument shows that the total probability contained in the exponential tail of vanishes extremely rapidly as , and is of order , where is a constant of order unity .moreover , the result of eq .( [ wigner ] ) implies that the persistence is given by for , and decays exponentially for .  in the limit , the determination of seems to be beyond the iia .indeed , let us assume for simplicity that decays faster than any exponential , , with .anticipating that is large , we need to evaluate for large negative . in this limit ,( [ asv],[asw ] ) leads to if , and using the iia expression for of eq .( [ pp1 ] ) , we obtain which leads to which grossly underestimate the divergence of , when .if decays as a gaussian ( ) , eq .( [ pp1 ] ) leads to which , again , behaves linearly with . finally ,if , we find that up to a computable multiplicative constant .let us now present exact bounds for which will lead to an exact asymptotics for , fully consistent with eq .( [ asthetap ] ) .we discretize time , with , and define . by definition , where the gaussian `` action '' has the quadratic form and where the matrix is the inverse of the correlation matrix defined by its matrix elements making the change of variables  ] . for small time , after making the change of variable and taking the limit , while using the fact that for small , we obtain the final exact result , up to a dimensional constant depending on , is proportional to the mean inverse first - passage time of the process at the level . by simulating the process , we have obtained which leads to this result has also been checked numerically for the process associated to . in fact , the constant appearing in eq .( [ iianumt2 ] ) was obtained exactly by wong , based on the study of the process of .their result leads to   hence , we find that the iia result of eq .( [ iiaalpha1 ] ) is not exact for marginally smooth processes with , although it predicts correctly that is a constant independent of , leading to a reasonably accurate estimate of this constant . for general , eq .( [ iiaalpha ] ) is certainly correct dimensionally speaking , and probably fairly accurate , in practice .we end this section by an approximate calculation of for the process , which does not reproduce the exact result of , obtained by a much more complex method .we make the approximation , where the factor ensures that both processes have the same mean square displacement .then , the original first - passage problem for becomes a standard first - passage problem for the usual brownian motion at the level , for which the first - passage time probability distribution is given by , and for which , finally , within this simple approximation , we find that , overestimating the value obtained in eq .( [ iianumt ] ) .in this work , we have considered the -crossing interval distributions and the persistence of a smooth non - markovian gaussian process .we have obtained exact results for the persistence exponents in the limit of a large crossing level , including exact bounds for and . in this limit, we have shown that the distributions of and intervals become universal , and are respectively given by the wigner and poisson distributions . for any value of , we have obtained the exact small time behavior of the interval distributions and the persistence .we have also derived these results within the independent interval approximation .quite surprisingly , the iia reproduces all these exact results , except for the large asymptotics of .in addition , the iia fails in reproducing the exact small time behavior of the interval distributions for marginally smooth processes , although it remains qualitatively correct and even quite accurate in this case . to the credit of the iia , it is the only method to provide precise approximate expressions of the interval distributions and the persistence for all times , and all values of the level , and thus to grant access to the distribution of extrema of a non - markovian gaussian process .in addition , the iia can be straightforwardly applied to any _ smooth non - gaussian process _ , for which the autocorrelation function ( or ) and the conditional number of crossings ( or ) are known analytically , or extracted from experimental or numerical data . for gaussian processes , simple forms of the derivative of these quantitieshave been obtained , which permits a simple and fast numerical implementation of the iia results .i. f. blake and w. c. lindsey , ieee trans .theory * 19 * , 295 ( 1973 ) ; s. o. rice , _ noise and stochastic processes _ , p. 133, ed . by n. wax ( dover , ny , 1954 ) ;c. w. helstrom , ire trans .theory * it-3 * , 232 ( 1957 ) ; j. a. mcfadden , ire trans .theory * it-4 * , 14 ( 1957 ) .j. krug , h. kallabis , s. n. majumdar , s. j. cornell , a. j. bray , and c. sire , phys .e * 56 * , 2702 ( 1997 ) ; s. n. majumdar and a. j. bray , phys .* 86 * , 3700 ( 2001 ) ; s. n. majumdar and c. dasgupta , phys .e * 73 * , 011602 ( 2006 ) .a. comtet , p. leboeuf , and s. n. majumdar , phys .* 98 * , 070404 ( 2007 ) ; d .- s .lee , phys .* 95 * , 150601 ( 2005 ) ; c. j. bolech and a. rosso , phys .lett . * 93 * , 125701 ( 2004 ) ; d. s. dean and s. n. majumdar , phys .e * 64 * , 046121 ( 2001 ) ; s. n. majumdar and p. l. krapivsky , phys .e * 62 * , 7735 ( 2000 ) .

Abstract: we review the properties of time intervals between the crossings at a level of a smooth stationary gaussian temporal signal . the distribution of these intervals and the persistence are derived within the independent interval approximation ( iia ) . these results grant access to the distribution of extrema of a general gaussian process . exact results are obtained for the persistence exponents and the crossing interval distributions , in the limit of large . in addition , the small time behavior of the interval distributions and the persistence is calculated analytically , for any . the iia is found to reproduce most of these exact results and its accuracy is also illustrated by extensive numerical simulations applied to non - markovian gaussian processes appearing in various physical contexts .


extreme value theory deals with the smallest or the largest of a set of random variables and has found numerous applications in physics , engineering , biology and finance . if the s are independent and identically distributed ( i.i.d . ) according to a continuous distribution , the probability of the maximum is given by ^(k)_n(x)= [ kmax ] where the cumulative distribution and is the beta function . a classic result in extreme value theory of i.i.d .random variables states that for large ( and ) , the distribution is of the following scaling form : ^(k)_n(x ) ^(k ) ( ) [ 1max ] in the above expression , while the location factor and the scale factor depend on the details of the parent distribution , the scaling function is determined by the large behavior of . for , the distribution can be related to by a transformation and the maximum value distribution itself can be one of the following three types :  \1 .frchet distribution if decays as a power law  \2 .weibull distribution if is bounded above  \3 .gumbel distribution if is unbounded above and decays faster than a power law  much less is known about the extreme value statistics when the random variables are not independent .it is interesting to investigate how the correlations affect the limiting distributions of the three i.i.d .classes mentioned above .if the correlations are strong , one may expect the extreme value distribution to be different from the i.i.d .limit laws and universal distributions may not even exist . for weakly correlated variables , on the other hand , the i.i.d .results may still hold . in this article, we study the extreme value statistics of random variables with weak correlations and identify the relevant parameters depending on which the extreme value distribution is seen to be either one of the suitably rescaled i.i.d .limit distributions or the ( rescaled ) parent distribution or a completely different distribution unrelated to these .extreme value statistics of correlated random variables has been studied for stationary gaussian process with correlations decaying logarithmically or faster and specific physical examples such as random energy model with correlated random potential , noise , directed polymer on cayley tree , fluctuating interfaces and mass transport models . herewe study this problem in the context of biological evolution where the question of extreme values naturally arises as a large population evolves to maximise its fitness . during the first step in evolution ,an individual in an initially unfit population may acquire one or more mutations .if the total population is infinitely large , the subpopulation with mutations is dominated by the _fittest _ mutant in this subpopulation .as several experiments have shown that the fitnesses are not completely random , we are led to study the extreme value statistics of correlated fitnesses .  in order to examine the implications of correlations amongst fitnesses , several models such as model , block model and rough mt .fuji - type model have been proposed . in this article, we will work with the block model which has been employed in recent years to address questions pertaining to evolutionary dynamics .the extreme value statistics in the block model has been studied for both strongly and weakly correlated fitnesses for an exponentially decaying parent distribution  . howeverthe question of the universality of the extreme value distributions was not addressed . herewe extend the previous study by considering a broad class of parent distributions and focus on weakly correlated but identically distributed fitnesses .the rest of the article is organised as follows .the block model of correlated fitnesses is defined in sec .[ model ] .the extreme value distributions are obtained when a single mutation is present in the initial fitness in sec .[ d1 ] and in the presence of multiple mutations in sec .finally a discussion and summary is given in sec .many biomolecules such as proteins , antibodies and enzymes have natural domains or partitions and can be modeled as a sequence of length divided into several blocks . in the simplestsetting , a sequence represented by a binary string of zeros and ones is divided into blocks of equal length ( see fig . [ fmodel ] ) .a block configuration is assigned a random fitness value regardless of its position in the sequence .these block fitnesses are chosen _ independently _ from a common distribution which is nonzero on the interval  ] approaches zero for and unity for if is kept fixed .then a nontrivial maximum value distribution is obtained if remains finite and we obtain p_n_d^(1 ) ( w ) ^(1 ) ( ) ,  1 < d however for , new extreme value distributions are expected for the reasons mentioned above .\(iii ) we finally consider block fitness distributions for which the scale factor ratio typically tends to a finite limit as .for considered in sec .[ d1 ] , the scale factor ratio . using stirling s approximation for ,we may write   ,  d/0 + 0 ,  d/0 , d/  + ,  d/ , d/   where and .thus we can not express in terms of the known functions even for when .but some progress is possible for which we discuss next .as we have already mentioned , for the scale factor is independent of and thus the scale ratio is a constant for any .however it is possible to find for all for exponentially distributed fitnesses .although the maximum value distribution for this case has been studied in a previous work , here we present a simpler derivation . for large , using ( [ expok ] ) for in ( [ g2 ] ) , we obtain _n_d^(1)(w ) & & _ 0 ^ 2 w df_0 ( 1-e^-2 w+f_0 ) s_d e^-f_0 e^-s_d e^-f_0 _ d=1^d_u _ 0 ^ 2 w df_d e^-s_d e^-2 w + f_d s_d e^- f_d e^-s_d e^-f_d [ pnd ] + & = & _ s_d e^-2 w^s_d df_0 e^-f_0 ( 1- ) _ d=1^d_u _ s_de^-2 w^s_d df_d e^-f_d e^-s_d s_d e^-2 w / f_d where we have used ( [ gs0 ] ) for term . in the limits , the above integrals are nontrivial if is finite and we get _n_d^(1)(w ) & & _ w_0^ df_0 e^-f_0 ( 1- ) _ d=1^d_u _ 0^ df_de^-f_d e^-w_d / f_d + & = & _ d=1^d_u 2 k_1 ( 2 ) [ err ] where is the modified bessel function of second kind . one can check that the above equation reduces to the cumulative distribution for obtained in sec .[ d1 ] on using in ( [ d1exact ] ) .we now consider the above cumulative distribution when is zero and when it is finite in the large limit .case a. since the distribution moves towards larger as increases ( see fig . [ expodlarge ] ) , we may try as a scaling variable to reduce the product in ( [ mult ] ) to a single function .for on using for large , we see that which is finite for all . thus unlike for the algebraically decaying or bounded block fitness distributions , all the factors in ( [ err ] ) contribute to the cumulative distribution .case b. for , the ratio goes to zero for nonzero and unity for vanishing :   = _ n=1^d_u - d ,  r-2 r > 0 [ e11 ] + 1 ,  r-2 r = 0 [ e22 ]  thus although each term in ( [ mult ] ) can be calculated ( unlike for other block fitness distributions discussed above ) , most of the terms contribute to the distribution . as shown in fig .[ expodlarge ] , the distribution s are quite well separated for small but get clustered at larger which is consistent with the behavior of discussed above .  [ b][b][2.5] [ l][l][2.5]  ( bold line ) for exponential parent distribution when and obtained using ( [ pnd ] ) .the other curves show the individual factors in the product in ( [ pnd ] ) with ( left to right).,title="fig : " ]a set of random variables obtained by summing a fixed number of i.i.d .random variables are correlated when they have at least one common term in the sum .such nonindependent variables may describe diverse quantities such as breeding value of an animal , fitness of a protein or antibody and energy of a directed polymer . for two different linear combinations of i.i.d .random variables , we showed that as the sum is over a set of independently distributed variables , it is possible to write the extreme value distribution for weakly correlated random variables as an integral involving the maximum value distribution of i.i.d .random variables which makes the problem amenable to analysis at least in some cases .interestingly even with weak correlations , a rich variety of extreme value distributions result : they can be highly nonuniversal parent distributions or universal extreme value distributions for i.i.d .random variables or novel distributions unrelated to these .when a single mutation occurs in the initial sequence , the limiting extreme value distribution ( up to a rescaling ) is found to be one of the i.i.d .distributions when the initial distribution decays faster than an exponential and parent distribution if decays slower than an exponential but a novel distribution appears for exponentially decaying .such a classification has also been observed in the context of near - extreme value statistics of i.i.d .random variables .the situation is more complex when multiple mutations are introduced in the initial sequence . when the number of mutations does not scale with so that for large , our analysis showed that and may exhibit different distributions . while the algebraically decaying parent distributions remain robust in that the universal frchet distribution holds for , for bounded distributions the nonuniversal parent distribution for single mutation problem changed to universal weibull distribution for . for unbounded , non - exponential parent distributions decaying faster than a power law also ,the extreme value distribution for single mutation case fail to hold for . for all the three classes of block fitness distributions considered ,novel extreme value distributions are expected if the number of mutations in both blocks is comparable to the block length . for exponentially distributed block fitnesses, we analysed the maximum value distribution for all and gave explicit expressions for two new extreme value distributions ( see ( [ d1exact ] ) and ( [ err ] ) ). it would be very interesting to see if they occur in other extreme value problems as well .david carpentier and pierre le doussal . glass transition of a particle in a random potential , front selection in nonlinear renormalization group , and entropic phenomena in liouville and sinh - gordon models ., 63(2):026110 , 2001 .y. v. fyodorov and j .- p .bouchaud . freezing and extreme - value statistics in a random energy model with logarithmically correlated potential . 41:372001 , 2008 ; y.v .fyodorov , p. le doussal , and a. rosso .statistical mechanics of logarithmic rem : duality , freezing and extreme value statistics of noises generated by gaussian free fields ., page p10005 , 2009 .t. aita , h. uchiyama , t. inaoka , m. nakajima , t. kokubo , and y. husimi .analysis of a local fitness landscape with a model of the rough mt .fuji - type landscape : application to prolyl endopeptidase and thermolysin ., 54:6479 , 2000 .

Abstract: we study the limit distribution of the largest fitness for weakly correlated and identically distributed random fitnesses . a fitness variable is obtained by taking a linear combination of a fixed number of independent random variables drawn from the same parent distribution and two fitnesses are correlated if they have at least one common term in the respective sum . we find that for certain class of parent distributions , the extreme value distribution for correlated random variables can be related either to one of the known limit laws for independent variables or the parent distribution itself . for other cases , new limiting distributions appear . the conditions under which these results hold are identified . 0.5 cm   # 1 ( [ eq:#1 ] ) # 1#1


the channel capacity is a crucial information - theoretic perspective that determines the fundamental limit on achievable information rates over a communication channel .however , since the channel capacity alone gives only the knowledge of the maximum achievable rate , a stronger form of the channel coding theorem has been pursued to determine the behavior of the error probability as a function of the codeword length and information rate  .the _ reliability function _ or the _ error exponent _ of a communication system is defined by  where is the average block error probability for the optimal block code of length and rate .the error exponent describes a decaying rate in the error probability as a function of the codeword length , and hence serves to indicate how difficult it may be to achieve a certain level of reliability in communication at a rate below the channel capacity .although it is difficult to find the exact error exponent , its classical lower bound is available due to gallager .this lower bound is known as the _ random coding error exponent _ or _gallager s exponent _ in honor of his discovery , and has been used to estimate the codeword length required to achieve a prescribed error probability .the random coding exponent was extensively studied for single - input single - output ( siso ) and single - input multiple - output ( simo ) flat - fading channels with average or peak power constraint . for simo block - fading channels ,the random coding exponent was derived in with perfect channel - state information ( csi ) at the receiver , where it has been shown that although the capacity is independent of the channel coherence time ( first asserted in and also recently addressed in and for multiple - antenna communication ) , the error exponent suffers a considerable decrease due to a reduction in the effective codeword length as the coherence time increases .therefore , this so - called _ channel - incurable effect _ reduces the communication reliability . while there are numerous prior investigations ( following the seminal work of ) on the capacity for multiple - input multiple - output ( mimo ) channels , only limited results are available for error exponents .the random coding exponent were given implicitly in ( without final analytical expressions ) for independent and identically distributed ( i.i.d . )rayleigh - fading mimo channels with a single - symbol coherence time , perfect receive csi , and gaussian inputs subject to the average power constraint .also , the random coding exponent was analyzed in for i.i.d.block-fading mimo channels with no csi and isotropically unitary inputs subject to the average power constraint .  in this paper , taking into account spatial fading correlation , we derive gallager s exponent for mimo channels .we consider a block - fading channel with gaussian inputs subject to the average power constraint and perfect csi at the receiver .our results resort to the methodology developed in and , which is based on the finite random matrix theory .the mimo exponent obtained in the paper provides insight into a fundamental tradeoff between the communication reliability and information rate ( below the channel capacity ) , enabling to determine the required codeword length for a prescribed error probability .it is interesting to note that as a special case of this _ reliability rate _ tradeoff , one can obtain the diversity multiplexing tradeoff of mimo channels , which is a scaled version of the asymptotic reliability rate tradeoff at high signal - to - noise ratio ( snr ) .we quantify the effects of the number of antennas , the channel coherence time , and the amount of spatial fading correlation on the mimo exponent .moreover , the general formulae for the ergodic capacity and cutoff rate are deduced from the exponent expressions . in particular, our capacity formula embraces all the previously known results for i.i.d . , one - sided correlated , and doubly correlated channels .this paper is organized as follows . in section [ sec : sec:2 ] , signal and channel models are presented .section [ sec : sec:3 ] derives the expression for the mimo random coding exponent .section [ sec : proof ] gives proofs of the main results stated in theorem [ thm : rcee ] . in section [ sec : sec:4 ] , some numerical results are provided to illustrate the reliability rate tradeoff in block - fading mimo channels .finally , section [ sec : sec:5 ] concludes the paper ._ notation : _ throughout the paper , we shall use the following notation . , , and denote the natural numbers and the fields of real and complex numbers , respectively .the superscripts and stand for the transpose and transpose conjugate , respectively . is the identity matrix and denotes the matrix with the entry .the trace operator of a square matrix is denoted by and .the kronecker product of matrices is denoted by . by , we denote is positive definite . for a hermitian matrix , denotes the eigenvalues of in decreasing order and denote the vector of the ordered eigenvalues , whose element is .also , denotes the number of distinct eigenvalues of , and and , , denote the distinct eigenvalues of in decreasing order and its multiplicity , respectively , that is , and .finally , we shall use the notation to denote that a random matrix is ( matrix - variate ) gaussian distributed with the probability density function ( pdf ) where , , and .we consider a mimo system with transmit and receive antennas , where the channel remains constant for symbol periods and changes independently to a new value for each coherence time , i.e. , every symbols . since the propagation coefficients independently acquire new values for every coherence interval , the channel is memoryless when considering a block length of symbols as one channel use with input and output signals of dimension and , respectively . for an observation interval of symbol periods ,the received signal is a sequence , each is given by where are the transmitted signal matrices , are the channel matrices , and are the additive white gaussian noise ( awgn ) matrices .[ fig:1 ] shows a communication link with transmit and receive antennas to communicate at a rate ( in bits or nats per symbol ) over independent -symbol coherence intervals .since the channel is memoryless with identical channel statistics for each coherence time interval , the index can be dropped .let be the input probability assignment for with covariance subject to the average power constraint of the form where is the positive semidefinite matrix and is the total transmit power over transmit antennas .taking into account spatial fading correlation at both the transmitter and the receiver , we consider the channel matrix is given by  where and are the transmit and receive correlation matrices , respectively , and is a matrix with i.i.d ., zero - mean , unit - variance , complex gaussian entries .the entry , , , of is a complex propagation coefficient between the transmit antenna and the receive antenna with .note that  . with perfect csi at the receiver, we have the transition pdf which completely characterizes a block - fading mimo channel .  in what follows, we define the random matrix as  ^ , & + ^ , &  which is a matrix quadratic form in complex gaussian matrices , denoted by  , where , , and  ( _ 1^m m,_2^n n ) =  ( _ , _ ) , & + ( _ , _ ) , & .the pdf of is given by  where , , is the complex multivariate gamma function , is the euler gamma function , and is the hypergeometric function of two hermitian matrices , defined by ( * ? ? ? * eq .this section is based on gallager s random coding bound on the error probability of maximum - likelihood ( ml ) decoding for a channel with continuous inputs and outputs .notably , the bound determines the behavior of the error probability as a function of the rate and the codeword length .hence , by determining gallager s exponent , we can obtain significant insight into the reliability rate tradeoff in communication over mimo channels and the required codeword length to achieve a certain level of reliable communication .in particular , the diversity multiplexing tradeoff of mimo channels is a special case of the reliability rate tradeoff as the snr goes to infinity .using the formulation developed in ( * ? ? ?* ch . 7 ) , we obtain the random coding bound on the error probability of ml decoding over block - fading mimo channels as is an matrix of complex variables that do not depend functionally on each other , if is hermitian , then . ] where and ^ 2       { p_{{{\pmb{x}}}}\left({{\pmb{x}}}\right ) }       d{\pmb{x}}. { } \end{gathered}\ ] ] the random coding exponent in is given by with                       }                     p\left (                         \left .{ \pmb{y } } \right|                         { \pmb{x } } ,                         { \pmb{h } }                      \right)^{1/\left(1+\rho\right ) }                   d{\pmb{x } }               \right)^{1+\rho }             d{\pmb{y } }           d{\pmb{h } }       \right\}. \ ] ] the parameter to be optimized may be viewed as a lagrange multiplier corresponding to the input - power constraint .   as in , we choose the capacity - achieving distribution for satisfying the power constraint , namely , with . under the power constraint to maximize the error exponent ( i.e. , to minimize the upper bound ) is a difficult task . ]although this choice of the gaussian input distribution is optimal only if the rate approaches the channel capacity , it makes the problem analytically tractable .[ thm : eogi ]  let be in for the gaussian input distribution of .then , we have   see appendix [ sec : appendix : i ] .  for the case of equal power allocation to each of transmit antennas ,i.e. , ( because the transmitter has no channel knowledge ) , becomes where is the average snr at each receive antenna .let us introduce a new variable where is restricted to the range to have a meaningful result in .then , we have where with maximization over  ] to obtain the tightest bound , we have the random coding exponent for gaussian codebooks and equal power allocation as follows :   [ thm : ob ]  let be the value of that maximizes defined in for each  ] and  ] .these singularities stem from the quantity , which can be alleviated using the following analysis .suppose that is a positive integer .then , using ( * ? ? ?* lemma 1 ) , we have where is a complex gaussian matrix statistically independent of , and the last equality follows from the fact that and have the same nonzero eigenvalues .if , then . hence , using ( * ? ? ? * theorem 9 ) , for the case of can be written as  [ eq : eorf : int ] _ 0^(,,n _ ) & = _ ( ) \ { _ k=1^n _ \ { _ n + _k^ ( ) _ 2}^-1 } + & = k_^-1 _ ( ) _ k=1^n _ \ { _ n + _ k^ ( ) _ 2 } ^-1 + & (  _ ( m - n_)(_1 ) +  ) _ 1i , j n _ ( _ j^i-1 ( ) ) d ( )  where and the entry of , , is given by   now , applying ( * ? ? ?* lemma 2 ) to yields  [ eq : eorf : int:1 ] _ 0^(,,n _ ) & = k_^-1 (  _ ( m - n_)(_1 ) + ( , )  )  where the entry of the constituent matrix is given by using the _ characteristic coefficients _ ( * ? ? ? * definition 6 ) , can be written as where is the characteristic coefficient of . finally , substituting intogives the first case of and hence , we complete the proof of the first part .the second and third parts can be obtained by differentiating in theorem [ thm : rcee].1 with respect to and , respectively , with the help of the logarithmic derivative of a determinant ( * ? ? ?* theorem 9.4 ) ( or more generally ( * ? ? ?* lemma 1 ) ) .in this section , we provide some numerical results to illustrate the reliability rate tradeoff in block - fading mimo channels . for spatial fading correlation , we consider an exponential correlation model with and , , in all examples .  to ascertain the effect of the channel coherence on the error exponent , figs .[ fig:2 ] and [ fig:3 ] , respectively , show the random coding exponent as a function of a rate for i.i.d .( , ) and exponentially correlated ( , ) mimo channels at db , where and ranges from to .we can see from the figures that the exponent at a rate below the ergodic capacity decreases with , while the ergodic capacity remains constant for all ( i.e. , and nats / symbol for figs .[ fig:2 ] and [ fig:3 ] , respectively ) .for example , the error exponents at rates for are reduced by roughly and for i.i.d . andexponentially correlated cases , respectively , compared with those for .this reduction in the exponent , consequently , requires using a longer code to achieve the same error probability .hence , we see that unlike the capacity ( with perfect receive csi ) , the channel coherence time plays a fundamental role in the error exponent or the reliability of communications .[ fig:4 ] demonstrates the effect of spatial fading correlation on the random coding exponent , where , db , , , and ranges from ( i.i.d . ) to .as seen from the figure , there exists a remarkable reduction in the exponent at the same rate due to correlation , especially for .the amount of reduction in the exponent at rates , relative to the i.i.d .mimo exponent , ranges from for to for , indicating that a longer code is required to achieve the same level of reliable communications .equivalently , a decrease in the information rate is required for more correlated channels to achieve the same value of the exponent .for example , the exponent at a rate nats / symbol are and for the i.i.d . and correlated ( ) channels , respectively .this implies that increase in the codeword length , due to spatial fading correlation , is required to achieve a rate nats / symbol with the same communication reliability .  to get more insight into the influences of the number of antennas , channel coherence time , and fading correlation on a coding requirement for mimo channels , the codeword length required to achieve at a rate bits / symbol ( nats / symbol )are investigated in tables  [ table:1][table:3 ] .the codeword lengths in the tables are calculated in such a manner as described in section [ sec : cr ] .table [ table:1 ] serves to demonstrate the effect of increasing the number of antennas on the coding requirement , in which the required codeword length is shown for i.i.d .mimo channels with .note that it is impossible to reliably communicate at a rate bits / symbol below the snr of db , db , and db for , , and , respectively , since these snr s are required to attain the ergodic capacity of bits / symbol in each of the cases . as seen from the table , with increasing the number of antennas at both transmit and receive sides ,the required codeword lengths are remarkably reduced .this is due to the advantages of the use of multiple antennas , e.g. , spatial multiplexing and diversity gains .for example , at db , increasing the number of antennas at both sides from to and reduces the corresponding codeword length to almost and of the amount required for two transmit and receive antennas , respectively , which is a tremendous reduction in the codeword length.  table [ table:2 ] shows the required codeword length for i.i.d . and exponentially correlated ( , ) mimo channels with at db when varies from to .it is clear from table [ table:2 ] that for each value of , the codeword lengths for correlated channels are much longer than those for i.i.d .for example , the increase in the required codeword length , due to exponential correlation ( , ) , ranges from for to for , which is a significant increase in required codeword length . also , when going from to , there is a considerable increase in the required codeword length , relative to that for the single - symbol coherence time , which ranges from to for the i.i.d .case and from to for the correlated case , respectively .table [ table:3 ] demonstrates the effect of correlation on the required code length , where , , , and db . the table contains the corresponding codeword lengths for from to .as seen from the table , the required codeword length for the case of exponential correlation is equal to times as long as for the i.i.d .channel ( ) .particularly , when , there exists a large amount of increase in required codeword length due to a stronger correlation . also , since the ergodic capacity is bits / symbol for at db , it is impossible to achieve reliable communications at a rate bits / symbol ( regardless of the codeword length ) , when .  finally , fig .[ fig:5 ] shows the cutoff rate in nats / symbol as a function of a correlation coefficient for exponentially correlated mimo channels with at db , where and varies from to .we see that the cutoff rate decreases with for all .while remains constant , monotonically decreases with , going to as ( see and with and ) .hence , these two measures diverge as increases and eventually , which coincides with the divergent behavior of the capacity and cutoff rate of a channel with block memory .this observation reveals that is more pertinent than as a figure of merit that reflects the quality of block - fading channels .in this paper , we derived gallager s random coding error exponent to investigate a fundamental tradeoff between the communication reliability and information rate in spatially correlated mimo channels .we considered a block - fading channel with perfect receive csi and gaussian codebooks .the required codeword lengths for a prescribed error probability were calculated from the random coding bound to aid in the assessment of the coding requirement on such mimo channels , taking into account the effects of the number of antennas , the channel coherence time , and the amount of spatial fading correlation .in addition , we obtained the general formulae for the ergodic capacity and cutoff rate , which encompass all the previous capacity results as special cases of our expressions . in parallel to the capacity cutoff rate divergence in a block - memory channel , we observed the channel - incurable effect : the monotonically decreasing property of the mimo exponent ( i.e. , communication reliability ) with the channel coherence time .note that by writing the trace of the quadratic form in the exponent of as            \left [               { \pmb{s } }               -               \left({\pmb{i}}_m + { \pmb{\sigma}}{\pmb{a}}\right)^{-1 }               { \pmb{m } }           \right]^\dag           -           \left (               { \pmb{a}}^{-1 } + { \pmb{\sigma } }           \right)^{-1 }           { \pmb{mm}}^\dag       \right\},\end{aligned}\ ] ] we get              \left [                 { \pmb{s } }                 -                 \left({\pmb{i}}_m + { \pmb{\sigma}}{\pmb{a}}\right)^{-1 }                 { \pmb{m } }             \right]^\dag         \right\ }       d{\pmb{s } }     } _ {       =       \pi^{mn }       \det\left (             { \pmb{a } }             +             { \pmb{\sigma}}^{-1 }         \right)^{-n }       } \end{aligned}\ ] ] from which follows readily ._ proof of proposition [ thm : eogi ] : _ using lemma [ le : cgl ] , we have      }     p\left (         \left . { \pmb{y } } \right|         { \pmb{x } } ,         { \pmb{h } }      \right)^{1/\left(1+\rho\right ) }     d{\pmb{x } }   \nonumber \\   & \quad   =     e^ {       -r        { { n_{\text{c}}}}\mathcal{p }      }     \left(\pi n_0\right)^{-{{n_{\mathrm{r}}}}{{n_{\text{c}}}}/\left(1+\rho\right ) }     \det\left (           { \pmb{i}}_{{n_{\mathrm{t}}}}-           r           { \pmb{q } }       \right)^{-{{n_{\text{c } } } } }     \det\left (           { \pmb{i}}_{{n_{\mathrm{r}}}}+           \frac {               { \pmb{h } }               \left (                   { \pmb{q}}^{-1 }                   -                   r                   { \pmb{i}}_{{n_{\mathrm{t}}}}\right)^{-1 }               { \pmb{h}}^\dag              }              {               n_0               \left(1+\rho\right )              }       \right)^{-{{n_{\text{c } } } } }   \nonumber \\   & \hspace{1.5 cm }   \times     \operatorname{\mathrm{etr}}\left\ {           -\frac{1}{n_0 \left(1+\rho\right ) }            \left (               { \pmb{i}}_{{n_{\mathrm{r}}}}+               \frac {                   { \pmb{h } }                   \left (                       { \pmb{q}}^{-1 }                       -                       r                       { \pmb{i}}_{{n_{\mathrm{t}}}}\right)^{-1 }                   { \pmb{h}}^\dag                 }                 {                   n_0                   \left(1+\rho\right )                 }            \right)^{-1 }            { \pmb{yy}}^\dag        \right\}. { } \end{aligned}\ ] ] substituting into and integrating over , we have            }           p\left (               \left . { \pmb{y } } \right|               { \pmb{x } } ,               { \pmb{h } }            \right)^{1/\left(1+\rho\right ) }           d{\pmb{x } }     \right\}^{1+\rho }     d{\pmb{y } }   \nonumber \\   & \quad   =     e^ {       -r        { { n_{\text{c}}}}\mathcal{p }        \left(1+\rho\right )     }     \det\left (           { \pmb{i}}_{{n_{\mathrm{t}}}}-           r           { \pmb{q } }       \right)^{-{{n_{\text{c}}}}\left(1+\rho\right ) }     \det\left (           { \pmb{i}}_{{n_{\mathrm{r}}}}+           \frac {               { \pmb{h } }               \left (                   { \pmb{q}}^{-1 }                   -                   r                   { \pmb{i}}_{{n_{\mathrm{t}}}}\right)^{-1 }               { \pmb{h}}^\dag              }              {               n_0               \left(1+\rho\right )              }       \right)^{-{{n_{\text{c}}}}\rho}. { } \end{aligned}\ ] ] finally , substituting into yields the result .we provide a sketch of the proof of proposition [ thm : ob ] using a similar approach in and . for notational simplicity ,let us denote .then , in can be rewritten as where . since is concave in , is a concave function of if is concave in for all  ] .hence , the maximum over occurs at for which \bigr|_{\beta={\beta^\ast\left(\rho\right)}}=0 ] and \bigr|_{\beta={{n_{\mathrm{t } } } } } \leq 0 ] in order to prove . since it is clear that . also , \right|_{\beta={{n_{\mathrm{t } } } } }   =     -\frac{{m}\rho}{{{n_{\mathrm{t } } } } }     -\frac{1}{{{n_{\text{c } } } } }      { \mathcal{l}_1^{-1}\left({\rho},{{{n_{\mathrm{t}}}}},{{{n_{\text{c}}}}}\right ) }      \left .\left [         \frac {             \partial             { \mathcal{l}_1^{}\left({\rho},{\beta},{{{n_{\text{c}}}}}\right ) }            }            {             \partial             \beta            }      \right ]      \right|_{\beta={{n_{\mathrm{t}}}}}. { } \end{aligned}\ ] ] since and      \right|_{\beta={{n_{\mathrm{t } } } } } , \ ] ] it follows that    \right|_{\beta={{n_{\mathrm{t } } } } }    \leq 0 . { } \end{aligned}\ ] ] thus , \bigr|_{\beta={{n_{\mathrm{t } } } } } \leq 0 ] +  +  + +  +  + + where +         + where is the generalized hypergeometric function of scalar argument + ( * ? ? ?* eq . ( 9.14.1 ) ) and is the meijer g - function ( * ? ? ?* eq . ( 9.301 ) ) .the detailed derivation of this + integral identity can be found in ( * ? ? ?* appendix a ) .c|c|c|c [ 0cm][0cm] & + & & & + & - & - & + & - & & + & - & & + & - & & + & & & + & & & + & & & +  .required codeword length as a function of channel coherence time for i.i.d . andexponentially correlated mimo channels at a rate bits / symbol for , , and  [ cols="^,^,^ " , ]       + & + & + & + & + & + & + & + & + & + & - +       in nats / symbol as a function of a correlation coefficient for for exponentially correlated mimo channels with when , , , , , , , , , and . and db ., scaledwidth=70.0% ]

Abstract: in this paper , we derive gallager s random coding error exponent for multiple - input multiple - output ( mimo ) channels , assuming no channel - state information ( csi ) at the transmitter and perfect csi at the receiver . this measure gives insight into a fundamental tradeoff between the _ communication reliability _ and _ information rate _ of mimo channels , enabling to determine the required codeword length to achieve a prescribed error probability at a given rate below the channel capacity . we quantify the effects of the number of antennas , channel coherence time , and spatial fading correlation on the mimo exponent . in addition , general formulae for the ergodic capacity and the cutoff rate in the presence of spatial correlation are deduced from the exponent expressions . these formulae are applicable to arbitrary structures of transmit and receive correlation , encompassing all the previously known results as special cases of our expressions .   shin and win : gallager s exponent for mimo channels : a reliability rate tradeoff  block fading , channel capacity , cutoff rate , multiple - input multiple - output ( mimo ) system , random coding error exponent , spatial fading correlation .


many applications such as electronic structure calculations require an efficient method to solve discretized linear differential equations which are generators of long range interactions , such as poisson s equation . in modern electronic structure calculations , except those using planewaves , solving poisson s equation takes up a significant amount of computational time .it is essential to produce an algorithm that has the capacity to solve efficiently problems involving such long range interactions .the multigrid algorithm is the state of the art in solving discretized linear differential equations describing long range interactions .it uses a basic iterative method , such as the weighted jacobi method , over a sequence of scales to iterate faster toward the exact solution . in some cases, even multigrid does not produce sufficiently rapid convergence , particularly in calculations using wavelet bases .multiresolution analysis , from wavelet theory , provides the ability to carry out calculations on non - uniform grids , focusing resolution only in regions where it is needed .it is especially useful if the local region which requires high resolution moves through space during the calculation , as when the atoms move in a molecule in an electronic structure calculation .however , a basis set alone does not provide computational efficiency .often , a finite element basis set provides the analytic framework for multigrid calculations . combining the advantages of finite element analysis and the multigrid algorithm with wavelet theory ,could provide a very powerful method to solve discretized linear differential equations .this present research aims to unify these approaches for use in a wide range of applications , especially in electronic structure calculations , where all of the aforementioned approaches prove to be very beneficial in various stages of the calculations individually .a basis set that combines wavelet theory with finite element analysis exists and is known as the interpolet basis ( interpolets is based upon deslauriers and dubuc scaling functions [ 4 ] , which play the role of both scaling functions and interpolets in the interpolet basis . see also [ 5 ] ) .the present research forms a natural bridge in the synthesis of the multigrid algorithm with this basis , thereby adding the power of multiresolution analysis to the multigrid algorithm . in particular , we introduce this new _ hopgrid _ algorithm for use in problems where an underlying wavelet or interpolet basis is necessitated by the problem at hand and application of multigrid algorithm could provide very useful in solving linear differential equations in the problem .  in section [ mgrid ] of the paper ,to establish a common notation , we give a brief overview of the traditional multigrid algorithm and the iterative methods that form the framework for this algorithm .section [ interp ] introduces the interpolet theory .section [ mra ] includes the description of non - orthogonal multiresolution analysis from wavelet theory in the interpolet basis .section [ img ] presents the basic ideas behind expressing the multigrid algorithm in the interpolet basis , and describes how such a union becomes more efficient by using multiresolution analysis .section [ hgrid ] introduces the new algorithm formed by a more impact synthesis of the multigrid algorithm and the interpolet basis , the hopgrid algorithm .the last section expresses the results obtained using this new algorithm .the appendix of this paper aims to familiarize the reader with the essentials of the multigrid algorithm with a thorough discussion of the basic iterative methods and the framework of this algorithm .this section briefly summarizes the multigrid algorithm [ 2 ] and the interpolation and restriction operators used in this algorithm .a more detailed description of the multigrid algorithm and these operators , aimed for the physicist , may be found in section [ appenmg ] . throughout this section, superscripts are used to refer to the scale on which a particular level of the problem is solved .the multigrid algorithm is used to solve discretized linear differential equations of the form , where is the linear operator , is the solution vector , and is the source vector .multigrid is based upon a basic iterative method , which is then applied over a sequence of scales in order to improve the convergence rate .one such iterative method used for this purpose is the _ weighted jacobi method _ , which provides the following recursion , here , is an appropriate weight ( usually chosen to be ) , is the initial guess , is the diagonal matrix composed of the diagonal elements of , and is the _ residual _ at step , defined as .we refer to application of this recursion as _relaxation_.  multigrid exploits this basic iterative method in a multiscale fashion through the following procedure .first , relaxations are performed on the scale on which the problem is defined .then , the residual is transfered up to a spatially coarser scale , using a linear _ restriction _ operator , . on this coarser scale ,one then solves the _ error equation _ , relaxing again times .( note that , as discussed in the appendix , the exact solution to the error equation , when summed with the current iterative solution , yields the exact solution to the problem . ) the procedure of passing up the residual to successively coarser scales continues until a predetermined coarsest scale is reached . at this coarsest scale ,one then transfers the solution vector down to the next finer scale using a linear _ interpolation _operator .the error equation is then relaxed on this scale times , using as the initial guess .this procedure is followed down to the finest scale , relaxing times at each scale with the initial guess , where finally one relaxes times on the original linear equation to obtain an approximate solution to the overall problem .the procedure described above is the _ v - cycle of the multigrid algorithm_. v - cycles may be repeated in succession to produce a solution of any desired accuracy .the interpolation and restriction operators should obey two conditions , known as the _ variational properties_. the first condition , the _ galerkin condition _, requires that it is a recipe for the linear operator to be used in relaxations on the next coarser scale .the second condition states that where is a scalar constant .this is the recipe for the restriction operator , up to an overall scalar constant , given the interpolation operator .in this section , we briefly review interpolet theory based on the discussion in lippert , arias and edelman [ 1 ] .  in interpolet theory , functions varying slowly over integer length scales can be closely approximated as linear combinations of interpolating functions , where the are the expansion coefficients and the are functions with compact support , also known as _interpolets are a basis set combining wavelet theory with finite element analysis . in place of the orthonormality condition common in wavelet theory , interpolets have cardinality and interpolation from finite element analysis as their characteristic properties .we will first discuss the properties which interpolets share with finite elements and then we will describe those properties which they share with traditional wavelets .cardinality means that the values of a function are zero at all integers except for zero , as a consequence of this condition , the function formed by the linear expansion in eq .( [ eqn : fifteen ] ) will match exactly the value of the original continuous function at the integer grid points when the expansion coefficients are taken to be the values of at the integers , , interpolation is the further condition that eq .( [ eqn : fifteen ] ) reproduces any polynomial , up to order l for all , where subscript denotes the order of interpolation .given the values of a function at the integers , interpolets then provide a compact estimate correct to order of the values of the function at the half integers through the matrix representations for the interpolation operators defined by this relation for both first order interpolets ( ) and third order interpolets ( ) are given below .note that the case gives the familiar linear interpolation procedure . and that the case is distinct from the interpolation given by traditional cubic .interpolet theory thereby provides guidance in selecting natural interpolation operators for use in the multigrid algorithm .   in addition to properties from finite element analysis , interpolets satisfy the central condition of wavelet theory ,the _ two - scale relation_. this relation states that every coarse scale interpolet is expressible as a linear combination of finer scale interpolets .interpolets therefore interpolate themselves _ exactly _, here , the is the coarse scale interpolet , the are fine scale interpolets , and the are the expansion coefficients .( note that from the cardinality property , we have that the are just the values of the interpolet at the half - integers . ) the significance of this condition is to ensure that a basis made from interpolets of varying scales always provides a very uniform description of space .this point is described in detail the next section ..values of interpolets of orders at various points within their compact support .values not shown are zero . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ]   the hopgrid calculations require matrix elements of the linear operator in the mra represenation .these elements are where is the scale of basis function .these values may be computed from the previous by appropriate change of integration variable and application of the two scale relation .multiplying a vector by this mra matrix may be performed using one of two different methods .the first is straight forward multiplication by the matrix .because of the density of the mra matrix mentioned above , the number of flops required to apply the operator within the method grows faster than linearly with the size of the problem .the second approach uses the two - scale relation to reduce the number of operations so that the effort in the matrix multiplication scales only linearly with the size of the problem [ 1 ] .figure [ nonst ] compares the number of needed by the standard and the non - standard multiplications as a function of problem size .we use this superior approach in all of our comparisons .figure [ compare ] shows how the convergence of the hopgrid algorithm to the exact solution compares with the convergence of the traditional multigrid algorithm for a problem of 1024 grid points . in figure[ compare]a , the convergence to the exact solution is plotted against _ work units _( wu ) . using this comparisonthe hopgrid proceesure is superior to the traditional multigrid algorithm .when the convergence rates of these two algorithms are compared in terms of per mflops , as in figure [ compare]b , we find that despite the increased density of the mra matrix ( by about a factor of ten ) , the two methods give comparable results on a uniform grid .next we investigate the dependence of these convergence rates on problem size . in terms of per ,figure [ convrate]a shows that although the convergence rate of the traditional multigrid algorithm remains nearly constant , the convergence rate of the hopgrid algorithm _ increases _ with increasing problem size . figure[ convrate]b shows that even for the uniform case , the efficiency of the hopgrid algorithm overcomes the increased complexity of the mra matrix and results in nearly the same convergence rate as the traditional method in terms of a direct floating point operations comparison .it is in the case of non - uniform grids , where the application of the mra matrix requires far fewer operations , where we expect the maximum from the new algorithm ,in this research , our aim was to develop a new algorithm that benefits from the best combination of a variety of approaches , in particular the multigrid algorithm , multiresolution analysis from wavelet theory , and finite element analysis .the application for the new algorithm which we have in mind is the numerical solution of problems involving long - range interactions where an underlying interpolet basis with its multiresolution properties is needed .we have shown that the interpolet theory , based upon multiresolution analysis from wavelet theory and interpolation properties from finite element theory , provides a natural and unique choice of interpolation and restriction operators and an underlying grid for the multigrid algorithm .then we showed how the operations of restriction and interpolation may be carried out efficiently , without the application of linear operators , by exploiting the properties of an mra basis .finally , we introduced the hopgrid algorithm , which draws upon these results to produce a multigrid - like method to solve discretized linear equations in problems expressed in multiresolution wavelet bases .we have seen that the convergence rate of the new algorithm in terms of work units is superior to that of the traditional multigrid apprach and that , even in the worse - case scenario of a uniform problem , the new algorithm performs as well as the traditional approach on a direct floating point operation basis despite the increase complexity of mra matrices .in this section , we give a detailed discussion of the traditional multigrid algorithm .this introduction to multigrid is a review of the technique as given in _a multigrid tutorial _ by william briggs [ 2 ] .assume that the discretized differential equation we wish to solve may be represented in matrix form as  is the linear operator , is the solution vector , and is the source vector . given an approximation to the solution , which we will refer to as , the error can be expressed as .another measure of error is the _ residual _ , this definition , equivalent to eq .( [ eqn : one ] ) .henceforth , we shall refer to eq .( [ eqn : three ] ) as the _ error equation_.  decomposing the initial matrix as , where , , are the diagonal , lower triangular and upper triangular elements of respectively , eq . ( [ eqn : one ] ) becomes equation may be solved iteratively with the following recursion , recursive algorithm can be improved by choosing an appropriate weight , and taking a weighted average of the initial guess , , and the solution from eq .( [ eqn : five ] ) , {(n ) } + wd^{-1}f.\end{aligned}\]]solving eq .( [ eqn : two ] ) for and substituting into eq .( [ eqn : six ] ) gives a simplified form for this recursion , in terms of the residual alone ,  is the residual at step .this simplified recursion is the core of the basic iterative method known as the _ weighted jacobi method_. the application of this method for solving a problem is referred to as _relaxation_.  the weighted jacobi method and similar basic iterative methods rapidly eliminate high frequency errors while leaving the smooth components of the error mostly unaffected .this is called the _ smoothing property_. the reason for this behavior can be explained when we look at the eigenvalues of the convergence operator .the eigenvalues of this operator determine how fast the error converges to zero .[ wjaceval ] shows that the eigenvalues of the convergence operator for the weighted jacobi method lie in various ranges .the bands are chosen for illustration purposes and an explanation for these choices are given in the main body of the paper .the errors that are multiplied by the eigenvalues in the slow convergence band correspond to eigenvectors varying slowly in space . because the convergence operator does not eliminate efficiently most of these low frequency errors , this basic iterative method approach the exact solution very slowly .a more efficient method would eliminate all the error modes at equal rates .the multigrid algorithm has this feature and produces a solution that approaches the analytic solution with a higher convergence rate .the transition from the weighted jacobi method to the multigrid algorithm is made by improving upon the basic jacobi method .one way to improve the efficiency of weighted jacobi relaxations is to start with a better initial guess .conceptually , one can do this by solving the problem on a spatially coarser scale and using this solution as the initial guess for the original problem .this procedure is referred to as _ coarse grid correction _ ( cgc ) . on the coarse scale , there are fewer grid points , and on this grid the smooth error modes from the fine scale now appear higher frequency modes . replacing the original problem with a smaller one of mostly high frequency errorsallows the application of the weighted jacobi method to solve the problem more efficiently .this way , the convergence rate for the total error is accelerated since both the high and the low frequency error modes are eliminated on the fine and the coarse scales .the multigrid algorithm simply extends the idea of cgc over a sequence of scales . throughout this section, superscripts are used to refer to the scale on which that particular level of the problem is solved .  in order to find an approximation to the solution of the linear equation , we first apply the weighted jacobi method as in eq .[ eqn : seven ] times ( do relaxations ) on the finest scale , to obtain an approximate solution and a residual .the residual calculated on this scale is then transferred to the next coarser scale using a linear operator , known as the _ restriction operator_. in general , takes a vector from a fine scale , and transfers it to the next coarse scale .( this operator is discussed later in the appendix . ) for the restriction step , we assume that the relaxation on the finest scale has eliminated most of the high frequency error components , which lie in the null - space of restriction , so that little information is lost in the restriction process .the restricted residual is now we solve the residual equation  is the linear operator on the coarse scale , performing relaxations , using as an initial guess .this relaxation produces an approximation to the error on the fine scale .we could add this error to to obtain an approximate solution for the fine scale problem , since , as previously explained .however , the multigrid algorithm continues to go down to coarser scales and uses cgc to find solutions for the error equations at each coarser scale to improve the convergence rate . following this prescription , we compute the residual on scale and transfer it to the next coarser scale , using the restriction procedure described .repetition of such relaxation and restriction steps continues until the predetermined coarsest scale is reached . during each step leading up to the coarsest scale ,the generated approximate solution vectors , the , and the residuals , the , are stored to be used later in the algorithm .although the notation used for the result of the relaxations at each scale is , it is important to realize that except for the finest scale on which the original problem is defined , all the vectors are approximations to the error since the error equation is used during relaxations .  at the coarsest scale , another linear operator , an interpolation operator transfers the approximate solution vector down to the next finer scale . in general , the interpolation operator , takes a vector from a coarse scale , and transfers it to the next fine scale .( this operator is also discussed later . )the interpolated approximate solution vector , , is the approximation to the error for scale .the sum of the approximate solution and , give the approximate solution on scale .however , the interpolation process may introduce high frequency errors into this solution .we eliminate these errors by relaxing times on the residual equation using the initial guess and the residual stored from the previous restriction steps as the source vector .this relaxation produces the final which is then interpolated to the next finer scale .this procedure is followed down to the finest scale . on the finest scale ,we relax times on the original equation , eq .( [ eqn : one ] ) , using the initial guess and as the source vector .the solution obtained at this final step is the result of one iteration of the _ v - cycle of multigrid algorithm _ and gives an approximate solution to the problem .the interpolation and the restriction operators in the multigrid algorithm obey two constraints , known as the _variational properties_. the first of these constraints is the _ galerkin condition _ , which specifies the form of the linear operator on a coarser scale .assume that the error vector on scale is in the range of interpolation , , for some vector on scale .the residual equation at scale , , then reads both sides , we find  is precisely the galerkin condition for the linear operator on the next coarse scale . by solving eq .( [ eqn : twelve ] ) with as the linear operator and the restricted residual , we determine the error vector on scale which when interpolated becomes the error vector we seek , .the second variational property specifies the restriction operator upto a scalar constant , given the interpolation operator , , where is a scalar constant .this condition gives the restriction operator the full weighting property .the full weighting restriction involves taking some weighted average of values with neighboring points , instead of simply down sampling , a process known as _injection_.   w.l .briggs ; _ a multigrid tutorial _( lancaster press , lancaster , penssylvania , 1987 ) i. daubechies ; _ ten lectures on wavelets _( siam press , philadelphia , 1992 ) g. deslauriers , s. dubuc ; _ fractals , dimensions non entieres et applications _ , ( masson , paris , 1987 ) .d.l . donoho ; interpolating wavelet transforms , ( stanford dept of statistics technical report 408 , nov 1992 ) .et al _ ; _ wavelets and their applications _ , ( jones and bartlett , boston , 1992 ) c.k .chui ; _ an introduction to wavelets _( academic press , boston , 1992 ) c.k .chui , ed ; _ wavelets : a tutorial in theory and applications _( academic press , boston , 1992 ) s.f .mccormick ; _ multigrid methods _ ( siam press , philadelphia , 1987 ) r.d .cook ; _ concepts and applications of finite element analysis _( wiley , new york , 1992 )

Abstract: the multigrid algorithm is a multilevel approach to accelerate the numerical solution of discretized differential equations in physical problems involving long - range interactions . multiresolution analysis of wavelet theory provides an efficient representation of functions which exhibit localized bursts of short length - scale behavior . applications such as computing the electrostatic field in and around a molecule should benefit from both approaches . in this work , we demonstrate how a novel interpolating wavelet transform , which in itself is the synthesis of finite element analysis and wavelet theory , may be used as the mathematical bridge to connect the two approaches . the result is a specialized multigrid algorithm which may be applied to problems expressed in wavelet bases . with this approach , interpolation and restriction operators and grids for the multigrid algorithm are predetermined by an interpolating multiresolution analysis . we will present the new method and contrast its efficiency with standard wavelet and multigrid approaches .   = 10000


the lattice boltzmann method ( lbm ) is a discrete velocity method primarily used in the numerical simulation of complex fluid problems .the essence of the method is that a finite number of populations stream and collide on a fixed computational lattice in such a manner that their populations velocity moments obey the navier stokes equations .there are a number of ways to derive the method  .historically , the method comes from lattice gas automata theory but it can also be derived directly by the overrelaxation discretization of boltzmann s kinetic transport equation .we prefer to describe the lbm without direct reference to boltzmann s equation .indeed , we prefer to describe the lbm as being generated wholly by the mechanical motion , of a single - particle distribution function , by free - flight and entropic involution . in thissetting , the lbm is the discrete dynamical system which arises by discretizing this motion in a particular manner .the discrete velocity set arises as approximation nodes of certain cubature in velocity space  , where these nodes are automorphisms of some underlying lattice .a consequence of this realisation is that the stability analysis ( and analysis of conservation laws ) of the lbm is more natural .one is able to clearly identify the main instability mechanisms of the lbm ( sect .[ sec2 ] ) , which are triggered when the continuous mechanical motion of free - flight and entopic involution is discretized .  a further consequence of viewing the lbm in this setting ( also in sect . [ sec2] ) is that a prescription of coupled steps is suggested to stabilise the method . after proceeding for one step of the usual overrelaxation lbm scheme ( lbgk ) ,populations are streamed and then replaced with their local equilibrium distribution .additional dissipation results but the scheme retains the second - order in time accuracy of lbgk . compared with the standard lbm , the proposed scheme of coupled steps constitutes no additional computational cost .each cubature rule gives rise to its own lbm , and multispeed lattice schemes can be readily concocted ( sect .[ sec3b ] ) .multispeed lattices are an attractive prospect because they promise greater flexibility in the attainable sound speed provided by the model , allow for the modelling of nonisothermic flows and the dynamics of higher moments as well .the literature sometimes contains statements whose sentiment is that some multispeed lattice lbm schemes are inherently numerically unstable ( see , e.g. ,  ) . on the contrary ,we observe that the aforementioned lbm of coupled steps is stable for a variety of multispeed lattices and we demonstrate this through the numerical simulation of a one - dimensional isothermal shock tube ( sect . [ sec4 ] ) .to describe the lbm as being generated by free - flight and entropic involution requires a certain amount of background knowledge and terminology which we now briefly introduce . for proofs and further justification of any statements ,the reader should consult  .the free - flight equation is given by where is a single - particle distribution function , is the space vector , is velocity .this equation preserves the value of a strictly concave entropy functional , .the default choice of entropy is   in addition , we have a fixed linear mapping to some macroscopic variables . for example , in hydrodynamic applications is the vector of five hydrodynamic fields ( ,, ) ( density momentum energy ) :   for each , the _ quasiequilibrium state _ is defined as the unique solution of the constrained optimisation problem : . for the hydrodynamic fields , the quasiequilibrium state is the well known local maxwellian where is particle mass , is boltzmann s constant and is kinetic temperature .  for every , a corresponding quasiequilibrium state is defined .the set of all quasiequilibrium states is parameterised by the macroscopic variables , , and defines the _ quasiequilibrium manifold _ , which we denote by .the quasiequilibrium approximation for the free - flight equation  is the following evolution equation for the macroscopic variables for the hydrodynamics fields  the system  is the compressible euler equations .we denote by the projector of a point onto : .let be the time shift transformation for the free - flight equation : .now , for a fixed time step , the following step provides a second - order in time step approximation to the solution of the conservative macroscopic equations : if we would like to model dissipative dynamics then we should follow the free - flight trajectory by some extra time : where .now ,  provides a second - order in time step approximation to the solution of the compressible navier stokes equations with dynamic viscosity and prandtl number .equations  and  constitute one step of an approximation to some conservative and dissipative macroscopic equations , respectively . to iterate and produce a step wise approximation to the macroscopic equations for time greater than we can employ _ ( partial ) entropic involution_.  it will be useful to define the _ film of nonequilibrium states _  as the manifold that is the trajectory ( forward and backward in time ) of the quasiequilibrium manifold . a point is naturally parameterised by : , where is the value of the macroscopic variables , and is the time shift from a quasiequilibrium state : is a quasiequilibrium state for some ( other ) value of .the quasiequilibrium manifold divides into two parts , , where and .now , the _ partial entropic involution _operator is defined as follows : for the point , for , is specified by the two conditions : if in this definition we refer to the operator as _ entropic involution _ and denote it by .the point , , is closer to the quasiequilibrium point ( with respect to entropy ) than .  if , then , after some initial steps , the following sequence gives a second - order in time step approximation of the compressible navier stokes equations ( as mentioned above ) with , : for entropic involution ( ) ,  provides a second - order in time step approximation of the compressible euler equations with time step .finally , to generate the standard lbm we must perform three tasks :  1 .transfer to a finite number of velocities with the same macroscopic equations ; 2 .transfer from space to a lattice , where these velocities are automorphisms ; 3 .transfer from dynamics and involution on to the whole space of states .the first two points will be addressed in sect .[ sec3a ] and sect .[ sec3b ] . to address the final point we replace the operator with the transformation if , for a given , the sequence  gives the sought after second - order in time step approximation of the macroscopic equations , then the sequence also gives a second - order approximation to the same equations .there are a number of sources of instability which are triggered when the operator is replaced with  .indeed , we identify three main instability mechanisms .firstly , a small shift in in the direction of the vector does not relax back for , and its relaxation is very slow for ( i.e. , for small viscosity ) .this effect is most easily illustrated by fig .[ nstab ] , where a perturbed chain of entropic involutions is shown .this instability , which we refer to as _ neutral instability _, causes one step oscillations to be triggered .neutral stability and one - step oscillations in a sequence successive entropic involutions .bold dotted line a perturbed motion ,  direction of neutral stability.[fig1],width=264,height=132 ]  secondly , there is a _ nonlinear instability _ due to the nonlinear nature of the projector .more specifically , for all of the accuracy estimates in the previous section , we implicitly use the assumption that is sufficiently close to .the sequences  and  give the same accuracy as one another , but a long chain of steps with the linearised operator  can lead far from the quasiequilibrium manifold and even from itself .  finally , there is a _ directional instability _ that can affect accuracy .there can be large deviation in the angle the vector makes with the tangent space to .the directional instability changes the structure of dissipation terms in the target macroscopic equations : accuracy is decreased to the first - order in and significant fluctuations of the prandtl number and viscosity coefficient may occur .   there is a strikingly simple prescription that simultaneously alleviates the effect of all three instabilities identified in the previous section .consider starting from a point , then evolve the state by and apply the operator  , as is usual .then , evolve by again and project back on to using : this _ coupled step _ with quasiequilibrium ends is illustrated in fig .[ coupledstep ] and gives a second - order in time approximation , to the shift in time , for the target macroscopic equations with ,  ] and we discretize this interval with uniformly spaced lattice sites .we choose the initial density ratio as 1:2 so that for we set else we set .  in all of our simulationswe use a lattice with unit spacing , a unit time step and we have chosen to consider the three - velocity gauss - hermite model , as well as five- and seven - velocity newton - cotes models ( see fig .[ fig3 ] for the specific lattices details ) .time steps using lbgk ( solid line ) and the coupled steps  ( dashed line ) with the lattice a ) ; b ) ; c ) , width=321 ]  the standard three - velocity lbgk discretization  is violently oscillatory in the immediate neighbourhood of the shock , the effects of which extend over much of the domain . the situation is worse for lbgk with five- and seven - velocities ; here the oscillations do not remain bounded ( we always implement the positivity rule  that prohibits negative densities ) .the scheme quickly becomes meaningless . in comparison , we find that the proposed scheme of coupled steps  gives stable behaviour on all tested lattices , by which we mean no blow - up and spurious post - shock oscillations are observed . in all caseswe do observe a small deviation near the shock front which may be an unavoidable gibbs effect .the amplitude of the gibbs effect in our experiments decreases by increasing the degree of approximation .by considering a realisation in which the lbm is wholly generated by free - flight and entropic involution , we have suggested a simple stabilisation procedure of coupled steps  which retains the second - order accuracy of the method without artificial viscosity at that order .the procedure effectively constitutes no additional computational cost and can be implemented in any existing lbm code with just a few changes .we have demonstrated that the oscillatory pattern of the lbm in the vicinity of shocks is not due to a lack of artificial diffusivity and are not pertinent to proper lattice boltzmann schemes . on the other hand, there exists a gibbs effect that we can not fully suppress without artificial viscosity or special ad hoc approximation schemes .the role of gauss hermite quadratures in lbm constructions is now overestimated ( here we fully agree with  ) .other quadratures require slightly more points for the same degree of accuracy , but can be realised on integer nodes ( and , more generally , are flexible with regards to the choice of nodes ) , which is more convenient for lbm needs .the five - velocity lattice can produce a stable lbgk based method , as can many other suspicious " lattices which were previously discussed by many authors as intrinsically unstable .these results can be immediately generalised onto high dimensional lattices ( - lattices , cubatures and equilibria as products of 1d lattices , quadratures and equilibria ) .finally , with the advent of stable multispeed lattice formulations comes the prospect of stable nonisothermic lbm realisations .this work is supported by engineering and physical sciences research council ( epsrc ) grant number gr / s95572/01 .

Abstract: we demonstrate how to produce a stable multispeed lattice boltzmann method ( lbm ) for a wide range of velocity sets , many of which were previously thought to be intrinsically unstable . we use non - gauss hermitian cubatures . the method operates stably for almost zero viscosity , has second - order accuracy , suppresses typical spurious oscillation ( only a modest gibbs effect is present ) and introduces no artificial viscosity . there is almost no computational cost for this innovation .   * disclaimer : additional tests and wide discussion of this preprint show that the claimed property of coupled steps : no artificial dissipation and the second - order accuracy of the method are valid only on sufficiently fine grids . for coarse grids the higher - order terms destroy coupling of steps and additional dissipation appears . *  the equations are true .