determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

local geometry and dynamical behavior on folded basic sets we study new phenomena associated with the dynamics of higher dimensional non invertible hyperbolic maps f on basic sets of saddle type the dynamics in this case presents important differences from the case of diffeomorphisms or expanding maps we show that the stable dimension i e the hausdorff dimension of the intersection between local stable manifolds and the basic set and the unstable dimension similar definition give a lot of information about the dynamical ergodic properties of endomorphisms on folded basic sets we prove a geometric flattening phenomenon associated to the stable dimension i e we show that if the stable dimension is zero at a point then the fractal must be contained in a submanifold and f is expanding on we characterize folded attractors and folded repellers as those basic sets with full unstable dimension respectively with full stable dimension we classify possible dynamical behaviors and establish when is the system f 1 sided or 2 sided bernoulli for certain equilibrium measures on folded basic sets for a class of perturbation maps

error analysis of waveform relaxation method for semi linear partial differential equations the waveform relaxation wr method is investigated for semi linear partial differential equations explicit error estimation is given for the iteration error a way to combine wr with convergent numerical methods is proposed the error of the combined method is analyzed and its convergence is proven the effect of the application of time windows is discussed numerical tests are presented to confirm the theoretical results

waveform relaxation for reaction diffusion equations abstract we report a new waveform relaxation wr algorithm for general semi linear reaction diffusion equations the superlinear rate of convergence of the new wr algorithm is proved and we also show the advantages of the new approach superior to the classical wr algorithms by the estimation on iteration errors the corresponding discrete wr algorithm for reaction diffusion equations is presented and further the parallelism of the discrete wr algorithm is analyzed moreover the new approach is extended to handle the coupled reaction diffusion equations numerical experiments are carried out to verify the effectiveness of the theoretic work

space time continuous analysis of waveform relaxation for the heat equation waveform relaxation algorithms for partial differential equations pdes are traditionally obtained by discretizing the pde in space and then splitting the discrete operator using matrix splittings for the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach however the bounds depend in general on the mesh parameter and convergence rates deteriorate as one refines the mesh motivated by the original development of waveform relaxation in circuit simulation where the circuits are split in the physical domain into subcircuits we split the pde by using overlapping domain decomposition we prove linear convergence of the algorithm in the continuous case on an infinite time interval at a rate depending on the size of the overlap this result remains valid after discretization in space and the convergence rates are robust with respect to mesh refinement the algorithm is in the class of waveform relaxation algorithms based on overlapping multisplittings our analysis quantifies the empirical observation by jeltsch and pohl siam j sci comput 16 1995 pp 40 49 that the convergence rate of a multisplitting algorithm depends on the overlap numerical results are presented which support the convergence theory

comment on a p calderon s paper on an inverse boundary value problem in seminar on numerical analysis and its applications to continuum physics rio de janeiro 1980 65 73 soc brasil mat rio de janeiro 1980 mr0590275 81k 35160

krylov subspace acceleration of waveform relaxation in this paper we describe and analyze krylov subspace techniques for accelerating the convergence of waveform relaxation for solving time dependent problems a new class of accelerated waveform methods convolution krylov subspace methods is presented in particular we give convolution variants of the cg algorithm and the gmres algorithm and analyze their convergence behavior we prove that the convolution krylov subspace algorithms for initial value problems have the same convergence bounds as their linear algebra counterparts analytical examples are given to illustrate the operation of convolution krylov subspace methods experimental results are presented which show the convergence behavior of traditional and convolution waveform methods applied to solving a linear initial value problem as well as the convergence behavior of static krylov subspace methods applied to solving the associated linear algebraic equation

using waveform relaxation methods to approximate neutral stochastic functional differential equation systems this paper extends the waveform relaxation wr method to neutral stochastic functional differential equations the linear convergence theory of the continuous time wr method is established in the mean square sense when the coefficients of stochastic differential equation systems satisfy the lipschitz condition and the contractive mapping the discrete time wr method based on the euler scheme which is used in an actual implementation was also studied it turns out that the sequence produced by this method converges linearly to the euler approximate solution which is convergent in addition we prove that the preceding methods are convergent superlinearly if the neutral term does not been split finally the theory is applied to a one dimensional model problem and checked against results obtained by numerical experiments

a note on windowing for the waveform relaxation method the technique of windowing has often been used in the implementation of the waveform relaxation method for solving ordinary differential equations and time dependent partial differential equations its efficiency depends upon the operator splitting and the coupling of the subsystems involved for two model problems estimates for the window length and convergence rate for the early phase of the iteration are derived the effectiveness of windowing is then investigated leading to some suggestions for the use of windowing the discussions are supported by the computational results

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

optimized schwarz waveform relaxation for the primitive equations of the ocean in this article we are interested in the derivation of efficient domain decomposition methods for the viscous primitive equations of the ocean we consider the rotating three dimensional incompressible hydrostatic navier stokes equations with free surface performing an asymptotic analysis of the system in the regime of small rossby numbers we compute an approximate dirichlet to neumann operator and build an optimized schwarz waveform relaxation algorithm we establish that the algorithm is well defined and provide numerical evidence of the convergence of the method

some remarks concerning quasiconvexity and strong convergence we show that the weak convergence of a sequence of functions in a sobolev space plus the convergence of appropriately quasiconvex energies imply in fact strong convergence this assertion makes rigorous for example the heuristic principle that quasiconvexity damps out oscillations in the gradients of minimising sequences in the calculus of variations

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

optimal use of the translation method and relaxations of variational problems the translation method has been used with great success in bounding the effective moduli of composite materials we consider here the analogous method for bounding the relaxations of variational problems we optimize the bound over the set of all available translations our method is to cast this in the form of a minmax problem using techniques of nonsmooth analysis we are able to identify the optimal translation bound meanwhile proving the existence of at least one optimal combination rank one convex quadratic and null lagrangian translation the optimal translation bound proves to be a better general lower bound on relaxations of variational problems than is the polyconvexification in three dimensions in two dimensions we discuss the negative result that the optimal translation bound is exactly the polyconvexification several examples of optimal applications of translation bounds to non convex nonlinear variational problems are given

symplectic waveform relaxation methods for hamiltonian systems in this literature a new method called symplectic waveform relaxation method is for the first time proposed to solve hamiltonian systems this method is based on waveform relaxation method which makes computation cheaper and makes use of symplectic method to determine its numerical scheme under the guidance of the symplectic method the discrete waveform relaxation method elegantly preserves the discrete symplectic form windowing technique is utilized to accelerate computation the windowing technique also makes it possible to advance in time window by window convergence results of continuous and discrete symplectic waveform relaxation methods are analyzed numerical results show that the symplectic waveform relaxation method with the windowing technique precisely preserves the hamiltonian function

unstable directions and fractal dimension for skew products with overlaps in fibers a unique feature of smooth hyperbolic non invertible maps is that of having different unstable directions corresponding to different prehistories of the same point in this paper we construct a new class of examples of non invertible hyperbolic skew products with thick fibers for which we prove that there exist uncountably many points in the locally maximal invariant set actually a cantor set in each fiber having different unstable directions corresponding to different prehistories also we estimate the angle between such unstable directions we discuss then the hausdorff dimension of the fibers of for these maps by employing the thickness of cantor sets the inverse pressure and also by use of continuous bounds for the preimage counting function we prove that in certain examples there are uncountably many points in with two preimages belonging to as well as uncountably many points having only one preimage in in the end we give examples which also from the point of view of hausdorff dimension are far from being homeomorphisms on as well as far from being constant to 1 maps on

block toeplitz preconditioning for static and dynamic linear systems abstract acceleration techniques for iterative methods for linear systems of both static qy b and dynamic y qy g t type are analyzed a new splitting q m n where m is block toeplitz is proposed in the static case considerable improvement is observed while in the dynamic case this preconditioning results only in a slightly faster waveform relaxation scheme than the traditional block jacobi dynamic iteration method in the case of linear systems approximating heat equation in one or two dimensions it is shown that the static analysis does not predict correctly the optimal value of the parameters introduced for the dynamic case but that a spectral analysis does

on the numerical analysis of non convex variational problems we discuss a numerical method for finding young measure valued minimizers of non convex variational problems to have any hope of a convergence theorem one must work in a setting where the minimizer is unique and minimizing sequences converge strongly this paper has two main goals i we specify a method for producing strongly convergent minimizing sequences despite the failure of strict convexity and ii we show how uniqueness of the young measure can be parlayed into a numerical convergence theorem the treatment of ii is done in the setting of two model problems one involving scalar valued functions and a multiwell energy the other from micromagnetics

remarks on waveform relaxation method with overlapping splittings by studying the superlinear convergence of waveform relaxation method on finite time intervals it has formerly been shown by using the theory of quasinilpotent operators that the convergence properties are largely determined by the graph properties of the splitting in this paper we show how the directed graphs associated to the decomposition are modified when overlapping splittings are used in particular we explain how overlapping should be used in order to best accelerate convergence of the iteration method

a uniqueness theorem for an inverse boundary value problem in electrical prospection we show that a near constant conductivity of a two dimensional body can be uniquely determined by steady state direct current measurements at the boundary mathematically we show that the coefficient in the operator is uniquely determined by its dirichlet integrals

waveform methods for space and time parallelism abstract waveform methods are a form of generalized picard iteration developed by electrical engineers for the numerical solution of the large systems of ordinary differential equations that arise in circuit simulation for vlsi design their effectiveness is principally due to computational considerations disk swapping simple organization of calculations for components with very different frequency spectra and potential use of parallelism in this paper we first examine the rate of convergence of simple extension of picard methods suitable for parallel computation for numerical computation the differential equation must be replaced by a finite dimensional equation for an approximation to its solution the second part of the paper examines the application of the iterative methods directly to finite difference approximations of the original differential equation this permits a larger class of methods than obtained by discretizing the differential equations satisfied by successive iterates

convergence of the parallel chaotic waveform relaxation method for stiff systems in this paper we establish several algorithms for parallel chaotic waveform relaxation methods for solving linear ordinary differential systems based on some given models under some different assumptions on the coefficient matrix a and its multisplittings we obtain corresponding sufficient conditions of convergence of the algorithms also a discussion on convergence speed comparison of synchronous and asynchronous algorithms is given

on the multigrid waveform relaxation method the multigrid waveform relaxation method is an efficient method for solving certain classes of time dependent partial differential equations pdes this paper studies the relationship between this method and the analogous multigrid method for steady state problems using a fourier laplace analysis practical convergence rate estimates of the multigrid waveform relaxation are obtained experimental results show that the analysis yields accurate performance prediction

relaxed notions of curvature and a lumped strain method for elastic plates the paper proposes an approximation method lumped strain method lsm for elastic plates based on generalized notions of the gaussian and mean curvatures and on a relaxation of the energy functional to the space of continuous piecewise linear functions the method adapts ideas from the theory of gamma convergence we restrict our attention to the case of the simply supported plate and state the proof of convergence of the method we give an application to the rhombic plate and compare the results with those obtained by standard finite element approximations

a relaxation theorem and its application to the equilibrium of gases keywords relaxed problem lower convex envelope gas equilibrium van der waal s equation reference caa article 1981 001doi 10 1007 bf00280643view record in web of science record created on 2008 11 25 modified on 2017 05 12

optimization of transmission conditions in waveform relaxation techniques for rc circuits waveform relaxation techniques have become increasingly important with the wide availability of parallel computers with a large number of processors a limiting factor for classical waveform relaxation however is the convergence speed for an important class of problems especially if long time windows are considered in contrast the optimized waveform relaxation algorithm discussed in this paper is well suited to address this problem today several numerical analyses have shown that optimized waveform relaxation algorithms can overcome slow convergence over long time windows however the optimized waveform relaxation techniques require the determination of optimized parameters in this paper we present a theoretical foundation for the determination of the optimized parameters for an important class of rc circuits

convergence results for continuous time waveform methods for volterra integral equations in this paper the waveform methods for solving systems of second kind volterra integral equations are introduced the convergence properties of continuous time waveform methods on finite length and infinite length time intervals are investigated with particular regard to parallel waveform relaxation methods

minimization based formulations of inverse problems and their regularization the conventional way of formulating inverse problems such as identification of a possibly infinite dimensional parameter is via some forward operator which is the concatenation of the observation operator with the parameter to state map for the underlying model recently all at once formulations have been considered as an alternative to this reduced formulation avoiding the use of a parameter to state map which would sometimes lead to overly restrictive conditions here the model and the observation are considered simultaneously as one large system with the state and the parameter as unknowns a still more general formulation of inverse problems containing not only the reduced and all at once formulations but also the well known and highly versatile variational approach also called the kohn vogelius functional approach as special cases is to formulate the inverse problem as a minimization problem instead of an equation for the state and parameter regularization can be incorporated via imp

contractivity of waveform relaxation runge kutta iterations and related limit methods for dissipative systems in the maximum norm contractivity properties of runge kutta methods are analyzed with suitable interpolation implemented using waveform relaxation strategy for systems of ordinary differential equations that are dissipative in the maximum norm in general this type of implementation which is quite appropriate in a parallel computing environment improves the stability properties of runge kutta methods as a result of this analysis a new class of methods is determined which is different from runge kutta methods but closely related to them and which combines its high order of accuracy and unconditional contractivity in the maximum norm this is not possible for classical runge kutta methods

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

estimating waveform relaxation convergence critical limitations of the waveform relaxation method concern the unsolved problems of estimating convergence and controlling the iteration in this paper waveform relaxation is applied to a linear second order model system and is studied in the laplace domain by placing the iteration in a weighted space computable estimates for a window of rapid convergence are developed numerical experiments illustrate the utility of the approach

a schwarz waveform moving mesh method an r refinement moving mesh method is considered for solving time dependent partial differential equations pdes the resulting coupled system consisting of the physical pde and a moving mesh pde is solved by a schwarz waveform relaxation method in particular the computational space time domain is decomposed into overlapping subdomains and the solution obtained by iteratively solving the system of pdes on each subdomain dirichlet boundary conditions are used to pass solution information between neighboring regions the efficacy of this approach is demonstrated for some model problems for problems where the solutions evolve on disparate time scales in different regions of the spatial domain this approach demonstrates the significant savings in computational time and effort which are possible

stochastic symplectic partitioned runge kutta methods for stochastic hamiltonian systems with multiplicative noise a new class of sprk methods and their order conditions are given symplectic conditions of the sprk methods are obtained stochastic generating functions of the ssprk methods are obtained it is proved that the ssprk methods can preserve the quadratic invariants some low stage ssprk methods are constructed some new stochastic partitioned runge kutta sprk methods are proposed for the strong approximation of partitioned stochastic differential equations sdes the order conditions up to strong global order 1 0 are calculated the sprk methods are applied to solve stochastic hamiltonian systems with multiplicative noise some conditions are captured to guarantee that a given sprk method is symplectic it is shown that stochastic symplectic partitioned runge kutta ssprk methods can be written in terms of stochastic generating functions in addition this paper also proves that the ssprk methods can conserve the quadratic invariants of original stochastic systems based on the order and symplectic conditions some low stage ssprk methods with strong global order 1 0 are constructed finally some numerical results are presented to demonstrate the efficiency of the ssprk methods

a multiscale theory for image registration and nonlinear inverse problems abstract in an influential paper tadmor et al 2004 42 introduced a hierarchical decomposition of an image as a sum of constituents of different scales here we construct analogous hierarchical expansions for diffeomorphisms in the context of image registration with the sum replaced by composition of maps we treat this as a special case of a general framework for multiscale decompositions applicable to a wide range of imaging and nonlinear inverse problems as a paradigmatic example of the latter we consider the calderon inverse conductivity problem we prove that we can simultaneously perform a numerical reconstruction and a multiscale decomposition of the unknown conductivity driven by the inverse problem itself we provide novel convergence proofs which work in the general abstract settings yet are sharp enough to prove that the hierarchical decomposition of tadmor nezzar and vese converges for arbitrary functions in l 2 a problem left open in their paper we also give counterexamples that show the optimality of our general results

quasiconvexification in 3 d for a variational reformulation of an optimal design problem in conductivity abstract we analyze a typical 3 d conductivity problem which consists in seeking the optimal layout of two materials in a given design domain r 3 by minimizing the l 2 norm of the electric field under a constraint on the amount on each material that we can use we utilize a characterization of the 3 d divergence free vector fields which is especially appropriate for a variational reformulation by using gradient young measures as a main tool we can give an explicit form of the constrained quasiconvexification of the cost density this result is similar to the one in the 2 d situation however the characterization of the divergence free vector fields introduces a certain nonlinearity in the problem that needs to be addressed properly

optimal design in conductivity under locally constrained heat flux this paper addresses the relaxation of an optimal design problem in conductivity under a point wise constraint on the heat flux by using a variational approach developed by the authors in a previous work we obtained a complete relaxation involving the explicit computation of constrained quasiconvex envelopes this relaxed formulation turns out to be very simple in the remarkable compliance case for which numerical simulations are performed those simulations show that our approach and results are interesting from a theoretical point of view as well as from a practical perspective the emphasis of our work is placed on the local constraint on the heat flux which seems to be a new ingredient for this kind of optimal design problems

optimal design via variational principles the three dimensional case the analysis of some three dimensional optimal design problems leads us to study variational principles under curl free and divergence free constraints simultaneously we explicitly exploit the relationship between curl free and div free restrictions in order to take advantage of the accumulated experience in the classical curl case by the introduction of potentials our discussion takes place in the three dimensional situation this is a first contribution in the sense that we only deal with the most basic issues

invariant measures satisfying an equality relating entropy folding entropy and negative lyapunov exponents in this paper we prove that for a c2 non invertible but non degenerate map on a compact manifold an invariant measure satisfies an equality relating entropy folding entropy and negative lyapunov exponents if and under a condition on the jacobian of the map only if the measure has absolutely continuous conditional measures on the stable manifolds

space time domain decomposition for parabolic problems we analyze a space time domain decomposition iteration for a model advection diffusion equation in one and two dimensions the discretization of this iteration is the block red black variant of the waveform relaxation method and our analysis provides new convergence results for this scheme the asymptotic convergence rate is super linear and it is governed by the diffusion of the error across the overlap between subdomains hence it depends on both the size of this overlap and the diffusion coefficient in the equation however it is independent of the number of subdomains provided the size of the overlap remains fixed the convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases the duration of the transient linear regime is proportional to the length of the time window for advection dominated problems the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases moreover it is independent of the size of the time window and of the number of subdomains numerical calculations illustrate our analysis

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

convergence analysis of waveform relaxation methods for neutral differential functional systems in this paper the problems of convergence and superlinear convergence of continuous time waveform relaxation method applied to volterra type systems of neutral functional differential equations are discussed under a lipschitz condition with time and delay dependent right hand side imposed on the so called splitting function more suitable conditions about convergence and superlinear convergence of continuous time wr method are obtained we also investigate the initial interval acceleration strategy for the practical implementation of the continuous time waveform relaxation method i e discrete time waveform relaxation method it is shown by numerical results that this strategy is efficacious and has the essential acceleration effect for the whole computation process

equilibrium measures on saddle sets of holomorphic maps on mathbb p 2 we consider the case of hyperbolic basic sets lambda of saddle type for holomorphic maps f mathbb p 2 mathbb c rightarrow mathbb p 2 mathbb c we study equilibrium measures mu phi associated to a class of holder potentials phi on lambda and find the measures mu phi of iterates of arbitrary bowen balls estimates for the pointwise dimension delta mu phi of mu phi that involve lyapunov exponents and a correction term are found and also a formula for the hausdorff dimension of mu phi in the case when the preimage counting function is constant on lambda for terminal minimal saddle sets we prove that an invariant measure nu obtained as a wedge product of two positive closed currents is in fact the measure of maximal entropy for the restriction f lambda this allows then to obtain formulas for the measure nu of arbitrary balls and to give a formula for the pointwise dimension and the hausdorff dimension of nu

multi symplectic preissmann scheme for a high order wave equation of kdv type the high order wave equation of kdv type which describes many important physical phenomena has been investigated widely in last several decades here the new multi symplectic formulations for the high order wave equation of kdv type are presented and the associated local conservation laws are shown the multi symplectic discretization of each formulation is exemplified by the multi symplectic preissmann scheme the numerical experiments are given and the results verify the efficiency of the multi symplectic preissmann scheme

timestep acceleration of waveform relaxation dynamic iteration methods for treating certain classes of linear systems of differential equations are considered it is shown that the discretized picard lindelof waveform relaxation iteration can be accelerated by solving the defect equations with a larger timestep or by using a recursive procedure based on a succession of increasing timesteps a discussion of convergence is presented including analysis of a discrete smoothing property maintained by symmetric multistep methods applied to linear wave equations numerical experiments indicate that the method can speed convergence

convergence of a reconstruction method for the inverse conductivity problem the inverse conductivity problem is that of reconstructing a spatially varying isotropic conductivity in the interior of some region by means of steady state measurements taken at the boundary reconstruction schemes including least squares type minimization methods have been widely studied and implemented but convergence analysis has been largely ignored this paper establishes the convergence of a well known least squares minimization scheme the levenberg marquardt method on a regularized formulation of the inverse conductivity problem

explicit computation of the relaxed density coming from a three dimensional optimal design problem we analyze the relaxation of a typical three dimensional optimal design problem in conductivity consisting of minimizing the square of the l2 norm of the underlying electric potential under a volume constraint via a variational reformulation of the problem our main achievement is the explicit characterization of the constrained quasiconvexification of the cost density

identifying conductivity in electrical impedance tomography with total variation regularization in this paper we investigate the problem of identifying the conductivity in electrical impedance tomography from one boundary measurement a variational method with total variation regularization is here proposed to tackle this problem we discretize the pde as well as the conductivity with piecewise linear continuous finite elements we prove the stability and convergence of this technique for the numerical solution we propose a projected armijo algorithm finally a numerical experiment is presented to illustrate our theoretical results

sor waveform relaxation methods for stochastic differential equations waveform relaxation for solving large scale systems of ordinary differential equations has extended to stochastic differential equations sdes this paper deals with the acceleration of the gauss sediel waveform relaxation method for sdes by successive overrelaxation sor techniques a sufficient condition of the linear convergence is obtained for the continuous time sor waveform relaxation method the discrete time sor waveform relaxation method used in an actual implementation was also studied we first prove that the general numerical schemes are convergent and then the particular method based on euler scheme linearly converge to the euler approximate solution of sdes at last the theory is applied to a model problem and checked against results obtained by numerical experiments

schwarz waveform relaxation with adaptive pipelining schwarz waveform relaxation swr methods have been developed to solve a wide range of diffusion dominated and reaction dominated equations the appeal of these methods stems primarily from their ability to use nonconforming space time discretizations swr methods are consequently well adapted for coupling models with highly varying spatial and time scales the efficacy of swr methods is questionable however since in each iteration one propagates an error across the entire time interval in this manuscript we introduce an adaptive pipeline approach wherein one subdivides the computational domain into space time blocks and adaptively selects the waveform iterates which should be updated given a fixed number of computational workers our method is complementary to existing space and time parallel methods and can be used to obtain additional speedup when the saturation point is reached for other types of parallelism we analyze these waveform relaxation with adaptive pipelining wrap methods to show c

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

optimal bounds for conduction in two dimensional multiphase polycrystalline media a complete geometrical characterization is given of the set of all possible effective conductivity tensors for two dimensional composites made of an arbitrary number of given anisotropic phases it is rigorously established that any polycrystalline composite formed from an arbitrary possibly infinite number of phases can be replaced by a composite formed from only two of the given phases without altering the effective conductivity tensor

comment on calderon s paper on an inverse boundary value problem calder6n determined a method to approximate the conductivity a of a con ducting body in rn for n 2 based on measurements of boundary data the approximation is good in the l norm provided that the conductivity is a small perturbation from a constant we calculate the approximation exactly for the case of homogeneous concentric conducting disks in r2 with different conductivities here the difference in the conductivities is the perturbation we show that the approximation yields precise information about the spatial variation of a even when the perturbation is large this ability to distinguish spatial regions with different conductivities is important for clinical monitoring applications

direct reconstructions of conductivities from boundary measurements the problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging geophysics and nondestructive testing a nachman ann of math 2 143 1996 pp 71 96 proved global uniqueness for the two dimensional inverse conductivity problem using a constructive method of proof based on this proof siltanen mueller and isaacson inverse problems 16 2000 pp 681 699 presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration the method was verified with nonnoisy rotationally symmetric examples in this paper the method is extended by introducing a new regularization scheme which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise

constrained quasiconvexity and structural optimization we propose an approach to general structural optimization problems based on variational techniques the analysis involves gradient young measures in the vector case and the notion of constrained quasiconvexity and depends on the appropriate use of stream functions in dimension two the case of several nonlinear materials and arbitrary cost functionals depending on the gradients of the equilibrium states can also be treated this generality is the main motivation for this work

the inverse problem of resistivity sounding the electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two layer systems this new representation is called the bilayer expansion for the green s function it enables the forward problem of resistivity sounding to be solved very efficiently even for complicated profiles also the bilayer expansion facilitates the solution of the resistivity inverse problem the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model specifically i consider models comprising uniformly conducting layers of equal thickness for a given finite data set a quadratic program can be used to find the best fitting model in this class for any specified thickness as the thickness is reduced models of this kind can approximate arbitrary profiles with unlimited accuracy if there is a model that satisfies

2 d and 3 d interpretation of electrical tomography measurements part 2 the inverse problem the interpretation of borehole to borehole electrical measurements requires solving an inverse problem for a given class of model geometries the usual approach to an inverse problem includes a model dependent task i e forward modeling and a generic task i e an optimization scheme we have developed an optimization algorithm using a nonlinear inversion technique this algorithm allows recovery of a possible resistivity distribution in an investigated zone between two boreholes or in the vicinity of them this resistivity distribution is defined as a set of 2 d or 3 d volumes of constant resistivity the inversion procedure minimizes a least squares term plus a damping term this latter term seeks to minimize the roughness of the solution an improved form of this smoothness term may enhance the spatial resolution of the resistivity image assuming that the resistivity contrast is known a priori this reconstruction algorithm has been tested for both 2 d and 3 d geometries these inversion tests we

high precision identification of an object optimality conditions based concept of imaging a class of inverse problems for the identification of an unknown geometric object from given measurements is considered a concept for object imaging based on optimality conditions and level sets is introduced which provides high resolution properties of the identification problem and stability to discretization and noise errors as a specific case the identification of the center of a test object of arbitrary shape and unknown boundary conditions from d boundary measurements in d spatial dimensions in the context of the helmholtz equation is described in detail for analysis and numerical realization methods from topology optimization generalized singular perturbations endowed with variational techniques and a petrov galerkin enrichment within generalized fem are used

waveform relaxation with overlapping splittings in this paper an extension of the waveform relaxation algorithm for solving large systems of ordinary differential equations is presented the waveform relaxation algorithm is well suited for parallel computation because it decomposes the solution space into several disjoint subspaces allowing the subspaces to overlap i e dropping the assumption of disjointedness an extension of this algorithm is obtained this new algorithm the so called multisplitting algorithm is also well suited for parallel computation as numerical examples demonstrate this overlapping of the subsystems heavily reduces the computation time

shape sensitivity analysis for an interface problem via minimax differentiability this paper deals with the shape sensitivity analysis for the conductivity problem with discontinuous coefficients we consider the geometric inverse problem of locating the interfaces where the jump of the coefficients occurs we propose a reconstruction method based on the minimization of the kohn vogelius functional the reformulation of the functional in a suitable saddle point problem form allows us to obtain the optimality conditions by using the differentiability of the minimax combined with the function space parametrization technique an important advantage of this approach is that it avoids the differentiability of the state variable with respect to the domain the reconstruction is then performed with an iterative algorithm using an appropriate quasi newton method we give some numerical examples showing the efficiency of the method

first order system least squares and electrical impedance tomography electrical impedance tomography eit belongs to a family of imaging methods that employ boundary measurements to distinguish interior spatial variation of an electromagnetic parameter the associated inverse problem is notoriously ill posed due to diffusive effects in the quasi static regime when electrical impedance reduces to its real part resistivity the standard approach to eit is output least squares ols for a set of applied normal boundary currents one minimizes the defect between the measured and computed boundary voltages associated respectively with the exact impedance and its approximation in minimizing a boundary functional ols implicitly imposes the governing poisson equation as an optimization constraint to reconstruct resistivity or equivalently conductivity we introduce a new first order system least squares fosls formulation that incorporates the elliptic pde as an interior functional in a global unconstrained minimization scheme we then establish equivalence of our functional to ols and to an existing interior least squares functional due to kohn and vogelius comm pure appl math 37 1984 pp 289 298 that the latter may be viewed as a special dual approach fosll is an interesting attribute of this equivalence because the fosls functional implicitly reflects the inherent loss of resolution away from the boundary relative to the set of applied boundary tests the need for artificial regularization may be avoided

uniform endomorphisms which are isomorphic to a bernoulli shift a uniformly p to one endomorphism is a measure preserving map with entropy log p which is almost everywhere p to one and for which the conditional expectation of each preimage is precisely 1 p the standard example of this is a one sided p shift with uniform i i d bernoulli measure we give a characterization of those uniformly finite to one endomorphisms conjugate to this standard example by a condition on the past tree of names which is analogous to very weakly bernoulli or loosely bernoulli as a consequence we show that a large class of isometric extensions of the standard example are conjugate to it

pseudospectra of waveform relaxation operators abstract the performance of the waveform relaxation method for solving systems of odes depends largely on the choices that are made for splitting size of time window and preconditioning although it is known that superlinear convergence is obtained on finite time windows the convergence may be slow in the first few iterations we propose the use of pseudospectra to analyze the convergence ratio of the first few iterations when waveform relaxation is applied to linear systems of odes through pseudospectral radii we can examine the effect of preconditioning and overlapping on the rate of convergence we may also use this to estimate a suitable size of the time window numerical experiments performed on a system of odes arising from the discretization of an advection diffusion equation confirm the validity of the obtained estimates

on the convergence of the discretized parallel chaotic waveform relaxation method in this paper we establish several algorithm of discretized parallel chaotic waveform relaxation methods for solving linear ordinary differential systems based on some given models under some different assumptions on the coefficient matrix a and its multisplittings we obtain corresponding sufficient conditions for convergence of the algorithms also a discussion on convergence speed comparison of synchronous and asynchronous algorithms is given

shape optimization for dirichlet problems relaxed solutions and optimality conditions we study an optimal design problem for the domain of an elliptic equation with dirichlet boundary conditions we introduce a relaxed formulation of the problem which always admits a solution and we prove some necessary conditions for optimality both for the relaxed and for the original problem

on multisplitting methods for band matrices we present new theoretical results on two classes of multisplitting methods for solving linear systems iteratively these classes are based on overlapping blocks of the underlying coefficient matrix a which is assumed to be a band matrix we show that under suitable conditions the spectral radius rho h of the iteration matrix h does not depend on the weights of the method even if these weights are allowed to be negative for a certain class of splittings we prove an optimality result for rho h with respect to the weights provided that a is an m matrix this result is based on the fact that the multisplitting method can be represented by a single splitting a m n which in our situation surprisingly turns out to be a regular splitting furthermore we show by numerical examples that weighting factors alpha not in 0 1 may considerably improve the convergence

a jacobi waveform relaxation method for odes a jacobi waveform relaxation wr method for solving initial value problems for ordinary differential equations odes is presented in each window the method uses a technique called dynamic fitting and a pair of continuous runge kutta rk formulas to produce the initial waveform after which a fixed number of waveform iterates are computed the reliability and efficacy of the method are demonstrated numerically by applying it to qualitatively different problems for linear tridiagonal odes

energy law preserving c0 finite element schemes for phase field models in two phase flow computations we use the idea in 33 to develop the energy law preserving method and compute the diffusive interface phase field models of allen cahn and cahn hilliard type respectively governing the motion of two phase incompressible flows we discretize these two models using a c 0 finite element in space and a modified midpoint scheme in time to increase the stability in the pressure variable we treat the divergence free condition by a penalty formulation under which the discrete energy law can still be derived for these diffusive interface models through an example we demonstrate that the energy law preserving method is beneficial for computing these multi phase flow models we also demonstrate that when applying the energy law preserving method to the model of cahn hilliard type un physical interfacial oscillations may occur we examine the source of such oscillations and a remedy is presented to eliminate the oscillations a few two phase incompressible flow examples are computed to show the good performance of our method