uniform concentration compactness for sobolev spaces on variable domains abstract we present a new method for proving existence results in shape optimization problems involving the eigenvalues of the dirichlet laplace operator this method brings together the convergence theory and the concentration compactness principle given a sequence of open sets a n n n in r n not necessarily bounded but of uniformly bounded measure we prove a concentration compactness result in l l 2 r n for the sequence of resolvent operators r a n n n where r a n l 2 r n h 1 0 a n r a n 1

numerical verification of solutions for a simplified signorini problem abstract we proposed some numerical methods for the automatic proof of existence of solutions for variational inequalities i e obstacle problems and elastoplastic torsion problems it is based on nakao s method by using finite element approximation and its explicit error estimates for the problem in this paper we present a numerical verification algorithm of exact solutions for a simplified signorini problem in a computer a numerical example for the signorini problem is presented

numerical minimization of dirichlet laplacian eigenvalues of four dimensional geometries we develop the first numerical study in four dimensions of optimal eigenmodes associated with the dirichlet laplacian we describe an extension of the method of fundamental solutions adapted to the four dimensional context based on our numerical simulation and a postprocessing adapted to the identification of relevant symmetries we provide and discuss the numerical description of the eighth first optimal domains

numerical verification of solutions for some unilateral boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for some unilateral boundary value problems using the finite element approximations and explicit a priori error estimates we construct in a computer a set of solutions which satisfies the hypothesis of schauder s fixed point theorem for a compact map on a certain sobolev space further the conditions of verifiability by this method are considered and some numerical examples are presented

destabilization by noise of transverse perturbations to heteroclinic cycles a simple model and an example from dynamo theory we show that transverse perturbations from structurally stable heteroclinic cycles can be destabilized by surprisingly small amounts of noise even when each individual fixed point of the cycle is stable to transverse modes a condition that favours this process is that the linearization of the dynamics in the transverse direction be characterized by a non normal matrix the phenomenon is illustrated by a simple two dimensional switching model and by a simulation of a convectively driven dynamo

explicit h2 estimates and pointwise bounds for solutions of second order elliptic boundary value problems abstract it is well known that the h 2 norm and the c 0 norm of a function u h 2 where r n is a bounded domain n 3 can be estimated in terms of a given uniformly elliptic second order differential operator l and some boundary operator b applied to u if certain regularity assumptions are satisfied if these bounds shall be used for numerical purposes the constants occurring in the estimates must be known explicitly the main goal of the present article is the computation of such explicit constants for simplicity of presentation we restrict ourselves to the case where l u u c x u as an application we prove an existence and inclusion result for nonlinear boundary value problems

asymptotic behaviour of optimal spectral planar domains with fixed perimeter we consider the problem of minimizing the kth dirichlet eigenvalue of planar domains with fixed perimeter and show that as k goes to infinity the optimal domain converges to the ball with the same perimeter we also consider this problem within restricted classes of domains such as n polygons and tiling domains for which we show that the optimal asymptotic domain is that which maximises the area for fixed perimeter within the given family i e the regular n polygon and the regular hexagon respectively physically the above problems correspond to the determination of the shapes within a given class which will support the largest number of modes below a given frequency

pseudospectra of the convection diffusion operator the spectrum of the simplest 1d convection diffusion operator is a discrete subset of the negative real axis but the pseudospectra are regions in the complex plane that approximate parabolas put another way the norm of the resolvent is exponentially large as a function of the peclet number throughout a certain parabolic region these observations have a simple physical basis and suggest that conventional spectral analysis for convection diffusion operators may be of limited value in some applications

pseudospectra of semiclassical pseudo differential operators the pseudo spectra or spectral instability of non selfadjoint operators is a topic of current interest in applied mathematics for example in computational fluid dynamics it affects the study of the stability of laminar flows in fact even for the most basic flows the computations entirely fails to predict what is observed in the experiments the explanation is that for non normal operators the resolvent could be very large far away from the spectrum which makes computation of the eigenvalues impossible the occurence of false eigenvalues is due to the existence of quasi modes i e approximate local solutions to the eigenvalue problem the quasi modes appear since the nirenberg treves condition psi is not satisfied for topological reasons less

solving obstacle problems with guaranteed accuracy abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for some simple obstacle problems using the finite element approximation and constructive error estimates we construct on a computer a set of solutions which satisfies the hypothesis of the schauder fixed point theorem for a compact map on a certain sobolev space we describe the numerical verification algorithm for solving a two dimensional obstacle problems and report some numerical results

two projection methods for the solution of signorini problems abstract two iterative methods based on projection and boundary element methods are considered for signorini problems the regularized problem with the projection boundary condition is first deduced from the signorini problem by using the equivalence between the signorini boundary condition and the projection fixed point problem our methods formulate the signorini boundary condition into a sequence of robin boundary conditions in the new boundary condition there is a penalty parameter the convergence speed of the first method is greatly influenced by the value of which is difficult to choose for individual problems to improve the performance of this method we present a self adaptive projection method which adjusts the parameter automatically per iteration based on the iterative data the main result of this work is to provide the convergence of the methods under mild assumptions as the iteration process is given by the potential and its derivative on the boundary of the domain the unknowns of the problem are computed explicitly by using the boundary element method both theoretical results and numerical experiments indicate efficiency of the methods proposed

numerical linear algebra and solvability of partial differential equations it was observed long ago that the obstruction to the accurate computation of eigenvalues of large non self adjoint matrices is inherent in the problem the basic idea is that the resolvent of a highly non normal operator can be very large far away from the spectrum this leads to an easily observable fact that algorithms for locating eigenvalues will typically find some false eigenvalues these false eigenvalues also explain one of the most surprising phenomena in linear pdes namely the fact discovered by hans lewy in 1957 in berkeley that one cannot always locally solve the pde p u f almost immediately after that discovery hormander provided an explanation of lewy s example showing that almost all operators with non constanct complex valued coefficients are not locally solvable in modern language that was done by considering the essentially dual problem of existence of non propagating singularities the purpose of this article is to review this work in the context of almost eigenvalues and from the point of view of semi classical analysis

a counterexample for two conjectures about stability one measure of the stability of a matrix a is the distance to the nearest unstable matrix b recently van loan presented an algorithm to compute b which depended on a conjecture about the location of its eigenvalues we provide a counterexample to this conjecture which shows that the algorithm may overestimate the distance to b by an arbitrary amount the same counterexample invalidates another conjecture and algorithm of the author

numerical inclusion methods of solutions for variational inequalities we consider a numerical method that enables us to verify the existence of solutions for variational inequalities this method is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates we present an effective verification procedure that through numerical computation generates a set which includes the exact solution copyright 2002 john wiley sons ltd

quantitative characterizations of exponential convergence property of nonlinear discrete dynamical systems in this technical note we consider the exponential convergence property of the trajectories of nonlinear discrete dynamical systems with exponential stability based on an invariant quantity derived from lyapunov functions of the systems the essential and quantitative relationship between exponential convergence property of trajectories and the lyapunov functions is revealed

meshless projection iterative analysis of signorini problems using a boundary element free method this paper presents a meshless method based on coupling a boundary element free method befm with a new projection iterative scheme for signorini problems using the new iterative scheme the nonlinear complementary signorini boundary conditions are transformed into neumann boundary conditions then mixed elliptic problems are formed and are solved by the befm in this method only a nodal data structure on the boundary is required and the coefficient matrices of the discrete system equations only have to be computed once in the whole iterative process convergence of the iterative scheme is discussed theoretically numerical examples involving beach percolation and an electropaint process are given to illustrate the convergence rate and high computational efficiency of the method

the computation of resonances in open systems using a perfectly matched layer in this paper we consider the problem of computing resonances in open systems we first characterize resonances in terms of improper eigen functions of the helmholtz operator on an unbounded domain the perfectly matched layer pml technique has been successfully applied to the computation of scattering problems we shall see that the application of pml converts the resonance problem to a standard eigenvalue problem still on an infinite domain this new eigenvalue problem involves an operator which resembles the original helmholtz equation transformed by a complex shift in the coordinate system our goal will be to approximate the shifted operator first by replacing the infinite domain by a finite computational domain with a convenient boundary condition and second by applying finite elements on the computational domain we shall prove that the first of these steps leads to eigenvalue convergence to the desired resonance values which is free from spurious computational eigenvalues provided that the size of computational domain is sufficiently large the analysis of the second step is classical finally we illustrate the behavior of the method applied to numerical experiments in one and two spatial dimensions

numerical verification of solutions for variational inequalities in this paper we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities this technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations using the finite element approximations and explicit a priori error estimates for obstacle problems we present an effective verification procedure that through numerical computation generates a set which includes the exact solution further a numerical example for an obstacle problem is presented

regularity for shape optimizers the nondegenerate case we consider minimizers of f 1 n 1 where f is a function strictly increasing in each parameter and k is the kth dirichlet eigenvalue of our main result is that the reduced boundary of the minimizer is composed of c1 graphs and exhausts the topological boundary except for a set of hausdorff dimension at most n 3 we also obtain a new regularity result for vector valued bernoulli type free boundary problems 2018 wiley periodicals inc

pseudospectra of the orr sommerfeld operator this paper investigates the pseudospectra and the numerical range of the orr sommerfeld operator for plane poiseuille flow a number z in bf c is in the epsilon pseudospectrum of a matrix or operator a if zi a 1 geq epsilon 1 or equivalently if z is in the spectrum of a e for some perturbation e satisfying e leq epsilon the numerical range of a is the set of numbers of the form au u where cdot cdot is the inner product and u is a vector or function with huh u 1 the spectrum of the orr sommerfeld operator consists of three branches it is shown that the eigenvalues at the intersection of the branches are highly sensitive to perturbations and that the sensitivity increases dramatically with the reynolds number the associated eigenfunctions are nearly linearly dependent even though they form a complete set to understand the high sensitivity of the eigenvalues a model operator is considered related to the airy equation tha

an existence result for a class of shape optimization problems given a bounded open subset of rn we prove the existence of a minimum point for a functional f defined on the family a of all quasiopen subsets of under the assumption that f is decreasing with respect to set inclusion and that f is lower semicontinuous on a with respect to a suitable topology related to the resolvents of the laplace operator with dirichlet boundary condition applications are given to the existence of sets of prescribed volume with minimal kth eigenvalue or with minimal capacity with respect to a given elliptic operator

pseudospectra of linear operators if a matrix or linear operator a is far from normal its eigenvalues or more generally its spectrum may have little to do with its behavior as measured by quantities such as an or exp ta more may be learned by examining the sets in the complex plane known as the pseudospectra of a defined by level curves of the norm of the resolvent zi a 1 five years ago the author published a paper that presented computed pseudospectra of thirteen highly nonnormal matrices arising in various applications since that time analogous computations have been carried out for differential and integral operators this paper a companion to the earlier one presents ten examples each chosen to illustrate one or more mathematical or physical principles

high accuracy periodic solutions to the sivashinsky equation the aim of this work is the accurate calculation of periodic solutions to the sivashinsky equation which models dynamics of the long wave instability of laminar premixed flame a highly accurate computational algorithm was developed in both one and two spatial dimensions and its crucial implementation details are presented the algorithm is based on the concept of saturated asymptotic approximations and can be straightforwardly extended to a wide variety of nonlinear integro differential equations the development of such an algorithm was motivated by difficulties in interpretation of the results of numerical experiments with the sivashinsky equation using spectral methods the computations carried out by the algorithm in question are in good agreement with the results obtained earlier by spectral methods analysis of the accuracy of obtained numerical solutions and of their stabilization to steady states supports the idea of the instability of the steady coalescent pole solutions with maximal possible number of poles to the sivashinsky equation in large domains through huge linear transient amplification of nonmodal perturbations of small but finite amplitudes

lipschitz regularity of the eigenfunctions on optimal domains we study the optimal sets omega ast subseteq mathbb r d for spectral functionals of the form f big lambda 1 omega ldots lambda p omega big which are bi lipschitz with respect to each of the eigenvalues lambda 1 omega lambda 2 omega ldots lambda p omega of the dirichlet laplacian on omega a prototype being the problem min big lambda 1 omega cdots lambda p omega omega subseteq mathbb r d omega 1 big we prove the lipschitz regularity of the eigenfunctions u 1 ldots u p of the dirichlet laplacian on the optimal set omega ast and as a corollary we deduce that omega ast is open for functionals depending only on a generic subset of the spectrum as for example lambda k omega our result proves only the existence of a lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved

minimization of the k th eigenvalue of the robin laplacian abstract the paper is concerned with the minimization of the k th eigenvalue of the laplace operator with robin boundary conditions among all open sets of r n satisfying a volume constraint we prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets the main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation one of the key difficulties for k 3 comes from the fact that the eigenvalues are critical points

minimization of the k th eigenvalue of the dirichlet laplacian for every k in mathbb n we prove the existence of a quasi open set minimizing the k th eigenvalue of the dirichlet laplacian among all sets of prescribed lebesgue measure moreover we prove that every minimizer is bounded and has a finite perimeter the key point is the observation that such quasi open sets are shape subsolutions for an energy minimizing free boundary problem

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

pseudo spectra theory of tensors and tensor polynomial eigenvalue problems abstract this paper is devoted to the extension of the pseudo spectra theory from matrices to tensors based on the definition of an eigenpair of real symmetric tensors and results on the pseudo spectrum of square matrices we first introduce the pseudo spectrum of a complex tensor and investigate its fundamental properties such as its computational interpretations and the link with the stability of its related homogeneous dynamical system we then extend the pseudo spectrum to the setting of tensor polynomial eigenvalue problems we further derive basic structure of the pseudo spectrum for tensor polynomial eigenvalue problems including the symmetry boundedness and number of connected components under suitable mild assumptions finally we discuss the implication of the pseudo spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial

numerical verification of solutions for elasto plastic torsion problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for the elasto plastic torsion problems governed by the variational inequality based upon the finite element approximations and the explicit a priori error estimates for a simple problem we present an effective verification procedure that through numerical computation generates a set which includes the exact solution this paper is an extension of the previous paper 1 in which we mainly dealt with the obstacle problems but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem a numerical example is illustrated

computation of pseudospectra by continuation the concept of pseudospectrum was introduced by trefethen to explain the behavior of nonnormal operators many phenomena for example hydrodynamic instability and convergence of iterative methods for linear systems cannot be accounted for by eigenvalue analysis but are more understandable by examining the pseudospectra the straightforward way to compute pseudospectra involves many applications of the singular value decomposition svd this paper presents several fast continuation methods to calculate pseudospectra of different types of matrices

can spectral value sets of toeplitz band matrices jump abstract the spectral value set speb ca of a bounded linear operator a on l2 is the union of the spectra of all operators of the form a bkc where k

solving overdetermined eigenvalue problems we propose a new interpretation of the generalized overdetermined eigenvalue problem left bf a lambda bf b right bf v approx 0 for two m times n m n matrices bf a and bf b its stability analysis and an efficient algorithm for solving it usually the matrix pencil bf a lambda bf b does not have any rank deficient member therefore we aim to compute lambda for which bf a lambda bf b is as close as possible to rank deficient i e we search for lambda that locally minimize the smallest singular value over the matrix pencil bf a lambda bf b practically the proposed algorithm requires cal o mn 2 operations for computing all the eigenpairs we also describe a method to compute practical starting eigenpairs the effectiveness of the new approach is demonstrated with numerical experiments a matlab based implementation of the proposed algorithm can be found at http www mat univie ac at neum software oeig

maximal spectral surfaces of revolution converge to a catenoid we consider a maximization problem for eigenvalues of the laplace beltrami operator on surfaces of revolution in r3 with two prescribed boundary components for every j we show there is a surface

numerical verification of solutions for signorini problems using newton like method we proposed a numerical method to verify the existence of solutions for a simplified signorini problem comput math appl 2000 40 1003 1013 using sequential iteration method we numerically constructed a set containing solutions that satisfies the hypothesis of schauder s fixed point theorem in a certain sobolev space it is difficult to apply this method to the problem of which associated operator is not retractive in a neighborhood of the solution in this paper in order to overcome such a difficulty we describe an alternative approach to this problem numerical examples are presented copyright 2007 john wiley sons ltd

existence of minimizers for eigenvalues of the dirichlet laplacian with a drift abstract this paper deals with the eigenvalue problem for the operator l x with dirichlet boundary conditions we are interested in proving the existence of a set minimizing any eigenvalue k of l under a suitable measure constraint suggested by the structure of the operator more precisely we prove that for any c 0 and k n the following minimization problem min k quasi open set e x 2 2 d x c has a solution

minimization of the third eigenvalue of the dirichlet laplacian for every k in mathbb n we prove the existence of a quasi open set minimizing the k th eigenvalue of the dirichlet laplacian among all sets of prescribed lebesgue measure moreover we prove that every minimizer is bounded and has a finite perimeter the key point is the observation that such quasi open sets are shape subsolutions for an energy minimizing free boundary problem

multiphase shape optimization problems this paper is devoted to the analysis of multiphase shape optimization problems which can formally be written as min g f 1 omega 1 dots f h omega h m bigcup i 1 h omega i omega i subset d omega i cap omega j emptyset where d subseteq mathbb r d is a given bounded open set omega i is the lebesgue measure of omega i and m is a positive constant for a large class of such functionals we analyze qualitative properties of the cells and the interaction between them each cell is itself a subsolution for a single phase shape optimization problem from which we deduce properties like finite perimeter inner density separation by open sets absence of triple junction points etc as main examples we consider functionals involving the eigenvalues of the dirichlet laplacian of each cell i e f i lambda k i

a numerical verification method for the existence of weak solutions for nonlinear boundary value problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for nonlinear two point boundary value problems bvp we formulate the problem as a fixed point of a newton like operator and present a verification algorithm by computer based on sadovskii s fixed point theorem in order to compute the newton type iterative sequences in infinite dimensional space two concepts the rounding and the rounding error are introduced also some numerical examples are illustrated we emphasize that since the present method uses the weak formulation of bvp and the finite element approximations instead of the existing validation techniques it is aslo applicable in principle to the bvp for partial differential equations as it stands

spectra and pseudospectra of waveform relaxation operators we describe the spectra and pseudospectra of continuous time and discrete time waveform relaxation operators since the spectrum of the finite interval waveform relaxation operator is only a singleton the spectrum of the infinite interval operator is typically used to describe the behavior of waveform relaxation algorithms here we show that the pseudospectrum is a more useful tool for analyzing convergence of waveform iterations on finite intervals and prove that the pseudospectrum of the infinite interval operator is in fact the limit of the pseudospectra of the finite interval operators

optimal spectral rectangles and lattice ellipses we consider the problem of minimizing the k th eigenvalue of rectangles with unit area and dirichlet boundary conditions this problem corresponds to finding the ellipse centred at the origin with axes on the horizontal and vertical axes with the smallest area containing k integer lattice points in the first quadrant we show that as k goes to infinity the optimal rectangle approaches the square and correspondingly the optimal ellipse approaches the circle we also provide a computational method for determining optimal rectangles for any k and relate the rate of convergence to the square with the conjectured error term for gauss s circle problem

extremal eigenvalues of the laplacian on euclidean domains and closed surfaces we investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the laplacian on euclidean domains of unit volume under dirichlet and neumann boundary conditions respectively in a second part we study sequences of extremal eigenvalues of the laplace beltrami operator on closed surfaces of unit area