on the condition numbers of large semidefinite toeplitz matrices abstract this paper is devoted to asymptotic estimates for the spectral or euclidean condition numbers t n a t n a t 1 n a of large n n toeplitz matrices t n a in the case where the symbol a is an l function and re a 0 almost everywhere we describe several classes of symbols a for which t n a increases like log n x n x or even e xn

numerical methods for porous medium equation by an energetic variational approach abstract we study numerical methods for porous media equation pme there are two important characteristics the finite speed propagation of the free boundary and the potential waiting time which make the problem difficult to handle based on different dissipative energy laws we develop two numerical schemes by an energetic variational approach firstly based on f log f as the total energy form of the dissipative law we obtain the trajectory equation and then construct a fully discrete scheme it is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy next based on 1 2 f as the total energy form of the dissipation law we construct a linear numerical scheme for the corresponding trajectory equation both schemes preserve the corresponding discrete dissipation law meanwhile under some smoothness assumption both schemes are second order convergent in space and first order convergent in time each scheme yields a good approximation for the solution and the free boundary no oscillation is observed for the numerical solution around the free boundary furthermore the waiting time problem could be naturally treated which has been a well known difficult issue for all the existing methods due to its linear nature the second scheme is more efficient

the spectral approximation of multiplication operators via asymptotic structured linear algebra abstract a multiplication operator on a hilbert space may be approximated with finite sections by choosing an orthonormal basis of the hilbert space multiplication operators with nonzero symbols defined on l 2 spaces of functions are never compact and then such approximations cannot converge in the norm topology instead we consider how well the spectra of the finite sections approximate the spectrum of the multiplication operator whose expression is simply given by the essential range of the symbol i e the multiplier we discuss the case of real orthogonal polynomial bases and the relations with the classical fourier basis whose choice leads to the well studied toeplitz case indeed the asymptotic approximation of the spectrum by the spectra of the associated toeplitz sections is possible only under precise geometric assumptions on the range of the symbol conversely the use of circulant approximations leads to constructive algorithms with o n log n complexity n number of sections working in general and generalizable to the separable multivariate and matrix valued cases as well

lagrangian numerical approximations to one dimensional convolution diffusion equations this work focuses on the numerical analysis of one dimensional nonlinear diffusion equations involving a convolution product first homogeneous friction equations are considered algorithms follow recent ideas on mass transportation methods and lead to simple schemes which can be proved to be stable to decrease entropy and to converge toward the unique solution of the continuous problem in particular for the first time homogeneous cooling states are displayed numerically further we present results on the more delicate fourth order thin film equation for which a nonnegativity preserving scheme is derived the dead core phenomenon is presented for the hele shaw cell

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

a note on singular values of cauchy toeplitz matrices abstract we consider the asymptotic behaviour of the smallest singular values of the n n sections of a general infinite cauchy toeplitz matrix

blow up dynamics for the aggregation equation with degenerate diffusion abstract we study radially symmetric finite time blow up dynamics for the aggregation equation with degenerate diffusion u t u m u k u in r d where the kernel k x is of power law form x depending on m d and the initial data the solution exhibits three kinds of blow up behavior self similar with no mass concentrated at the core imploding shock solution and near self similar blow up with a fixed amount of mass concentrated at the core computations are performed for different values of m d and using an arbitrary lagrangian eulerian method with adaptive mesh refinement

on the extreme eigenvalues of hermitian block toeplitz matrices abstract we are concerned with the behavior of the minimum maximum eigenvalue 0 n n n of an n 1 n 1 hermitian toeplitz matrix tn where is an integrable real valued function kac murdoch and szego widom parter and r h chan obtained that 0 n min o 1 n2k in the case where c2k at least locally and inf has a zero of order 2k we obtain the same result under the second hypothesis alone moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices proving that the rate of convergence of 0 n to inf depends only on the order not necessarily even or integer or finite of the zero of inf with the help of this tool we derive an absolute lower bound for the minimal eigenvalues of toeplitz matrices generated by nonnegative l1 functions and also an upper bound for the associated euclidean condition numbers finally these results are extended to the case of hermitian block toeplitz matrices with toeplitz blocks generated by a bivariate integrable function

solution of systems with toeplitz matrices generated by rational functions abstract we consider toeplitz matrices t n t i j n i j 0 where t j z j is a formal laurent series of a rational function r z a criterion is given for t n to be invertible in terms of the nonvanishing of a determinant d n involving the zeros of r z and of order and form independent of n i e n enters into d n as a parameter and not so as to complicate d n as n increases explicit formulas involving similar determinants are given for the solution of the system t n x y in the case where t n is invertible formulas are also given for t 1 n in the case where t n 1 and t n are both convertible suggestions concerning possible computational procedures based on the results are included

on lagrangian schemes for porous medium type generalized diffusion equations a discrete energetic variational approach abstract in this paper we present a systematic framework to derive a variational lagrangian scheme for porous medium type generalized diffusion equations by employing a discrete energetic variational approach such discrete energetic variational approaches are analogous to energetic variational approaches 39 25 in a semidiscrete level which provide a basis of deriving the semi discrete equations and can be applied to a large class of partial differential equations with energy dissipation laws and kinematic relations the numerical schemes derived by this framework can inherit the variational structure from the continuous energy dissipation law as an illustration we develop two variational lagrangian schemes for the multidimensional porous medium equations pme based on two different energy dissipation laws we focus on the numerical scheme based on the energy dissipation law with 1 2 u 2 d x as the dissipation functional several numerical experiments demonstrate the accuracy of this scheme as well as its ability in capturing the free boundary and estimating the waiting time for the pme in both 1d and 2d

a new interaction potential for swarming models abstract we consider a self propelled particle system which has been used to describe certain types of collective motion of animals such as fish schools and bird flocks interactions between particles are specified by means of a pairwise potential repulsive at short ranges and attractive at longer ranges the exponentially decaying morse potential is a typical choice and is known to reproduce certain types of collective motion observed in nature particularly aligned flocks and rotating mills we introduce a class of interaction potentials that we call quasi morse for which flock and rotating mills states are also observed numerically however in that case the corresponding macroscopic equations allow for explicit solutions in terms of special functions with coefficients that can be obtained numerically without solving the particle evolution we compare the obtained solutions with long time dynamics of the particle systems and find a close agreement for several types of flock and mill solutions

a discontinuous galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials abstract we consider a class of time dependent second order partial differential equations governed by a decaying entropy the solution usually corresponds to a density distribution hence positivity non negativity is expected this class of problems covers important cases such as fokker planck type equations and aggregation models which have been studied intensively in the past decades in this paper we design a high order discontinuous galerkin method for such problems if the interaction potential is not involved or the interaction is defined by a smooth kernel our semi discrete scheme admits an entropy inequality on the discrete level furthermore by applying the positivity preserving limiter our fully discretized scheme produces non negative solutions for all cases under a time step constraint our method also applies to two dimensional problems on cartesian meshes numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics

identification of asymptotic decay to self similarity for one dimensional filtration equations the objective of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one dimensional filtration equations it is based on an alternative formulation of the problem using the pseudoinverse of the density s repartition function in particular the numerical approximations can be proven to satisfy a contraction property for a wasserstein metric various numerical results illustrate the ability of this numerical process to capture the time asymptotic decay towards self similar solutions even for fast diffusion equations

a mass transportation approach to a one dimensional fluid mechanics model with nonlocal velocity abstract we consider a one dimensional transport model with nonlocal velocity given by the hilbert transform and develop a global well posedness theory of probability measure solutions both the viscous and non viscous cases are analyzed both in original and in self similar variables we express the corresponding equations as gradient flows with respect to a free energy functional including a singular logarithmic interaction potential existence uniqueness self similar asymptotic behavior and inviscid limit of solutions are obtained in the space p 2 r of probability measures with finite second moments without any smallness condition our results are based on the abstract gradient flow theory developed by ambrosio et al 2005 2 an important byproduct of our results is that there is a unique up to invariance and translations global in time self similar solution with initial data in p 2 r which was already obtained by deslippe et al 2004 17 and biler et al 2010 6 by different methods moreover this self similar solution attracts all the dynamics in self similar variables the crucial monotonicity property of the transport between measures in one dimension allows to show that the singular logarithmic potential energy is displacement convex we also extend the results to gradient flow equations with negative power law locally integrable interaction potentials

on the limiting set of singular values of toeplitz matrices abstract this paper is devoted to the description of the limiting set of teoplitz matrices generated by functions being locally normal over qc

point dynamics in a singular limit of the keller segel model 2 formation of the concentration regions this paper continues the analysis started in the first part of this article cf j j l velazquez siam j appl math 64 2004 pp 1198 1223 it was seen there using the method of matched asymptotics that a regularized version of the keller segel system admits for a suitable asymptotic limit solutions with some regions of high concentrations for the cell density this paper considers the relation between the phenomenon of blow up for the limit problem and the dynamics of the concentration regions described in j j l velazquez siam j appl math 64 2004 pp 1198 1223 in particular this paper analyzes the precise way in which the regularization introduced in the keller segel system stops the aggregation process and yields the formation of concentration regions

preconditioned hss methods for the solution of non hermitian positive definite linear systems and applications to the discrete convection diffusion equation we study the role of preconditioning strategies recently developed for coercive problems in connection with a two step iterative method based on the hermitian skew hermitian splitting hss of the coefficient matrix proposed by bai golub and ng for the solution of nonsymmetric linear systems whose real part is coercive as a model problem we consider finite differences fd matrix sequences an a p n discretizing the elliptic convection diffusion problem with being a plurirectangle of rd with a x being a uniformly positive function and p x denoting the reynolds function here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes more precisely in connection with preconditioned hss gmres like methods we consider the preconditioning sequence pn a n pn a dn1 2 a an 1 0 dn1 2 a where dn a is the suitably scaled main diagonal of an a 0 if a x is positive and regular enough then the preconditioned sequence shows a strong clustering at unity so that the sequence pn a n turns out to be a superlinear preconditioning sequence for an a 0 n where an a 0 represents a good approximation of re an a p namely the real part of an a p the computational interest is due to the fact that the preconditioned hss method has a convergence behavior depending on the spectral properties of pn 1 a re an a p n pn 1 a an a 0 n therefore the solution of a linear system with coefficient matrix an a p is reduced to computations involving diagonals and to the use of fast poisson solvers for an 1 0 n some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

index calculus for approximation methods and singular value decomposition abstract we introduce two classes of sequences a n of approximation operators moore penrose and fredholm sequences which correspond in a sense to normally solvable and fredholm operators respectively for moore penrose sequences we derive a certain qualitative behavior of the singular values of the approximation operators a n and we describe this behavior quantitatively for fredholm sequences by defining the number and the index of an approximation method

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

self similar blowup solutions to an aggregation equation in r n we present numerical simulations of radially symmetric finite time blowup for the aggregation equation ut u k u where the kernel k x x the dynamics of the blowup exhibits self similar behavior in which zero mass concentrates at the core at the blowup time computations are performed in r n for n between 2 and 10 using a method based on characteristics in all cases studied the self similarity exhibits second kind anomalous scaling

dimensionality of local minimizers of the interaction energy in this work we consider local minimizers in the topology of transport distances of the interaction energy associated with a repulsive attractive potential we show how the dimensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin

singular values of cauchy toeplitz matrices abstract the behavior of singular values of matrices a n 1 i j g n i j 1 with n is investigated for any real g which is not integer it is proved that the singular values are clustered at sin g which is their upper boundary the only o n singular values are those which lie outside a given e neighborhood of the clustering point o n n 0 as n o n o ln 2 n holds if g 1 2 also proved is that the minimum singular values of a n g tend to zero provided that g 1 2

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

discretization of functionals involving the monge ampere operator gradient flows in the wasserstein space have become a powerful tool in the analysis of diffusion equations following the seminal work of jordan kinderlehrer and otto jko the numerical applications of this formulation have been limited by the difficulty to compute the wasserstein distance in dimension geqslant 2 one step of the jko scheme is equivalent to a variational problem on the space of convex functions which involves the monge ampere operator convexity constraints are notably difficult to handle numerically but in our setting the internal energy plays the role of a barrier for these constraints this enables us to introduce a consistent discretization which inherits convexity properties of the continuous variational problem we show the effectiveness of our approach on nonlinear diffusion and crowd motion models

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

global in time weak measure solutions and finite time aggregation for nonlocal interaction equations in this paper we provide a well posedness theory for weak measure solutions of the cauchy problem for a family of nonlocal interaction equations these equations are continuum models for interacting particle systems with attractive repulsive pairwise interaction potentials the main phenomenon of interest is that even with smooth initial data the solutions can concentrate mass in finite time we develop an existence theory that enables one to go beyond the blow up time in classical norms and allows for solutions to form atomic parts of the measure in finite time the weak measure solutions are shown to be unique and exist globally in time moreover in the case of sufficiently attractive potentials we show the finite time total collapse of the solution onto a single point for compactly supported initial measures our approach is based on the theory of gradient flows in the space of probability measures endowed with the wasserstein metric in addition to classical tools we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations

strongly consistent online forecasting of centered gaussian processes an estimator e spl circ d sub n n of the conditional expectation e x sub n 1 x sub n x n d sub n 1 in a centered stationary and ergodic gaussian process x sub i sub i with absolutely summable wold coefficients is constructed on the basis of having observed x sub 1 x sub n for a suitable choice of the length dn spl rarr spl infin n spl rarr spl infin of the past covered by the conditional expectation it is established that e spl circ d sub n n e x sub n 1 x sub n x n d sub n 1 spl rarr 0 with probability 1 in addition sufficient conditions for e x sub n 1 x sub n x sub n 1 e x sub n 1 x sub n x n d sub n 1 spl rarr 0 to hold with probability 1 are given that is conditions under which e spl circ d sub n n can be used as a strongly consistent forecaster for e x sub n 1 x sub n x sub n 1

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms we propose a numerical algorithm for solving nonlinear continuity equations written in lagrangian coordinates this transformation is intimately related to variational approaches for the well posedness of gradient flows of energy functionals with respect to the quadratic transportation distance in optimal transport theory these schemes allow the numerical approximation of both diffusive and aggregation regimes of different models positivity energy decreasing and mesh adaptation are built in in the numerical scheme and thus we are capable of capturing blow up densities and of dealing with vacuum regions and merging of mass patches in a natural way

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

nonlocal interactions by repulsive attractive potentials radial ins stability abstract we investigate nonlocal interaction equations with repulsive attractive radial potentials such equations describe the evolution of a continuum density of particles in which they repulse resp attract each other in the short resp long range we prove that under some conditions on the potential radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state we also investigate under which condition it is possible for a non radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface finally we provide a detailed analysis of the specific case of the repulsive attractive power law potential as well as numerical results

a clustering heuristic to detect staircase structures in large scale linear programming models abstract the staircase structure is one of the more frequently found structures in large scale linear programs it is known that the staircase structured problems lend themselves to efficient solution through decomposition based procedures however in many large scale problems any underlying structure may be obscured by getting imbedded into other more obvious structures this paper addresses the identification of the underlying staircase structure in the rectangular matrices of large scale linear programs lslps the problem is known to be computationally hard to solve this problem we propose a heuristic algorithm based on the rank order clustering procedure the results from the application of this algorithm to a number of test problems of known structure indicate that it runs it linear time of the problem size we demonstrate the effectiveness of the algorithm through its ability to identify the correct number of diagonal subproblems that make up the staircase structure in the coefficient matrices of the lslps of two test sets the first collected from the public domain software server netlib and the second randomly generated