multiwavelets for second kind integral equations we consider a galerkin method for an elliptic pseudodifferential operator of order zero on a two dimensional manifold we use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis using this basis we can compress the stiffness matrix from n2 to o n log n nonzero entries and still obtain up to log n terms the same convergence rates as for the exact galerkin method

a numerical comparison of chebyshev methods for solving fourth order semilinear initial boundary value problems in solving semilinear initial boundary value problems with prescribed non periodic boundary conditions using implicit explicit and implicit time stepping schemes both the function and derivatives of the function may need to be computed accurately at each time step to determine the best chebyshev collocation method to do this the accuracy of the real space chebyshev differentiation spectral space preconditioned chebyshev tau real space chebyshev integration and spectral space chebyshev integration methods are compared in the l 2 and w 2 2 norms when solving linear fourth order boundary value problems and in the l 0 t l 2 and l 0 t w 2 2 norms when solving initial boundary value problems we find that the best chebyshev method to use for high resolution computations of solutions to initial boundary value problems is the spectral space chebyshev integration method which uses sparse matrix operations and has a computational cost comparable to fourier spectral discretization

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

error estimates for multivariate regression on discretized function spaces in this paper we will discuss the discretization error for the regression setting and derive error bounds relying on the approximation properties of the discretized space furthermore we will point out how the sampling error and the discretization error interact and how they can be balanced appropriately we will present two examples based on tensor product spaces sparse grids hyperbolic crosses which provide a suitable approach in the case of large sample sets in moderate dimensions

a fast sparse grid based space time boundary element method for the nonstationary heat equation this article presents a fast sparse grid based space time boundary element method for the solution of the nonstationary heat equation we make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation this integral equation is discretized by galerkin s method with respect to the sparse tensor product of the spatial and temporal ansatz spaces by employing the mathcal h matrix and toeplitz structure of the resulting discretized operators we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization nevertheless the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

a piecewise constant algorithm for weighted l 1 approximation over bounded or unbounded regions in r s using smolyak s construction s a smolyak dokl akad nauk sssr 4 1963 pp 240 243 we derive a new algorithm for approximating multivariate functions over bounded or unbounded regions in r s with the error measured in a weighted l1 norm we provide upper bounds for the algorithm s cost and error for a class of functions whose mixed first order partial derivatives are bounded in the l1 norm in particular we prove that the error and the cost measured in terms of the number of function evaluations satisfy the relation error le frac s exp big frac1 12 s 1 big s 1 pi left frac e ln mbox cost s 1 sqrt 2 ln 2 right 2 s 1 frac1 cost whenever the cost is sufficiently large relative to the number s of variables more specifically the inequality holds when q ge2 s 1 where q is a special parameter defining the refinement level in smolyak s algorithm and hence the number of function evaluations used by the algorithm we also discuss extensions of the results to the spaces with the derivatives bounded in lp norms

numerical approximations to a new phase field model for two phase flows of complex fluids abstract we derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized onsager principle a novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures through a delicate explicit implicit numerical discretization we develop a decoupled linear scheme for a simplified version of the phase field model as well as a coupled nonlinear scheme for the full model both schemes are shown as unconditionally energy stable with consistent discrete dissipative energy laws several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it

space time approximation with sparse grids in this article we introduce approximation spaces especially suited for the approximation of solutions of parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time proper truncation then leads to so called space time sparse grid spaces for a uniform discretization of the spatial space of dimension d with o nd degrees of freedom these spaces involve for d 1 also only o nd degrees of freedom for the discretization of the whole space time problem but they provide the same approximation rate as classical space time finite element spaces which need o nd 1 degrees of freedoms this makes these approximation spaces well suited for conventional parabolic and time dependent optimization problems we analyze the approximation properties and the dimension of these sparse grid space time spaces for general stable multiscale bases we then restrict ourselves to an interpolatory multiscale basis i e a hierarchical basis here to be able to handle also complicated spatial domains omega we construct the hierarchical basis from a given spatial finite element basis as follows first we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method then we derive interpolatory prolongation operators between the respective coarse and fine grid points by a least squares approach this way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our space time sparse grid approach we give numerical results on the convergence rate of the interpolation error of these spaces for various space time problems with two spatial dimensions implementational issues data structures and questions of adaptivity also are addressed to some extent

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

an implicit midpoint spectral approximation of nonlocal cahn hilliard equations the paper is concerned with the convergence analysis of a numerical method for nonlocal cahn hilliard equations the temporal discretization is based on the implicit midpoint rule and a fourier spectral discretization is used with respect to the spatial variables the sequence of numerical approximations in shown to be bounded in various norms uniformly with respect to the discretization parameters and optimal order bounds on the global error of the scheme are derived the uniform bounds on the sequence of numerical solutions as well as the error bounds hold unconditionally in the sense that no restriction on the size of the time step in terms of the spatial discretization parameter needs to be assumed

on the computation of the eigenproblems of hydrogen helium in strong magnetic and electric fields with the sparse grid combination technique abstract we introduce the combination technique for the numerical solution of d dimensional eigenproblems on sparse grids here o d log n d 1 different problems each of size o n have to be solved independently this is in contrast to the one problem of size o n d for a conventional finite element discretization where n denotes the number of grid points in one coordinate direction therefore also higher dimensional eigenvalue problems can be treated by our sparse grid combination approach we apply this method to solve the three dimensional schrodinger equation for hydrogen one electron problem and the six dimensional schrodinger equation for helium two electron problem in strong magnetic and electric fields

unconditionally energy stable large time stepping method for the l2 gradient flow based ternary phase field model with precise nonlocal volume conservation abstract we consider numerical approximations for solving a new allen cahn type ternary phase field model we first formulate a phase field model from an energetic variational formulation based on the l 2 gradient flow and add three nonlocal lagrange multipliers to the system to conserve the volume for each phase then by combining the sav approach with the stabilization technique where a crucial linear stabilization term is added to enhance the stability and keep the required accuracy thus allowing large time steps we arrive at a decoupled linear non iterative and volume conserved scheme this scheme is unconditionally energy stable and requires solving only four linear second order elliptic equations with constant coefficients we further prove energy stability and present numerous 2d and 3d numerical simulations to show accuracy and stability

the fourier collocation method for the cahn hilliard equation abstract in this paper a fourier collocation method for numerically solving cahn hilliard equations with periodic boundary conditions is developed we establish their semidiscrete and fully discrete schemes that inherit the energy dissipation property and mass conservation property from the associated continuous problem we prove existence and uniqueness of the numerical solution and derive the optimal error bounds we perform some numerical experiments which confirm our results

a fast collocation method for solving stochastic integral equations based on sparse grid multiscale piecewise polynomial bases we develop a fast collocation method for solving fredholm integral equations of the second kind with stochastic loading terms it is proved that the proposed method preserves the optimal rate of convergence and has linear up to a logarithmic factor computational complexity the reduction in computational costs come from three aspects the adoption of the collocation principle the use of the sparse grid for the solution space and the employment of a truncation strategy for the coefficient matrix of the resulting linear system numerical experiments confirm the theoretical results on approximation accuracy and computational efficiency of the proposed method

sparse grids for boundary integral equations the potential of sparse grid discretizations for solving boundary integral equations is studied for the screen problem on a square in mathbb r 3 theoretical and numerical results on approximation rates preconditioning adaptivity and compression for piecewise constant and linear sparse grid spaces are obtained

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

time dependent hermite galerkin spectral method and its applications a time dependent hermite galerkin spectral method thgsm is investigated in this paper for the nonlinear convection diffusion equations in the unbounded domains the time dependent scaling factor and translating factor are introduced in the definition of the generalized hermite functions ghf as a consequence the thgsm based on these ghf has many advantages not only in theoretical proofs but also in numerical implementations the stability and spectral convergence of our proposed method have been established in this paper the korteweg de vries burgers kdvb equation and its special cases including the heat equation and the burgers equation as the examples have been numerically solved by our method the numerical results are presented and it surpasses the existing methods in accuracy our theoretical proof of the spectral convergence has been supported by the numerical results

spectral approximation of pattern forming nonlinear evolution equations with double well potentials of quadratic growth this paper is concerned with the analysis of a numerical algorithm for the approximate solution of a class of nonlinear evolution problems that arise as l 2 gradient flow for the modica mortola regularization of the functional v bv t d 1 1 e v 2 td v 1 2 k d k v k 2 here is the interfacial energy per unit length or unit area t d is the flat torus in d and is a nonnegative fourier multiplier that is continuous on d symmetric in the sense that for all d and that decays to zero at infinity such functionals feature in mathematical models of pattern formation in micromagnetics and models of diblock copolymers the resulting evolution equation is discretized by a fourier spectral method with respect to the spatial variables and a modified crank nicolson scheme in time optimal order a priori bounds are derived on the global error in the l 0 t l 2 t d norm

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed

sparse tensor spherical harmonics approximation in radiative transfer the stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five dimensional phase space to obtain a well posed problem boundary conditions have to be prescribed on the inflow part of the domain boundary we solve the equation with a multi level galerkin fem in physical space and a spectral discretization with harmonics in solid angle and show that the benefits of the concept of sparse tensor products known from the context of sparse grids can also be leveraged in combination with a spectral discretization our method allows us to include high spectral orders without incurring the curse of dimension of a five dimensional computational domain neglecting boundary conditions we find analytically that for smooth solutions the convergence rate of the full tensor product method is retained in our method up to a logarithmic factor while the number of degrees of freedom grows essentially only as fast as for the purely spatial problem for the case with boundary conditions we propose a splitting of the physical function space and a conforming tensorization numerical experiments in two physical and one angular dimension show evidence for the theoretical convergence rates to hold in the latter case as well

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

a wavelet based energetic approach for the analysis of biomedical signals application to the electroencephalogram and electro oculogram wavelet transform has emerged over recent years as a favoured tool for the investigation of biomedical signals which are highly non stationary by their nature a relevant wavelet based approach in the analysis of biomedical signals exploits the capability of wavelet transform to separate the signal energy among different frequency bands i e different scales realizing a good compromise between temporal and frequency resolution the rationale of this paper is twofold i to present a mathematical formalization of energy calculation from wavelet coefficients in order to obtain uniformly time distributed atoms of energy across all the scales ii to show two different applications of the wavelet based energetic approach to biomedical signals one application concerns the study of epileptic brain electrical activity with the aim of identifying typical patterns of energy redistribution during the seizure results obtained from this method provide interesting indications on the complex spatio temporal dynamics of the seizure the other application concerns the electro oculographic tracings with the purpose of realizing an automatic detector of a particular type of eye movements slow eye movements important to identify sleep phases the algorithm is able to identify this eye movement pattern efficiently characterizing it in rigorous energetic terms the energetic approach built within the framework of the multiresolution decomposition appears as a powerful and versatile tool for the investigation and characterization of transient events in biomedical signals

an energy stable linear diffusive crank nicolson scheme for the cahn hilliard gradient flow abstract we propose and analyze a linearly stabilized semi implicit diffusive crank nicolson scheme for the cahn hilliard gradient flow in this scheme the nonlinear bulk force is treated explicitly with two second order stabilization terms this treatment leads to linear elliptic system with constant coefficients and provable discrete energy dissipation rigorous error analysis is carried out for the fully discrete scheme when the time step size and the space step size are small enough second order accuracy in time is obtained with a prefactor controlled by some lower degree polynomial of 1 e here e is the thickness of the interface numerical results together with an adaptive time stepping are presented to verify the accuracy and efficiency of the proposed scheme

two scale boolean galerkin discretizations for fredholm integral equations of the second kind in this paper some two scale boolean galerkin discretizations are proposed and analyzed for a class of fredholm integral equations of the second kind in multidimensions it is shown by both theory and numerics that this type of multiscale discretization algorithm not only significantly reduces the number of degrees of freedom but also produces very accurate approximations

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

the h p boundary element method for solving 2 and 3 dimensional problems abstract in this survey paper we report on recent developments of the hp version of the boundary element method we consider weakly singular and hypersingular integral equations on polygons and polyhedral surfaces we show that the galerkin solutions computed with the hp version on geometric meshes converge exponentially fast towards the exact solutions of the integral equations the implementation of the hp version of the boundary element method is discussed and adaptive algorithms are given model problems for the coupling of finite elements and boundary elements are also considered numerical results are presented which underline the theoretical results

multiphase allen cahn and cahn hilliard models and their discretizations with the effect of pairwise surface tensions in this paper the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an n component mixture are studied for the general choice of phase variables the unisolvent property of the coefficient matrix involved in the n phase models based on the pairwise surface tensions is established moreover the symmetric positive definite property of the coefficient matrix on an n 1 n 1 dimensional hyperplane which is of fundamental importance to the well posedness of the models can be proved equivalent to some physical condition for pairwise surface tensions the n phase allen cahn and n phase cahn hilliard equations can then be derived from the free energy functional a natural property is that the resulting dynamics of concentrations are independent of phase variables chosen finite element discretizations for n phase models can be obtained as a natural extension of the existing discretizations for the two phase model the discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

on additive schwarz preconditioners for sparse grid discretizations based on the framework of subspace splitting and the additive schwarz scheme we give bounds for the condition number of multilevel preconditioners for sparse grid discretizations of elliptic model problems for a bxp like preconditioner we derive an estimate of the optimal ordero 1 and for a hb like variant we obtain an estimate of the ordero k 2 2 k 2 wherek denotes the number of levels employed furthermore we confirm these results by numerically computed condition numbers

efficient energy stable numerical schemes for a phase field moving contact line model in this paper we present two efficient energy stable schemes to solve a phase field model incorporating moving contact line the model is a coupled system that consists of incompressible navier stokes equations with a generalized navier boundary condition and cahn hilliard equation in conserved form in both schemes the projection method is used to deal with the navier stokes equations and stabilization approach is used for the non convex ginzburg landau bulk potential by some subtle explicit implicit treatments we obtain a linear coupled energy stable scheme for systems with dynamic contact line conditions and a linear decoupled energy stable scheme for systems with static contact line conditions an efficient spectral galerkin spatial discretization method is implemented to verify the accuracy and efficiency of proposed schemes numerical results show that the proposed schemes are very efficient and accurate we proposed two linear energy stable numerical schemes for a phase field moving contact line model discrete energy laws that are consistent to the continuous ones are proved the efficiency and fast convergence of the proposed schemes with spectral spatial approximations are numerically verified

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

decoupled energy stable schemes for phase field models of two phase complex fluids we consider in this paper numerical approximations of phase field models for two phase complex fluids we first reformulate the phase field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws then we construct two classes stabilized and convex splitting of decoupled time discretization schemes for the coupled nonlinear systems these schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step furthermore these elliptic equations are linear for the stabilized version stability analysis and ample numerical simulations are presented

convergence and error analysis for the scalar auxiliary variable sav schemes to gradient flows we carry out convergence and error analysis of the scalar auxiliary variable sav methods for l 2 and h 1 gradient flows with a typical form of free energy we first derive h 2 bounds under certain assumptions suitable for both the gradient flows and the sav schemes which allow us to establish the convergence of the sav schemes under mild conditions we then derive error estimates with further regularity assumptions we also discuss several other gradient flows which cannot be cast in the general framework used in this paper but for which convergence and error analysis can still be established using a similar procedure

sparse adaptive finite elements for radiative transfer the linear radiative transfer equation a partial differential equation for the radiation intensity u x s with independent variables email protected email protected r n in the physical domain d of dimension n 2 3 and angular variable email protected s 2 email protected r 3 y 1 is solved in the n 2 dimensional computational domain dxs 2 we propose an adaptive multilevel galerkin finite element method fem for its numerical solution our approach is based on a a stabilized variational formulation of the transport operator b on so called sparse tensor products of two hierarchic families of finite element spaces in h 1 d and in l 2 s 2 respectively and c on wavelet thresholding techniques to adapt the discretization to the underlying problem an a priori error analysis shows under strong regularity assumptions on the solution that the sparse tensor product method is clearly superior to a discrete ordinates method as it converges with essentially optimal asymptotic rates while its complexity grows essentially only as that for a linear transport problem in r n numerical experiments for n 2 on a set of example problems agree with the convergence and complexity analysis of the method and show that introducing adaptivity can improve performance in terms of accuracy vs number of degrees even further

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

numerical approximations for a phase field moving contact line model with variable densities and viscosities we consider the numerical approximations of a two phase hydrodynamics coupled phase field model that incorporates the variable densities viscosities and moving contact line boundary conditions the model is a nonlinear coupled system that consists of incompressible navierstokes equations with the generalized navier boundary condition and the cahnhilliard equations with moving contact line boundary conditions by some subtle explicitimplicit treatments to nonlinear terms we develop two efficient unconditionally energy stable numerical schemes in particular a linear decoupled energy stable scheme for the system with static contact line condition and a nonlinear energy stable scheme for the system with dynamic contact line condition an efficient spectral galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes various numerical results show that the proposed schemes are efficient and accurate

on the construction of sparse tensor product spaces let 1 rn1 and 2 rn2 be two given domains and consider on each domain a multiscale sequence of ansatz spaces of polynomial exactness r1 and r2 respectively in this paper we study the optimal construction of sparse tensor products made from these spaces in particular we derive the resulting cost complexities to approximate functions with anisotropic and isotropic smoothness on the tensor product domain 1 2 numerical results validate our theoretical findings

rational von neumann lattices a rational von neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers zeros of harmonic oscillator functions in the kq representation on such lattices are found it is shown that the number of zeros of the kq function determines the number of states by which a set on a von neumann lattice is overcomplete interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation

optimized general sparse grid approximation spaces for operator equations this paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order based on the framework of tensor product biorthogonal wavelet bases and stable subspace splittings we construct operator adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces provided that certain additional regularity assumptions on the solution are fulfilled specifically for operators of positive order their dimension is o 2 j independent of the dimension n of the problem compared to o 2 jn for the full grid space also for operators of negative order the overall cost is significantly in favor of the new approximation spaces we give cost estimates for the case of continuous linear information we show these results in a constructive manner by proposing a galerkin method together with optimal preconditioning the theory covers elliptic boundary value problems as well as boundary integral equations

adaptive multilevel bem for acoustic scattering we study the h and p versions of the galerkin boundary element method for integral equations of the first kind in 2d and 3d which result from the scattering of time harmonic acoustic waves at hard or soft scatterers we derive an abstract a posteriori error estimate for indefinite problems which is based on stable multilevel decompositions of our test and trial spaces the galerkin error is estimated by easily computable local error indicators and an adaptive algorithm for h or p adaptivity is formulated the theoretical results are illustrated by numerical examples for hard and soft scatterers in 2d and 3d

integral operators on sparse grids in this paper we are concerned with the construction and use of wavelet approximation spaces for the fast evaluation of integral expressions the spaces are based on biorthogonal anisotropic tensor product wavelets we introduce sparse grid hyperbolic cross approximation spaces which are adapted not only to the smoothness of the kernel but also to the norm in which the error is measured furthermore we introduce compression schemes for the corresponding discretizations numerical examples for the laplace equation with dirichlet boundary conditions and an additional integral term with a smooth kernel demonstrate the validity of our theoretical results