one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

conformal invariance and self avoiding walks in restricted geometries the prediction of conformal invariance for the statistics of self avoiding random walks restricted to both semi infinite and wedge shaped geometries are tested by extrapolating exact enumerations close agreement is found both for the angular distribution of the end to end vector and for the dependence of the critical exponent gamma 2 on the opening angle of the wedge

contractivity of wasserstein metrics and asymptotic profiles for scalar conservation laws abstract the aim of this paper is to analyze contractivity properties of wasserstein type metrics for one dimensional scalar conservation laws with nonnegative l and compactly supported initial data and its implications on the long time asymptotics the flux is assumed to be convex and without any growth condition at the zero state we propose a time parameterized family of functions as intermediate asymptotics and prove the solutions after a time depending scaling converge toward this family in the d wasserstein metric this asymptotic behavior relies on the aforementioned contraction property for conservation laws in the space of probability densities metrized with the d wasserstein distance finally we also give asymptotic profiles for initial data whose distributional derivative is a probability measure

regularity and large time behaviour of solutions of a conservation law without convexity a l aide de la methode des caracteristiques generalisees on etudie la regularite et le comportement a grand des solutions faibles admissibles d une seule loi de conservation a 1 variable d espace avec un point d inflexion

well posedness of convex maximization problems on stiefel manifolds and orthogonal tensor product approximations problems of best tensor product approximation of low orthogonal rank can be formulated as maximization problems on stiefel manifolds the functionals that appear are convex and weakly sequentially continuous it is shown that such problems are always well posed even in the case of non compact stiefel manifolds as a consequence problems of finding a best orthogonal strong orthogonal or complete orthogonal low rank tensor product approximation and problems of best tucker format approximation to any given tensor are always well posed even in spaces of infinite dimension the best rank one approximation is a special case of all of them in addition the well posedness of a canonical low rank approximation with bounded coefficients can be shown the proofs are non constructive and the problem of computation is not addressed here

effects of surface wedge corner and mixed boundary conditions on the local critical behaviour critical exponents for the o n spin model local field near a surface an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or comer is formed by mutually perpendicular d 1 dimensional walls each having von neumann special transition or dirichlet ordinary transitions boundary conditions the combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order e e 4 d the case of two or more von neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities

bv solutions for the cauchy problem of a quasilinear hyperbolic equation with σ finite borel measure and nonlinear source the aim of this paper is to discuss the cauchy problem of the quasilinear hyperbolic equation of the form ut um x up with nonnegative finite borel measures as initial conditions where m 1 0 p 1 are some given real numbers in particular the existence of bv solutions for the above problem is obtained

very singular solution and short time asymptotic behaviors of nonnegative singular solutions for heat equation with nonlinear convection abstract in this paper we study the following cauchy problem u t u x x u n x x t r 0 u x 0 0 x 0 where parameter n 0 its nonnegative solution is called singular solution when u x t satisfies the equation in the sense of distribution initial conditions in the classical sense and also u x t exhibits a singularity at the origin 0 0 as we know the singular solution is called source type solution if the initial is m x where x is dirac measure and constant m 0 the solution is called very singular solution if it possesses more singularity than that of source type solution at the origin here we focus on what happens in the interactive effect between the diffusion and convection in a whole physical process we find critical values n 2 n 1 n 0 such that there exists unique source type solution in the exponent range of 0 n n 0 while there exists no nonnegative singular solution if n n 0 only in the case of n 2 n n 1 there exists a very singular solution but in the case of n n 1 or n n 2 there is no solution that exhibits more singular than source type solution at the origin furthermore we describe the short time asymptotic behavior of the singular solutions when such cauchy problem is solvability for source type solution or very singular solution

a seminorm regularized alternating least squares algorithm for canonical tensor decomposition abstract the regularization method could deal with the swamp effect of alternating least squares als algorithms for tensor decomposition usually the regularization term is a norm of the difference between the solution and the current iterate in this paper we show that the norm could be weakened to a seminorm so the selection of the regularization term could be more flexible to overcome the swamp effect and avoid the drawback that the hessian of the subproblem may get close to singular in the iterative process we propose a seminorm regularized als algorithm for solving the canonical tensor decomposition moreover in the new algorithm we introduce a novel extrapolation in the update of each mode factor which makes an immediate impression on the update of subsequent ones by assuming the boundness of the infinite sequence of iterates generated by the new algorithm we establish the global convergence and the weakly linear convergence rate of the sequence of iterates numerical experiments on synthetic and real world problems illustrate that the new method is efficient and promising

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

critical behaviour at an edge the critical behaviour of a magnetic system with o n spin symmetry bounded by two d 1 dimensional hyperplanes meeting at an angle alpha is studied within mean field theory and in d 4 epsilon dimensions new exponents emerge for correlation functions and magnetisation and susceptibilities for spins close to the edge they can be expressed in terms of known bulk and surface exponents together with a single new edge exponent which depends however on the angle alpha this exponent is computed to first order in epsilon

existence and uniqueness of bv solutions for a conservation law withσ finite borel measures as initial conditions abstract the aim of this paper is to discuss the existence and uniqueness of the cauchy problem for a conservation law of the form u t u m x 0 m 1 with finite borel measures as initial conditions in particular we also obtain a non existence theorem which implies that our results are optimal

convergence rate analysis for the higher order power method in best rank one approximations of tensors a popular and classical method for finding the best rank one approximation of a real tensor is the higher order power method hopm it is known in the literature that the iterative sequence generated by hopm converges globally while the convergence rate can be superlinear linear or sublinear in this paper we examine the convergence rate of hopm in solving the best rank one approximation problem of real tensors we first show that the iterative sequence of hopm always converges globally and provide an explicit eventual sublinear convergence rate the sublinear convergence rate estimate is in terms of the dimension and the order of the underlying tensor space then we examine the concept of nondegenerate singular vector tuples and show that if the sequence of hopm converges to a nondegenerate singular vector tuple then the global convergence rate is r linear we show that for almost all tensors in the sense of lebesgue measure all the singular vector tuples are nondegenerate and so the hopm typically exhibits global r linear convergence rate moreover without any regularity assumption we establish that the sequence generated by hopm always converges globally and r linearly for orthogonally decomposable tensors with order at least 3 we achieved this by showing that each nonzero singular vector tuple of an orthogonally decomposable tensor with order at least 3 is nondegenerate

musings on multilinear fitting abstract we show that the problems of approximating tensors and multivariate functions as a sums of tensor products of vectors functions can be considered in a unified framework thus exposing their common multilinear structure we study the alternating least squares algorithm within this framework from the orthogonal projection and gradient perspectives we then use these perspectives to study its convergence behavior with and without regularization finally we formulate the infinite dimensional version of this problem and an algorithm to compute in that context

tensor decomposition for signal processing and machine learning tensors or multiway arrays are functions of three or more indices i j k ldots similar to matrices two way arrays which are functions of two indices r c for row column tensors have a rich history stretching over almost a century and touching upon numerous disciplines but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing statistics data mining and machine learning this overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors as such it focuses on fundamentals and motivation using various application examples aiming to strike an appropriate balance of breadth and depth that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and or developing tensor algorithms and software some background in applied optimization is useful but not strictly required the material covered includes tensor rank and rank decomposition basic tensor factorization models and their relationships and properties including fairly good coverage of identifiability broad coverage of algorithms ranging from alternating optimization to stochastic gradient statistical performance analysis and applications ranging from source separation to collaborative filtering mixture and topic modeling classification and multilinear subspace learning

behaviour of the velocity of one dimensional flows in porous media the one dimensional flow of gas of density u through a porous medium obeys the equation u um x where m 1 x e r and t 0 we prove that the local velocity of the gas given by v mu 2u1 not only is bounded for t t 0 but approaches an n wave profile as t oo n waves are the typical asymptotic profiles for some first order conservation laws a class of nonlinear hiperbolic equations the case m 0 0 2 u x 0 u0 x forx e r where m is a positive constant and u0 satisfies 0 3 u0 e l r u0 0 a e u0 0 apart from its mathematical interest as a simple model of nonlinear evolution equation of possibly degenerate or singular parabolic type equation 0 1 and its n dimensional analogue appear in a number of applications cf 18 for a reference in the typical application to the flow of gas through a porous medium we have m 1 and u stands for the density of the gas writing 0 1 as a conservation law for the mass

an oleinik type estimate for a convection diffusion equation and convergence to n waves in this article we propose an oleinik type estimate for sign changing solutions to a convection diffusion equation ut u x uxx u x 0 u0 x u x r 1 0 since the oleinik entropy inequality holds for nonnegative solutions or inviscid case 0 only the theoretical progress for the case was limited in this paper we show that its solution satisfies an oleinik type estimate t2 ux c 1 0 where c c u0 0 using this estimate the convergence to an n wave is proved for sign changing solutions and the theoretical gap in asymptotic convergence of the corresponding problem is filled

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to n waves abstract the goal of this article is to develop a new technique to obtain better asymptotic estimates for scalar conservation laws general convex flux f u 0 is considered with an assumption lim u 0 uf u f u 1 we show that under suitable conditions on the initial value its solution converges to an n wave in l1 norm with the optimal convergence order of o 1 t the technique we use in this article is to enclose the solution with two rarefaction waves we also show a uniform convergence order in the sense of graphs a numerical example of this phenomenon is included

asymptotics in the critical case for whitham type equations abstract consider the cauchy problem for nonlinear dissipative evolution equations u t n u u l u 0 x r t 0 u 0 x u 0 x x r where l is the linear pseudodifferential operator l u f x l u and the nonlinearity is a quadratic pseudodifferential operator n u u f x r a t y u t y u t y d y u f x u is direct fourier transformation let the initial data u 0 h 0 h 0 1 2 are sufficiently small and have a non zero total mass m u 0 x d x 0 here h n m l 2 x m i x n x l 2 is the weighted sobolev space then we prove that the main term of the large time asymptotics of solutions in the critical case is given by the self similar solution defined uniquely by the total mass m of the initial data

finding a low rank basis in a matrix subspace for a given matrix subspace how can we find a basis that consists of low rank matrices this is a generalization of the sparse vector problem it turns out that when the subspace is spanned by rank 1 matrices the matrices can be obtained by the tensor cp decomposition for the higher rank case the situation is not as straightforward in this work we present an algorithm based on a greedy process applicable to higher rank problems our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation and then computes a matrix with that rank using the method of alternating projections we provide local convergence results and compare our algorithm with several alternative approaches applications include data compression beyond the classical truncated svd computing accurate eigenvectors of a near multiple eigenvalue image separation and graph laplacian eigenproblems

critical behaviour at an edge for the saw and ising model the authors have studied the behaviour of two and three dimensional self avoiding walks confined to a wedge of wedge angle alpha series have been obtained and analysed for the angular dependent critical exponents characterising various edge susceptibilities in terms of a general scaling form for the edge free energy fe approximately t d 2 nu psi e ht y0 nu h1t y1 nu h2t y2 nu they find for the two dimensional case the following scaling indices y0 91 48 y1 3 8 y2 alpha 5 pi 8 alpha they argue that these results are exact from which follow all exponents for the bulk surface and edge problem in three dimensions the authors obtain y0 approximately 2 488 y1 0 65 or 0 02 y2 alpha alpha b pi alpha where a 0 51 or 0 04 b 0 847 or 0 017 which for y2 is precisely of the functional form given by mean field theory y2 alpha 1 2 pi 2 they argue that a 1 2 for all three dimensional o n models this simple angular dependence of y2 is different from that suggested by cardy s one loop epsilon expansion 1983 for the square lattice they have also studied the case in which the wedge is rotated through an angle of pi 4 and find that the various exponents are unchanged for the three dimensional ising model in a wedge analogy with the saw results plus mean field results in conjunction with rg and series work yield y0 approximately 2 485 y1 0 71 or 0 02 and y2 a b pi alpha with a 1 2 and b 0 79 or 0 02

percolation in an edge percolation in a three dimensional lattice bounded by two free plane surfaces meeting at an angle chi is studied within a real space renormalisation scheme the edge critical behaviour at various surface and bulk transitions is analysed the author does not find evidence for a first order edge transition at higher values of alpha as has recently been observed for an ising model at an edge dependence of the edge scaling power ye on the opening angle alpha has been confirmed

asymptotic estimates of solutions ofut 12δu u in r rd d 2 abstract we express the solution u of u t 1 2 u u in r r d d 2 u 0 through a functional of a d dimensional bessel process where is invariant by rotations and profiled this allows us to estimate the asymptotic behaviour of u t 1 and u t as t goes to infinity

on the regularity of the electronic schrodinger equation in hilbert spaces of mixed derivatives the electronic schrodinger equation describes the motion of electrons under coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry the present article is devoted to the regularity properties of the corresponding wavefunctions that are compatible with the pauli principle it is shown that these wavefunctions possess certain square integrable mixed weak derivatives of order up to n 1 with n the number of electrons across the singularities of the interaction potentials the result is of particular importance for the analysis of approximation methods that are based on the idea of sparse grids or hyperbolic cross spaces it indicates that such schemes could represent a promising alternative to current methods for the solution of the electronic schrodinger equation and that it may even be possible to reduce the computational complexity of an n electron problem to that of a one electron problem

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

existence and uniqueness for a quasilinear hyperbolic equation with σ finite borel measures as initial conditions abstract the aim of this paper is to discuss the cauchy problem for a quasilinear hyperbolic equation of the form u t u m x u p 0 m 1 p 0 with finite borel measures as initial conditions in particular the existence and uniqueness of global bv solutions with any nonnegative finite borel measures as initial conditions are obtained in the case m p m 1

n waves in elasticity we study the large time behavior of solutions of the one dimensional equations of elasticity typically a nonconvex system of conservation laws we show that solutions with compact initial support converge in l1 to a superposition of n waves at algebraic rate likewise we show that the perturbation expansion of weakly nonlinear geometric optics is uniformly valid to second order our analysis uses approximate scalar laws together with l1 stability of scalar solutions and decay in total variation of the wave speed

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

the hyperbolic cross space approximation of electronic wavefunctions the electronic schrodinger equation describes the motion of electrons under coulomb interaction forces in the field of clamped nuclei and forms the basis of quantum chemistry the present article continues the author s recent work numer math 98 731 759 2004 on the regularity of its solutions the electronic wavefunctions it was shown in the mentioned article that these wavefunctions possess square integrable high order mixed weak derivatives as they are needed to justify and underpin sparse grid and hyperbolic cross space approximation techniques theoretically how fast the norms of these derivatives can increase with the number of electrons is studied in the present article provided that all quantities are properly related to a characteristic lengthscale of the considered atomar or molecular system the norms of the derivatives remain bounded independent of the number of electrons and the number the positions and the charges of the nuclei

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

asymptotic behavior of solutions of a conservation law without convexity conditions in this paper we study the asymptotic behavior for the hyperbolic conservation law u i 0 t 0 oo x co 1 1 u x 0 uo x al x co 1 2 with riemann like data for x large the function f is a smooth nonlinear function of u in general eq 1 1 does not have a continuous solution for all time shock curves appear after finite time we will consider a piecewise continuous weak solution of 1 1 19 lo it is well known that across a discontinuity line x x t the solution satisfies the rankine hugoniot condifion r h and the entropy condition e 111

study of the hopf functional equation for turbulence duhamel principle and dynamical scaling we consider a formulation for the hopf functional differential equation which governs statistical solutions of the navier stokes equations by introducing an exponential operator with a functional derivative we recast the hopf equation as an integro differential functional equation by the duhamel principle on this basis we introduce a successive approximation to the hopf equation as an illustration we take the burgers equation and carry out the approximations to the leading order scale invariance of the statistical navier stokes equations in d dimensions is formulated and contrasted with that of the deterministic navier stokes equations for the statistical navier stokes equations critical scale invariance is achieved for the characteristic functional of the d th derivative of the vector potential in d dimensions the deterministic equations corresponding to this choice of the dependent variable acquire the linear fokker planck operator under dynamic scaling in three dimensions it is the vorticity gradient that behaves like a fundamental solution more precisely source type solution of deterministic navier stokes equations in the long time limit physical applications of these ideas include study of a self similar decaying profile of fluid flows moreover we reveal typical physical properties in the late stage evolution by combining statistical scale invariance and the source type solution this yields an asymptotic form of the hopf functional in the long time limit improving the well known hopf titt solution in particular we present analyses for the burgers equations to illustrate the main ideas and indicate a similar analysis for the navier stokes equations

sparse grid spaces for the numerical solution of the electronic schrodinger equation this article complements the author s recent work numer math 98 731 759 2004 on the regularity of the electronic schrodinger equation in hilbert spaces of mixed derivatives it has been shown there that the solutions of this equation are surprisingly smooth and possess square integrable mixed weak derivatives of order up to n 1 with n the number of electrons across the singularities of the interaction potentials and it has been claimed that this result can help to break the complexity barriers in computational quantum mechanics using correspondingly antisymmetrized sparse grid trial functions a construction of this kind that can be interpreted as a sparse grid sampling theorem is sketched here

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

ising model at an edge a position space renormalisation group approach the critical behaviour of a three dimensional ising model bounded by two plane surfaces meeting at an angle alpha is studied using the migdal kadanoff and two terminal cluster renormalisation methods the authors obtain expressions for the edge fixed point and for the independent edge exponent yhe at the various transitions they find that yhe is not completely universal but depends on the angle alpha and that it shows some unexpected features

randomized interpolative decomposition of separated representations we introduce an algorithm to compute tensor interpolative decomposition dubbed ctd id for the reduction of the separation rank of canonical tensor decompositions ctds tensor id selects for a user defined accuracy a near optimal subset of terms of a ctd to represent the remaining terms via a linear combination of the selected terms ctd id can be used as an alternative to or in combination with the alternating least squares als algorithm we present examples of its use within a convergent iteration to compute inverse operators in high dimensions we also briefly discuss the spectral norm as a computational alternative to the frobenius norm in estimating approximation errors of tensor id we reduce the problem of finding tensor ids to that of constructing interpolative decompositions of certain matrices these matrices are generated via randomized projection of the terms of the given tensor we provide cost estimates and several examples of the new approach to the reduction of separation rank

potential comparison and asymptotics in scalar conservation laws without convexity abstract two kinds of optimal convergence orders in l 1 norm to a self similar solution are proved or conjectured for various evolutionary problems so far the first convergence order is of the magnitude of the similarity solution itself and the second one is of order 1 t employing a potential comparison technique to scalar conservation laws we may easily see that these asymptotic convergence orders are related to space and time translation of potentials we present the technique clearly in the simple setting of scalar conservation laws in one space dimension

regularizing effects for ut aϑ u 0 in l1 abstract various initial boundary value problems and cauchy problems can be written in the form du dt a u 0 where r r is nondecreasing and a is the linear generator of strongly continuous nonexpansive semigroup e ta in an l1 space for example if a subject perhaps to suitable boundary conditions we obtain equations arising in flow in a porous medium or plasma physics depending on the choice of while if a x acting in l1 r we have a scalar conservation law in this paper we show that if m m 0 and m 2 m 2 where 1 1 then roughly speaking the norm of t du dt may be estimated in terms of the initial data u0 in l1 such estimates give information about the regularity of solutions asymptotic behaviour etc in applications side issues such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator a are also dealt with these considerations are of independent interest

dynamics in the fundamental solution of a non convex conservation law there is a huge jump in the theory of conservation laws if the convexity assumption is dropped in this paper we study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution three extra shock types are introduced other than the usual genuine shock which are left right and double sided contacts there are three kinds of phenomena of these shocks which are called branching merging and transforming all of these shocks and phenomena can be observed if the flux function has two inflection points a comprehensive picture of a global dynamics of a nonconvex flux is discussed in terms of characteristic maps and dynamical convex concave envelopes

optimized general sparse grid approximation spaces for operator equations this paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order based on the framework of tensor product biorthogonal wavelet bases and stable subspace splittings we construct operator adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces provided that certain additional regularity assumptions on the solution are fulfilled specifically for operators of positive order their dimension is o 2 j independent of the dimension n of the problem compared to o 2 jn for the full grid space also for operators of negative order the overall cost is significantly in favor of the new approximation spaces we give cost estimates for the case of continuous linear information we show these results in a constructive manner by proposing a galerkin method together with optimal preconditioning the theory covers elliptic boundary value problems as well as boundary integral equations