multiscale resolution in the computation of crystalline microstructure this paper addresses the numerical approximation of microstructures in crystalline phase transitions without surface energy it is shown that branching of different variants near interfaces of twinned martensite and austenite phases leads to reduced energies in finite element approximations such behavior of minimizing deformations is understood for an extended model that involves surface energies moreover the closely related question of the role of different growth conditions of the employed bulk energy is discussed by explicit construction of discrete deformations in lowest order finite element spaces we prove upper bounds for the energy and thereby clarify the question of the dependence of the convergence rate upon growth conditions and lamination orders for first order laminates the estimates are optimal

the z spl pi m algorithm a method for interferometric image reconstruction in sar sas this paper presents an effective algorithm for absolute phase not simply modulo 2 spl pi estimation from incomplete noisy and modulo 2 spl pi observations in interferometric aperture radar and sonar insar insas the adopted framework is also representative of other applications such as optical interferometry magnetic resonance imaging and diffraction tomography the bayesian viewpoint is adopted the observation density is 2 spl pi periodic and accounts for the interferometric pair decorrelation and system noise the a priori probability of the absolute phase is modeled by a compound gauss markov random field cgmrf tailored to piecewise smooth absolute phase images we propose an iterative scheme for the computation of the maximum a posteriori probability map absolute phase estimate each iteration embodies a discrete optimization step z step implemented by network programming techniques and an iterative conditional modes icm step spl pi step accordingly the algorithm is termed z spl pi m where the letter m stands for maximization an important contribution of the paper is the simultaneous implementation of phase unwrapping inference of the 2 spl pi multiples and smoothing denoising of the observations this improves considerably the accuracy of the absolute phase estimates compared to methods in which the data is low pass filtered prior to unwrapping a set of experimental results comparing the proposed algorithm with alternative methods illustrates the effectiveness of our approach

numerical analysis for a macroscopic model in micromagnetics the macroscopic behavior of stationary micromagnetic phenomena can be modeled by a relaxed version of the landau lifshitz minimization problem in the limit of large and soft magnets omega it is reasonable to exclude the exchange energy and convexify the remaining energy densities the numerical analysis of the resulting minimization problem begin align min e 0 bf m text amongst bf m omega to mathbb r d text with bf m x le1 text for almost every x in omega end align for d 2 3 faces difficulties caused by the pointwise side constraint bf m le1 and an integral over the whole space mathbb r d for the stray field energy this paper involves a penalty method to model the side constraint and reformulates the exterior maxwell equation via a nonlocal integral operator mathcal p acting on functions exclusively defined on omega the discretization with piecewise constant discrete magnetizations leads to edge oriented boundary integrals the implementation of which and related numerical quadrature are discussed as are adaptive algorithms for automatic mesh refinement a priori and a posteriori error estimates provide a thorough rigorous error control of certain quantities three classes of numerical experiments study the penalization empirical convergence rates and performance of the uniform and adaptive mesh refining algorithms

approximation of a laminated microstructure for a rotationally invariant double well energy density we give error estimates for the approximation of a laminated microstructure which minimizes the energy int omega phi nabla v x dx for a rotationally invariant double well energy density phi a we present error estimates for the convergence of the deformation in l 2 the convergence of directional derivatives of the deformation in the twin planes the weak convergence of the deformation gradient the convergence of the microstructure or young measure of the deformation gradients and the convergence of nonlinear integrals of the deformation gradient

on the rate of convergence to equilibrium in the becker doring equations abstract we provide an explicit rate of convergence to equilibrium for solutions of the becker doring equations using the energy energy dissipation relation the main difficulty is the structure of equilibria of the becker doring equations which do not correspond to a gaussian measure such that a logarithmic sobolev inequality is not available we prove a weaker inequality which still implies for fast decaying data that the solution converges to equilibrium as e ct1 3

a level set approach to the computation of twinning and phase transition dynamics a computational method is proposed for the dynamics of solids capable of twinning and phase transitions in a two dimensional sharp interface model of twinning the stored energy function is a nonconvex potential with multiple wells the evolution of twin interfaces is governed by field equations and jump conditions of momentum balance and by a kinetic relation expressing the interface velocity as a function of the local driving traction and interfacial orientation a regularized version of the model is constructed based on the level set method a level set function which changes signs across the interface is introduced the evolution of this function is described by a hamilton jacobi equation whose velocity coefficient is determined by the kinetic relation jump conditions are thereby eliminated allowing finite difference discretization numerical simulations exhibit complex evolution of the interface including cusp formation needle growth spontaneous tip splitting and topological changes that result in microstructure refinement the results capture experimentally observed phenomena in martensitic crystals

numerical approximation of young measures in non convex variational problems summary in non convex optimisation problems in particular in non convex variational problems there usually does not exist any classical solution but only generalised solutions which involve young measures in this paper first a suitable relaxation and approximation theory is developed together with optimality conditions and then an adaptive scheme is proposed for the efficient numerical treatment the young measures solving the approximate problems are usually composed only from a few atoms this is the main argument our effective active set type algorithm is based on the support of those atoms is estimated from the weierstrass maximum principle which involves a hamiltonian whose good guess is obtained by a multilevel technique numerical experiments are performed in a one dimensional variational problem and support efficiency of the algorithm

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

the simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation we study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type mathcal p 432 to mathcal p 222 the free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank one connected we consider the energy minimization problem with dirichlet boundary data compatible with an arbitrary but fixed simple laminate we first show that for all but a few isolated values of transformation strains this problem has a unique young measure solution solely characterized by the boundary data that represents the simply laminated microstructure we then present a theory of stability for such a microstructure and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers

face recognition using scale adaptive directional and textural features a novel approach to face recognition problem using directional and texture information from face images is proposed in this paper in order to capture the directionality specially designed using local polynomial approximation technique scale adaptive digital filters are used for texture features extraction a low dimensional and computationally effective local descriptor is utilized textural and directional features are captured at the holistic and part based levels resulting in a robust face descriptor the proposed method is tested on a number of standard test face datasets orl xm2vts extended yale cmu pie ar and feret for different scenarios and its performance is compared with several state of the art techniques

estimates for numerical approximations of rank one convex envelopes summary we present a convergence analysis of an algorithm for the numerical computation of the rank one convex envelope of a function f mathrm m m times n rightarrow mathbb r a rate of convergence for the scheme is established and numerical experiments are presented to illustrate the analytical results and applications of the algorithm

asymptotic behavior of solutions to the coagulation fragmentation equations ii weak fragmentation the discrete coagulation fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones the assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important our results on the asymptotic behavior of solutions generalize the corresponding results of ball carr and penrose for the becker doring equation

on the invariant distribution of a one dimensional avalanche process we consider an interacting particle system t t 0 with values in 0 1 z in which each vacant site becomes occupied with rate 1 while each connected component of occupied sites become vacant with rate equal to its size we show that such a process admits a unique invariant distribution which is exponentially mixing and can be perfectly simulated we also prove that for any initial condition the avalanche process tends to equilibrium exponentially fast as time increases to infinity finally we consider a related mean field coagulation fragmentation model we compute its invariant distribution and we show numerically that it is very close to that of the interacting particle system

on the computation of the rank one convex hull of a function we report on several numerical experiments where the rank one convexification of an energy density is computed the explicit examples cover a whole spectrum of typical situations one may encounter one of those is especially relevant for the computation of microstructures in crystalline solids

phase unwrapping via graph cuts phase unwrapping is the inference of absolute phase from modulo 2pi phase this paper introduces a new energy minimization framework for phase unwrapping the considered objective functions are first order markov random fields we provide an exact energy minimization algorithm whenever the corresponding clique potentials are convex namely for the phase unwrapping classical lp norm with pges1 its complexity is kt n 3n where k is the length of the absolute phase domain measured in 2pi units and t n m is the complexity of a max flow computation in a graph with n nodes and m edges for nonconvex clique potentials often used owing to their discontinuity preserving ability we face an np hard problem for which we devise an approximate solution both algorithms solve integer optimization problems by computing a sequence of binary optimizations each one solved by graph cut techniques accordingly we name the two algorithms puma for phase unwrapping max flow min cut a set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with state of the art phase unwrapping algorithms

the relaxation of non quasiconvex variational integrals we show that the steepest descent algorithm in connection with wiggly energies yields minimizing sequences that converge to a global minimum of the associated non quasiconvex variational integrals we introduce a multi level infinite dimensional variant of the steepest descent algorithm designed to compute complex microstructures by forming non smooth minimizers from the smooth initial guesses we apply this multi level method to the minimization of the variational problems associated with martensitic branching

polynomial phase estimation by least squares phase unwrapping estimating the coefficients of a noisy polynomial phase signal is important in fields including radar biology and radio communications one approach attempts to perform polynomial regression on the phase of the signal this is complicated by the fact that the phase is wrapped modulo 2 and must be unwrapped before regression can be performed in this paper we consider an estimator that performs phase unwrapping in a least squares manner we call this the least squares unwrapping lsu estimator the lsu estimator can be computed in a reasonable amount of time for data sets of moderate size using existing general purpose algorithms from algebraic number theory under mild conditions on the distribution of the noise we describe the asymptotic properties of this estimator showing that it is strongly consistent and asymptotically normally distributed a key feature is that the lsu estimator is accurate over a far wider range of parameters than many popular existing estimators monte carlo simulations support our theoretical results and demonstrate the excellent statistical performance of the lsu estimator when compared with existing state of the art estimators

gradient polyconvex material models and their numerical treatment abstract gradient polyconvex materials are nonsimple materials where we do not assume smoothness of the elastic strain but instead regularity of minors of the strain is required this allows for a larger class of admissible deformations than in the case of second grade materials we describe a possible implementation of gradient polyconvex elastic energies in nonlinear finite strain elastostatics besides a new geometric interpretation of gradient polyconvexity is given and it is compared with standard second grade materials finally we demonstrate application of the proposed approach using two different models namely a saint venant kirchhoff material and a double well stored energy density

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

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer

numerical approximation of non homogeneous non convex vector variational problems in this paper we study a numerical scheme for non convex vector variational problems allowing for microstructure based on the approximation of gradient young measures we present a convergence result and some numerical experiments

constitutive theory for some constrained elastic crystals abstract some of the observations of a 15 superconductors near cubic tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints here we begin to develop a theory of this kind

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

a discontinuous finite element method for solving a multiwell problem many physical materials of practical relevance can attain several variants of crystalline microstructure the appropriate energy functional is necessarily nonconvex and the minimization of the functional becomes a challenging problem a new numerical method based on discontinuous finite elements and a scaled energy functional is proposed it exhibits excellent convergence behavior for the energy second order as well as other crucial quantities of interest for general spatial meshes contrary to standard non conforming methods both theoretical analyses and numerical test calculations are presented and contrasted to other current finite element methods for this problem

absolute phase image reconstruction a stochastic nonlinear filtering approach this paper formulates and proposes solutions to the problem of estimating reconstructing the absolute not simply modulo 2 spl pi phase of a complex random field from noisy observations of its real and imaginary parts this problem is representative of a class of important imaging techniques such as interferometric synthetic aperture radar optical interferometry magnetic resonance imaging and diffraction tomography we follow a bayesian approach then not only a probabilistic model of the observation mechanism but also prior knowledge concerning the phase image to be reconstructed are needed we take as prior a nonsymmetrical half plane autoregressive nshp ar gauss markov random field gmrf based on a reduced order state space formulation of the linear nshp ar model and on the nonlinear observation mechanism a recursive stochastic nonlinear filter is derived the corresponding estimates are compared with those obtained by the extended kalman bucy filter a classical linearizing approach to the same problem a set of examples illustrate the effectiveness of the proposed approach

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

numerical analysis of oscillations in a nonconvex problem related to image selective smoothing we study some numerical properties of a nonconvex variational problem which arises as the continuous limit of a discrete optimization method designed for the smoothing of images with preservation of discontinuities the functional that has to be minimized fails to attain a minimum value instead minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional the oscillations of the gradient exhibit analogies with microstructures in ordered materials the pattern of the oscillations is analysed numerically by using discrete parametrized measures

on algorithms for nonconvex optimization in the calculus of variations a simple algorithm for the computation of local minima of nonconvex problems in one dimension is proposed this algorithm avoids certain well known local minima that are distant from the global minimum and can be used to improve the minima obtained using classical descent methods the calculated minima have mesh scale oscillations typical of weakly converging sequences it is shown that these local minima converge weakly to the correct limit for certain simple problems however numerical evidence suggests that the gradients may not converge to the correct weak limits and this is verified using asymptotic expansions extensions to multiple dimensions and some numerical examples in two dimensions are considered

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

approximation of martensitic microstructure with general homogeneous boundary data we consider the approximation of martensitic microstructure for a class of martensitic transformations we model such microstructures by multi well energy minimization problems with general homogeneous boundary data under our assumptions on such boundary data the underlying microstructure can be nonunique we first show that any energy minimizing sequence converges strongly to a unique macroscopic deformation that is precisely the homogeneous deformation in the boundary condition we then prove a series of estimates for the approximation of admissible deformations to the unique macroscopic deformation of the microstructure and for the closeness of the gradients of admissible deformations to the energy wells

gelation time in the discrete coagulation fragmentation equations with a bilinear coagulation kernel the discrete coagulation fragmentation equations with a bilinear coagulation kernel describing the evolution of clusters of particles undergoing binary reactions are studied we compute the gelation time and post gelation mass for the pure coagulation equation a method of characteristics is developed to solve numerically the partial differential equation satisfied by a moment generating function for a product coagulation kernel and a constant fragmentation rate the accuracy of the numerical method is verified by comparison of the numerical results with an exact solution for the number density of monomers for a given coagulation rate a critical value of the fragmentation rate values greater than which lead to mass conservation is identified

effective simulation of a macroscopic model for stationary micromagnetics the effective behaviour of stationary micromagnetic phenomena is modelled by a convexified landau lifshitz minimization problem for the limit of large and soft magnets x without the exchange energy the numerical simulation of the resulting minimization problem has to overcome difficulties caused by the pointwise side restriction jmj 6 1 and the stray field energy on the unbounded domain r d a penalty method models the side restriction and the exterior maxwell equation is recast via a nonlocal integral operator p as shown in numer anal macrosc model micromagnet submitted for publication analysis numerik und simulation eines relaxierten modellproblems zum mikromagnetismus doctoral thesis vienna university of technology 2003 the discretization leads to a nonlocal problem with piecewise constant ansatz and test functions and dense stiffness matrices with closed form formula for their entries this paper addresses the numerical solution with newton raphson schemes and the scientific computation of effective micromagnetic simulations 2004 published by elsevier b v

existence and uniqueness of steady state solution to a singular coagulation fragmentation equation abstract in this article we prove the existence of an equilibrium solution to the coagulation fragmentation equation with i a singular coagulation kernel in the form k x y 1 x y x y where 0 1 2 and 0 1 and ii a fragmentation kernel in the form f x y b 0 b x y m where b 0 0 and b m 0 importantly in the entire study the detailed balance condition is not assumed it is shown that the equilibrium solution of the problem under consideration is unique in the space of all continuous functions on 0 the proposed results are demonstrated by a numerical illustration

approximated convex envelope of a function the goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope such a problem arises in various physical situations where the function considered is some energy that has to be minimized this study is a first step toward understanding how to approximate the quasi convex envelope of a function the importance of this issue is due to the various applications that are encountered in particular in the field of material science 1994 copyright society for industrial and applied mathematics

exponential decay for the fragmentation or cell division equation abstract we consider a classical integro differential equation that arises in various applications as a model for cell division or fragmentation in biology it describes the evolution of the density of cells that grow and divide we prove the existence of a stable steady distribution first positive eigenvector under general assumptions in the variable coefficients case we also prove the exponential convergence for large times of solutions toward such a steady state

the appearance of microstructures in problems with incompatible wells and their numerical approach the goal of the paper is to analyze the creation of microstructure in problems of calculus of variations with wells more precisely we consider a case with strong incompatibility between the wells this forces the minimizing sequences to use other gradients than the wells in a puzzling way using a p 1 method we are then able to single out discrete minimizing sequences and to give energy estimates in terms of the mesh size

hill s class of compressible elastic materials and finite bending problems exact solutions in unified form abstract hill 1978 proposed a natural extension of hooke s law to finite deformations with all seth hill finite strains hill s natural extension presents a broad class of compressible hyperelastic materials over the whole deformation range we show that a number of known hookean type finite hyperelasticity models are included as particular cases in hill s class and that bell s and ericksen s constraints may be derived as natural consequences from hill s class subjected to internal constraints also we present a unified study of finite bending problems for elastic hill materials to date exact results are available for certain particular classes of compressible elastic materials which do not cover hill s class here with a novel idea of circumventing the strong nonlinearity we show that it is possible to derive exact solutions in unified form for the whole class of elastic hill materials reduced results are also given for cases subjected to internal constraints

finite element analysis of a class of stress free martensitic microstructures this work is concerned with the finite element approximation of a class of stress free martensitic microstructures modeled by multi well energy minimization finite element energy minimizing sequences are first constructed to obtain bounds on the minimum energy over all admissible finite element deformations a series of error estimates are then derived for finite element energy minimizers

exponential trend to equilibrium for discrete coagulation equations with strong fragmentation and without a balance condition the coagulation fragmentation equation describes the concentration f i t of particles of size i in nn 0 at time t geq 0 in a spatially homogeneous infinite system of particles subjected to coalescence and break up we show that when the rate of fragmentation is sufficiently stronger than that of coalescence f i t i in nn 0 tends to an unique equilibrium as t tends to infinity although we suppose that the initial datum is sufficiently small we do not assume a detailed balance or reversibility condition the rate of convergence we obtain is furthermore exponential

numerical approach to double well problems this paper deals with two sided numerical energy estimates of a solution to two dimensional double well problems these estimates are obtained by computing the energy appropriate for the polyconvex envelope and an upper estimate of the energy appropriate for the rank one convex envelope of the energy density examples of computed laminated microstructures are shown

nonconforming finite element approximation of crystalline microstructure we consider a class of nonconforming nite element approxima tions of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can un dergo either an orthorhombic to monoclinic double well or a cubic to tetrag onal triple well transformation we rst establish a series of error bounds in terms of elastic energies for the l 2 approximation of derivatives of the de formation in the direction tangential to parallel layers of the laminate for the l 2 approximation of the deformation for the weak approximation of the de formation gradient for the approximation of volume fractions of deformation gradients and for the approximation of nonlinear integrals of the deformation gradient we then use these bounds to give corresponding convergence rates for quasi optimal nite element approximations

numerical analysis of a nonconvex variational problem related to solid solid phase transitions the description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems in this paper a two dimensional model of such materials is studied due to the fact that the corresponding functional has two symmetry related martensitic energy wells the numerical approximation of the deformation gradient does not converge but tends to oscillate between the two wells as the size of the mesh is refined these oscillations may be interpreted in terms of microstructures using a nonconforming mathcal p 1 finite element an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient near one of the two martensitic wells

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

analysis of the finite element approximation of microstructure in micromagnetics the solution to nonconvex variational problems is often characterized by microstructure the variational problem for the magnetization field in micromagnetics has a nonconvex constraint and energy minimizing sequences of magnetization fields can have oscillations whose scale converges to zero but whose amplitude remains finite it is shown that the finite element approximation of the magnetization field for this variational problem does not converge pointwise as the mesh is refined but that nonlinear functions of the approximate magnetization fields converge weakly or equivalently all local spatial averages of nonlinear functions of the approximate magnetization field converge as the mesh is refined which implies the convergence as the mesh is refined of the probability distribution of the approximate magnetization fields in local spatial domains a norm is given to measure the convergence of this microstructure and we prove a rate of convergence in this norm

the discrete coagulation fragmentation equations existence uniqueness and density conservation the discrete coagulation fragmentation equation describes the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones these models have many applications in pure and applied science ranging from cluster formation in galaxies to the kinetics of phase transformations in binary alloys our results relate to existence uniqueness density conservation and continuous dependence and they generalise the corresponding results in ref 2 for the becker doring equations for which the processes are restricted to clusters gaining or shedding one particle examples are given which illustrate the role of the assumptions on the kinetic coefficients and show the rich set of analytic phenomena supported by the general discrete coagulation fragmentation equations

phase unwrapping of mr phase images using poisson equation the authors have developed a technique based on a solution of the poisson equation to unwrap the phase in magnetic resonance mr phase images the method is based on the assumption that the magnitude of the inter pixel phase change is less than spl pi per pixel therefore the authors obtain an estimate of the phase gradient by wrapping the gradient of the original phase image the problem is then to obtain the absolute phase given the estimate of the phase gradient the least squares ls solution to this problem is shown to be a solution of the poisson equation allowing the use of fast poisson solvers the absolute phase is then obtained by mapping the ls phase to the nearest multiple of 2 k from the measured phase the proposed technique is evaluated using mr phase images and is proven to be robust in the presence of noise an application of the proposed method to the 3 point dixon technique for water and fat separation is demonstrated

on the convergence of finite element approximations of a relaxed variational problem the accuracy of finite element approximations to a convex but not strictly convex variational problem is considered convergence is proved for a finite element approximation of a particular vector field related to the solution in a special one dimensional case 0 h convergence is shown for a piecewise linear approximation of the derivative h denotes the size of each element domain numerical results are also presented for this one dimensional case

linear programming approach to nonconvex variational problems in nonconvex optimization problems in particular in nonconvex variational problems there usually does not exist any classical solution but only generalized solutions which involve young measures in this paper after reviewing briefly the relaxation theory for such problems an iterative scheme leading to a sequential linear programming slp scheme is introduced and its convergence is proved by a banach fixed point technique then an approximation scheme is proposed and analyzed and calculations of an illustrative 2d broken extremal example are presented

phase local approximation phasela technique for phase unwrap from noisy data the local polynomial approximation lpa is a nonparametric regression technique with pointwise estimation in a sliding window we apply the lpa of the argument of cos and sin in order to estimate the absolute phase from noisy wrapped phase data using the intersection of confidence interval ici algorithm the window size is selected as adaptive pointwise varying this adaptation gives the phase estimate with the accuracy close to optimal in the mean squared sense for calculations we use a gauss newton recursive procedure initiated by the phase estimates obtained for the neighboring points it enables tracking properties of the algorithm and its ability to go beyond the principal interval pi pi and to reconstruct the absolute phase from wrapped phase observations even when the magnitude of the phase difference takes quite large values the algorithm demonstrates a very good accuracy of the phase reconstruction which on many occasion overcomes the accuracy of the state of the art algorithms developed for noisy phase unwrap the theoretical analysis produced for the accuracy of the pointwise estimates is used for justification of the ici adaptation algorithm

an estimation algorithm for 2 d polynomial phase signals we consider nonhomogeneous 2 d signals that can be represented by a constant modulus polynomial phase model a novel 2 d phase differencing operator is introduced and used to develop a computationally efficient estimation algorithm for the parameters of this model the operation of the algorithm is illustrated using an example

analysis of a class of nonconforming finite elements for crystalline microstructures an analysis is given for a class of nonconforming lagrange type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure these elements were first proposed and analyzed in 1992 by rannacher and turek for the stokes equation our analysis highlights the features of these elements which make them effective for the computation of microstructure new results for superconvergence and numerical quadrature are also given

adaptive approximation of young measure solutions in scalar nonconvex variational problems this paper addresses the numerical approximation of young measures appearing as generalized solutions to scalar nonconvex variational problems we prove a priori and a posteriori error estimates for a macroscopic quantity the stress for a scalar three well problem we show convergence of other quantities such as young measure support and microstructure region numerical experiments indicate that the computational effort in the solution of the large optimization problem is significantly reduced by using an adaptive mesh refinement strategy based on a posteriori error estimates in combination with an active set strategy due to carstensen and roubicek numer math 84 2000 pp 395 414

phase difference based multichannel phase unwrapping this work addresses the derivation of the phase difference based maximum likelihood ml phase unwrapping algorithm to this end we derive the joint statistics of the phase differences on a two dimensional grid for the multichannel case where several scaled wrapped phase values are available subsequently we determine and study the structure of the phase difference based ml estimator and compare it to known phase unwrapping techniques this work allows us to frame single and multichannel algorithms in a common formulation moreover among the known single channel phase difference based procedures we identify those attaining an ml solution we also show that multichannel phase difference based and recently proposed phase based ml algorithms achieve equivalent solutions

young measure approximation in micromagnetics modeling of micromagnetic phenomena typically faces the minimization of a non convex problem which gives rise to highly oscillatory magnetization structures mathematically this necessitates to extend the notion of lebesgue type solutions to young measure valued solutions the present work proposes and analyzes a conforming finite element method that is based on an active set strategy to compute efficiently discrete solutions of the generalized minimization problem computational experiments are given to show the efficiency of the scheme

finite element analysis of microstructure for the cubic to tetragonal transformation martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetry related rotationally invariant energy wells we give estimates for the numerical approximation of a first order laminate for such martensitic crystals we give bounds for the l2 convergence of directional derivatives in the twin plane for the l2 convergence of the deformation for the weak convergence of the deformation gradient for the convergence of the microstructure and for the convergence of nonlinear integrals of the deformation gradient

convergence of a reduced integration method for computing microstructures recent interest in materials which form microstructures has led to efforts to determine the material configurations computationally an approximation method using bilinear finite elements with a reduced integration scheme and a local postprocessing projection has been found in numerical experiments to have the flexibility to represent microstructures on various scales and for various orientations with respect to the grid and has produced results which agree with the theory in this paper we give a rigorous proof of the convergence of the microstructure for this method

numerical analysis of relaxed micromagnetics by penalised finite elements some micromagnetic phenomena in rigid ferro magnetic materials can be modelled by a non convex minimisation problem typically minimising sequences develop finer and finer oscillations and their weak limits do not attain the infimal energy solutions exist in a generalised sense and the observed microstructure can be described in terms of young measures a relaxation by convexifying the energy density resolves the essential macroscopic information the numerical analysis of the relaxed problem faces convex but degenerated energy functionals in a setting similar to mixed finite element formulations the lowest order conforming finite element schemes appear instable and nonconforming finite element methods are proposed an a priori and a posteriori error analysis is presented for a penalised version of the side restriction that the modulus of the magnetic field is bounded pointwise residual based adaptive algorithms are proposed and experimentally shown to be efficient

the becker doring cluster equations basic properties and asymptotic behaviour of solutions existence and uniqueness results are established for solutions to the becker doring cluster equations the density is shown to be a conserved quantity under hypotheses applying to a model of a quenched binary alloy the asymptotic behaviour of solutions with rapidly decaying initial data is determined denoting the set of equilibrium solutions byc 0 s the principal result is that if the initial density 0 s then the solution converges strongly toc o while if 0 s the solution converges weak toc s in the latter case the excess density 0 s corresponds to the formation of larger and larger clusters i e condensation the main tools for studying the asymptotic behaviour are the use of a lyapunov function with desirable continuity properties obtained from a known lyapunov function by the addition of a special multiple of the density and a maximum principle for solutions

approximation of infima in the calculus of variations abstract the goal of this paper is to give numerical estimates for some problems of the calculus of variations in the nonhomogeneous scalar case the stored energy function considered is then a function r n r we try to compare the infimum of the energy defined by on a sobolev space with the infimum of the same energy on a finite element space in terms of the mesh size