imposing boundary and interface conditions in multi resolution wavelet galerkin method for numerical solution of helmholtz problems abstract in this paper we propose a new formulation for numerical solution of boundary value problems using mesh free multi resolution wavelet galerkin wg method a difficulty with the wg method is imposing boundary condition constraints single scale wavelet basis functions together with a jump function approach using cubic polynomial functions have been used in solving some boundary problems in the literature ramp and cubic polynomial functions have been also used in the jump function approach described in element free galerkin context and it is proven ramp functions exactly satisfy the desirable property for the derivative of jump but the cubic polynomial functions do not in this paper a multi resolution wg method is employed and both ramp and cubic polynomial jump functions are applied obtained results in one dimension are compared with analytical solutions results obtained with ramp jump functions in two dimensions are also reported the results obtained with the wg method manifest boundary and interface conditions are accurately imposed when the ramp jump functions are applied numerical results are in good agreement with accurate solutions

a general fictitious domain method with immersed jumps and multilevel nested structured meshes this study addresses a new fictitious domain method for elliptic problems in order to handle general and possibly mixed embedded boundary conditions e b c robin neumann and dirichlet conditions on an immersed interface the main interest of this fictitious domain method is to use simple structured meshes possibly uniform cartesian nested grids which do not generally fit the interface but define an approximate one a cell centered finite volume scheme with a non conforming structured mesh is derived to solve the set of equations with additional algebraic transmission conditions linking both flux and solution jumps through the immersed approximate interface hence a local correction is devised to take account of the relative surface ratios in each control volume for the robin or neumann boundary condition then the numerical scheme conserves the first order accuracy with respect to the mesh step this opens the way to combine the e b c method with a multilevel mesh refinement solver to increase the precision in the vicinity of the interface such a fictitious domain method is very efficient the l 2 and l norm errors vary like o h l where h l is the grid step of the finest refinement level around the interface until the residual first order discretization error of the non refined zone is reached the numerical results reported here for convection diffusion problems with dirichlet robin and mixed dirichlet and robin boundary conditions confirm the expected accuracy as well as the performances of the present method

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

on the nature of fermi golden rule for open quantum systems we consider a general class of models consisting of a small quantum system s interacting with a reservoir r we compare three applications of 2nd order perturbation theory the fermi golden rule to the study of such models 1 the van hove weak coupling limit for the dynamics reduced to s 2 the fermi golden rule applied to the liouvillean an argument that was used in recent papers on the return to equilibrium 3 the fermi golden rule applied to the so called c liouvillean these three applications lead to three level shift operators as our main result we prove that if the reservoir r is thermal if it satisfies the kms condition then the level shift operator obtained in 1 often called the davies generator and the level shift operator constructed in 2 are connected by a similarity transformation we also show that the davies generator coincides with the level shift operator obtained in 3 for a general r

geometric and measure theoretical structures of maps with mostly contracting center we show that every diffeomorphism with mostly contracting center direction exhibits a geometric combinatorial structure which we call skeleton that determines the number basins and supports of the physical measures furthermore the skeleton allows us to describe how the physical measures bifurcate as the diffeomorphism changes in particular we use this to construct examples with any given number of physical measures with basins densely intermingled and to analyse how these measures collapse into each other through explosions of their basins as the dynamics varies this theory also allows us to prove that in the absence of collapses the basins are continuous functions of the diffeomorphism

wavelet solutions for the dirichlet problem a modified classical penalty method for solving a dirichlet boundary value problem is presented this new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains it is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast approximate calculation of boundary integral the compact support and orthonormality of the basis are essential for representing the boundary measure numerically and therefore for implementing this methodology

ergodic theory of generic continuous maps we study the ergodic properties of generic continuous dynamical systems on compact manifolds as a main result we prove that generic homeomorphisms have convergent birkhoff averages under continuous observables at lebesgue almost every point in spite of this when the underlying manifold has dimension greater than one generic homeomorphisms have no physical measures a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics similar results hold for generic continuous maps

wavelet and finite element solutions for the neumann problem using fictitious domains this paper presents a new fictitious domain formulation for the solution of a strongly elliptic boundary value problem with neumann boundary conditions for a bounded domain in a finite dimensional euclidean space with a smooth possibly only lipschitz boundary this extends the domain to a larger rectangular domain with periodic boundary conditions for which fast solvers are available the extended solution converges on the original domain in the appropriate function spaces as the penalty parameter approaches zero both wavelet galerkin and finite elements numerical approximation schemes are developed using this methodology the convergence rates of both schemes are comparable and the use of finite elements requires a parameterization of the boundary while the wavelet galerkin method admits an implicit description of the boundary in terms of a wavelet representation of the boundary measure defined as the distributional gradient of the characteristic function of the interior the accuracy of both methods is investigated and compared both theoretically and for numerical test cases the conclusion is that the methods are comparable and that the wavelet method allows the use of more general boundaries which are not explicitly parametrized which would be of greater advantage in higher dimensions the numerical tests are carried out in two dimensions

distributed lagrange multiplier methods for incompressible viscous flow around moving rigid bodies abstract in this article we discuss the application of a distributed lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modelled by the navier stokes equations around moving bodies we suppose the rigid bodies motion known a priori the solution method combines finite element approximations time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method numerical experiment results for two dimensional flow around a moving disk are presented

renormalization group analysis of spectral problems in quantum field theory abstract in this paper we present a self contained and detailed exposition of the new renormalization group technique proposed in 1 2 its main feature is that the renormalization group transformation acts directly on a space of operators rather than on objects such as a propagator the partition function or correlation functions we apply this renormalization transformation to a hamiltonian describing the physics of an atom interacting with the quantized electromagnetic field and we prove that excited atomic states turn into resonances when the coupling between electrons and field is nonvanishing

brain tumor interaction biophysical models for medical image registration state of the art algorithms for deformable image registration are based on the minimization of an image similarity functional that is regularized by adding a penalty term on the deformation map the penalty function typically represents a smoothness regularization in this article we use a constrained optimization formulation in which the image similarity functional is coupled to a biophysical model this formulation is pertinent when the data have been generated by imaging tissue that undergoes deformations due to an actual biophysical phenomenon such is the case of coregistering tumor bearing brain images from the same individual we present an approximate model that couples tumor growth with the mechanical deformations of the surrounding brain tissue we consider primary brain tumors in particular gliomas glioma growth is modeled by a reaction advection diffusion pde with a two way coupling with the underlying tissue elastic deformation tumor bulk infiltration and subsequent mass effects are not regarded separately but are captured by the model itself in the course of its evolution our formulation allows for updating the tumor diffusion coefficient following structural displacements caused by tumor growth infiltration our forward problem implementation builds on the petsc library of argonne national laboratory our reformulation results in a very small parameter space and we use the derivative free optimization library appspack of sandia national laboratories we test the forward model and the optimization framework by using landmark based similarity functions and by applying it to brain tumor data from clinical and animal studies state of the art registration algorithms fail in such problems due to excessive deformations we compare our results with previous work in our group and we observed up to 50 improvement in landmark deformation prediction we present preliminary validation results in which we were able to reconstruct deformation fields using four degrees of freedom our study demonstrates the validity of our formulation and points to the need for richer datasets and fast optimization algorithms

spectral theory of pauli fierz operators abstract we study spectral properties of pauli fierz operators which are commonly used to describe the interaction of a small quantum system with a bosonic free field we give precise estimates of the location and multiplicity of the singular spectrum of such operators applications of these estimates which will be discussed elsewhere concern spectral and ergodic theory of non relativistic qed our proof has two ingredients the feshbach method which is developed in an abstract framework and mourre theory applied to the operator restricted to the sector orthogonal to the vacuum

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

wavelets and the numerical solution of partial differential equations we present a new numerical method for the solution of partial differential equations in nonseparable domains the method uses a wavelet galerkin solver with a nontrivial adaptation of the standard capacitance matrix method the numerical solutions exhibit spectral convergence with regard to the order of the compactly supported daubechies wavelet basis furthermore the rate of convergence is found to be independent of the geometry we solve the helmholtz equation since for variations in the parameter the solutions have qualitative properties that well illustrate the applications of our method

triangular wavelet based finite elements via multivalued scaling equations this paper develops a class of triangular finite elements based on l2 rn orthonormal compactly supported wavelets as in the case of recently derived tensor product wavelet elements the triangular elements are defined by deriving an algebraic eigenvalue problem characterizing the entries of the elemental matrices these entries are uniquely defined by normalization conditions derived from the polynomial reproducing properties of the wavelet scaling functions the work herein generalizes the development of the tensor product wavelet elements in that the geometry of the triangular elemental domain is characterized using a multivalued scaling equation the multivalued scaling function is used to represent the self similar tiling of rn onto itself by triangular domains

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

parametrizing smooth compactly supported wavelets in this paper a concrete parameter space for the compactly supported wavelet systems of daubechies is constructed for wavelet systems with n generic nonvanishing coefficients the parameter space is a closed convex set in r n 2 2 which can be explicitly described in the fourier transform domain the moment free wavelet systems are subsets obtained by the intersection of the parameter space and an affine subspace of r n 2 2

adaptive multiscale wavelet galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization the multiscale galerkin formulation of two dimensional elasticity problems is presented for easy interpolation and boundary handlings as well as efficient adaptive analysis two dimensional interpolation wavelets are used as the multiscale trial functions in the galerkin formulation after the validity of the present multiscale adaptive method is verified with some benchmark problems the present wavelet based method is applied to the multiscale topology optimization that progresses design resolution levels dyadically from low to high levels by this application we show the potential of the multiscale method and the possibility of developing a fully integrated analysis and topology design optimization in the multiscale multiresolution setting

partial hyperbolicity lyapunov exponents and stable ergodicity we present some results and open problems about stable ergodicity of partially hyperbolic diffeomorphisms with non zero lyapunov exponents the main tool is local ergodicity theory for non uniformly hyperbolic systems

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

robustly topological mixing of kan s map abstract in 1994 i kan constructed a smooth map on the annulus admitting two physical measures whose basins are intermingled in this paper we prove that kan s map is c 2 robustly topologically mixing

analysis of finite element domain embedding methods for curved domains using uniform grids we analyze the error of a finite element domain embedding method for elliptic equations on a domain omega with curved boundary the domain is embedded in a rectangle r on which uniform mesh and linear continuous elements are employed the numerical scheme is based on an extension of the differential equation from omega to r by regularization with a small parameter epsilon for neumann and robin problems or penalty with a large parameter epsilon 1 for the dirichlet problem or a mixture of these for a mixed boundary value problem for neumann and robin problems we prove that when epsilon le h the mesh size the error in the h 1 omega norm is of the optimal order mathcal o h for the dirichlet problem when epsilon le h 1 2 the error is mathcal o h 1 2 that is not optimal if the mesh is adjusted around partial omega to fit it then the optimal convergence rate mathcal o h holds for the dirichlet problem if omega is convex and epsilon le h if omega is not convex then the convergence rate can only be improved to mathcal o h 2 3 by such mesh adjustment with the parameter being epsilon h 2 3 in this latter case a parameter smaller than h 2 3 thwarts the convergence rate which is verified by a numerical result

fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains abstract a tensor based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains it provides an accurate solution which is explicit with respect to parameters defining the shape of the domain thus allowing efficient a posteriori probabilistic or parametric analyses in the proposed method a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain yielding a weak formulation in a tensor product space product of space functions and parametric functions the paper is limited to the case of neumann conditions on uncertain parts of the boundary the proper generalized decomposition method is then introduced for the construction of a tensor product approximation separated representation of the solution it can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution this tensor based method is made computationally tractable by introducing separated representations of variational forms resulting from separated representations of the parameterized indicator function of the uncertain domain for this purpose a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well posedness of problems associated with approximate indicator functions moreover a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations

robust ergodic properties in partially hyperbolic dynamics we study ergodic properties of partially hyperbolic systems whose central direction is mostly contracting earlier work of bonatti and viana 2000 about existence and finitude of physical measures is extended to the case of local diffeomorphisms moreover we prove that such systems constitute a c 2 open set in which statistical stability is a dense property in contrast all mostly contracting systems are shown to be stable under small random perturbations

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

the computation of wavelet galerkin approximation on a bounded interval this paper describes exact evaluations of various finite integrals whose integrands involve products of daubechies compactly supported wavelets and their derivatives and or integrals these finite integrals play an essential role in the wavelet galerkin approximation of differential or integral equations on a bounded interval

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

weak material approximation of holes with traction free boundaries consider the solution of a boundary value problem for steady linear elasticity in which the computational domain contains one or several holes with traction free boundaries the presence of holes in the material can be approximated using a weak material that is the relative density of material rho is set to 0 epsilon rho ll1 in the hole region the weak material approach is a standard technique in the so called material distribution approach to topology optimization in which the inhomogeneous relative density of material is designated as the design variable in order to optimize the spatial distribution of material the use of a weak material ensures that the elasticity problem is uniquely solvable for each admissible value rho in epsilon 1 of the design variable a finite element approximation of the boundary value problem in which the weak material approximation is used in the hole regions can be viewed as a nonconforming but convergent approximation of a version of the original problem i

wavelet calculus and finite difference operators this paper shows that the naturally induced discrete differentiation operators induced from a wavelet galerkin finite dimensional approximation to a standard function space approximates differentiation with an error of order 0 h2d 2 where d is the degree of the wavelet system the degree of a wavelet system is defined as one less than the degree of the lowest order nonvanishing moment of the fundamental wavelet we consider in this paper compactly supported wavelets of the type introduced by daubechies in 1988 the induced differentiation operators are described in terms of connection coefficients which are intrinsically defined functional invariants of the wavelet system defined as l2 inner products of derivatives of wavelet basis functions with the basis functions themselves these connection coefficients can be explicitly computed without quadrature and they themselves have key moment vanishing properties proved in this paper which are dependent upon the degree of the wavelet system this is the basis for the proof of the principal results concerning the degree of approximation of the differentiation operator by the wavelet galerkin discrete differentiation operator

a fictitious domain approach with spread interface for elliptic problems with general boundary conditions the aim of this article is to solve second order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach the main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain as the spatial discretization is being performed in the fictitious domain this method allows the use of structured meshes the discretization is not boundary fitted to the original physical domain this paper describes several ways to impose dirichlet robin or neumann boundary conditions on a spread immersed interface without locally modifying the numerical scheme and without using lagrange multipliers the numerical applications focus on difiusion and convection problems in the unit disk with dirichlet or robin boundary conditions for such problems analytical solutions can be determined for a correctly chosen source term the numerical resolution is performed using a q1 finite element scheme the spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary the l2 norm of the errors is computed in order to evaluate the capability of the method immersed boundaries are found in many industrial applications like two phase flow simulations fluid structure interaction etc this article represents a first step towards the simulation of these kinds of applications

stationary van hove limit the weak coupling van hove limit of one parameter groups of contractions is studied by the stationary approach we show that the resolvent of the properly renormalized and rescaled generator of a contractive semigroup has a limit as the coupling constant goes to zero this limit is the resolvent of the generator of a certain contractive semigroup our results can be viewed as a stationary counterpart to the well known results about the weak coupling limit obtained by the time dependent approach due to davies we compare both approaches

element free galerkin solutions for helmholtz problems fomulation and numerical assessment of the pollution effect abstract the element free galerkin method efgm a particular case of the meshless methods is examined in its application to acoustic wave propagation addressed by the helmholtz equation dispersion and pollution effects two problems encountered by the classical numerical methods are reviewed numerical tests on two dimensional problems focus on the parameters governing the formulation of the efgm they also demonstrate that the efgm is affected by dispersion and pollution effects as well as fem but these effects are rather low showing that the efgm is a promising method

solution of cartesian and curvilinear quantum equations via multiwavelets on the interval abstract it is shown how orthogonal compact support multiwavelets may be used for the solution of quantum mechanical eigenvalue problems subject to specific boundary conditions special scaling functions and wavelets with convenient limiting behaviors at the edges of an interval are constructed in analogy to earlier work on single wavelet families all of the integrals required for hamiltonian matrix elements involving both regular and edge functions are calculated efficiently through use of recursion and quadrature methods it is demonstrated through accurate eigenvalue determination that both cartesian and curvilinear degrees of freedom are readily accommodated with such a basis using as examples the particle in a box and the hydrogen atom in spherical polar coordinates

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd