Adaptive Wavelet Galerkin Research Articles

HT-AWGM: a hierarchical Tucker–adaptive wavelet Galerkin method for high-dimensional elliptic problems

This paper is concerned with the construction, analysis, and realization of a numerical method to approximate the solution of high-dimensional elliptic partial differential equations. We propose a new combination of an adaptive wavelet Galerkin method (AWGM) and the well-known hierarchical tensor (HT) format. The arising HT-AWGM is adaptive both in the wavelet representation of the low-dimensional factors and in the tensor rank of the HT representation. The point of departure is an adaptive wavelet method for the HT format using approximate Richardson iterations and an AWGM for elliptic problems. HT-AWGM performs a sequence of Galerkin solves based upon a truncated preconditioned conjugate gradient (PCG) algorithm in combination with a tensor-based preconditioner. Our analysis starts by showing convergence of the truncated conjugate gradient method. The next step is to add routines realizing the adaptive refinement. The resulting HT-AWGM is analyzed concerning convergence and complexity. We show that the performance of the scheme asymptotically depends only on the desired tolerance with convergence rates depending on the Besov regularity of low-dimensional quantities and the low-rank tensor structure of the solution. The complexity in the ranks is algebraic with powers of four stemming from the complexity of the tensor truncation. Numerical experiments show the quantitative performance.

An adaptive wavelet Galerkin scheme for solving contact problems based on elliptic variational inequalities of the first kind

PurposeThe real challenge in the solution of contact problems is the lack of an optimal adaptive scheme. As the contact zone is a priori unknown, successive refinement and iterative method are necessary to obtain a high-accuracy solution. The purpose of this paper is to provide an optimal adaptive scheme based on second-generation finite element wavelets for the solution of non-linear variational inequality of the contact problem.Design/methodology/approachTo generate an elementary multi-resolution mesh, the authors used hierarchical bases (HB) composed of Lagrange finite element interpolation functions. These HB functions are customized using second-generation wavelet techniques for a fast convergence rate. At each step of the algorithm, the active set method along with mesh adaptation is used for solving the constrained minimization problem of contact case. Wavelet coefficients-based error indicators are used, and computation is focused on mesh zones with a high error indication. The authors take advantage of the wavelet transform to develop a parameter-free adaptive scheme to generate an appropriate and optimal mesh.FindingsAdaptive wavelet Galerkin scheme (AWGS), a newly developed method for multi-scale mesh adaptivity in this work, is a combination of the second-generation wavelet transform and finite element method and significantly improves the accuracy of the results without approximating an additional problem of error estimation equations. A comparative study is performed taking a solution on a highly refined mesh and results are generated using AWGS.Practical implicationsThe proposed adaptive technique can be utilized in the simulation of mechanical and biomechanical structures where multiple bodies come into contact with each other. The algorithm of the method is easy to implement and found to be successful in producing a sufficiently accurate solution with relatively less number of mesh nodes.Originality/valueAlthough many error estimation techniques have been developed over the past several years to solve contact problems adaptively, because of boundary non-linearity development, a reliable error estimator needs further investigation. The present study attempts to resolve this problem without having to recompute the entire solution on a new mesh.

Adaptive piecewise tensor product wavelets scheme for Laplace-interface problems

A Laplace type boundary value problem is considered with a generally discontinuous diffusion coefficient. A domain decomposition technique is used to construct a piecewise tensor product wavelet basis that, when normalized w.r.t. the energy-norm, has Riesz constants that are bounded uniformly in the jumps. An adaptive wavelet Galerkin method is applied to solve the boundary value problem with the best nonlinear approximation rate from the basis, in linear computational complexity. Although the solutions are far from smooth, numerical experiments in two dimensions show rates as for a one-dimensional smooth solution, the latter being possible because of the tensor product construction.

An efficient space-time adaptive wavelet Galerkin method for time-periodic parabolic partial differential equations

We introduce a multitree-based adaptive wavelet Galerkin algorithm for space-time discretized linear parabolic partial differential equations, focusing on time-periodic problems. It is shown that the method converges with the best possible rate in linear complexity and can be applied for a wide range of wavelet bases. We discuss the implementational challenges arising from the Petrov-Galerkin nature of the variational formulation and present numerical results for the heat and a convection-diffusion-reaction equation.

An adaptive Wavelet-Galerkin method for parabolic partial differentia equations

In this paper an Adaptive Wavelet-Galerkin method for the solution of parabolic partial differential equations modeling physical problems with different spatial and temporal scales is developed. A semi-implicit time difference scheme is applied and B-spline multiresolution structure on the interval is used. As in many cases these solutions are known to present localized sharp gradients, local error estimators are designed and an efficient adaptive strategy to choose the appropriate scale for each time is developed. Finally, experiments were performed to illustrate the applicability and efficiency of the proposed method.

Adaptive Wavelet Methods for Linear and Nonlinear Least-Squares Problems

The adaptive wavelet Galerkin method for solving linear, elliptic operator equations introduced by Cohen et al. (Math Comp 70:27---75, 2001) is extended to nonlinear equations and is shown to converge with optimal rates without coarsening. Moreover, when an appropriate scheme is available for the approximate evaluation of residuals, the method is shown to have asymptotically optimal computational complexity. The application of this method to solving least-squares formulations of operator equations $$G(u)=0$$G(u)=0, where $$G:H \rightarrow K'$$G:HźKź, is studied. For formulations of partial differential equations as first-order least-squares systems, a valid approximate residual evaluation is developed that is easy to implement and quantitatively efficient.

ウェーブレットガラーキン法による三次元構造のアダプティブ解析

ウェーブレットガラーキン法による三次元構造のアダプティブ解析

Piecewise tensor product wavelet bases by extensions and approximation rates

DIn this chapter, we present some of the major results that have been achieved in the context of the DFG-SPP project Adaptive Wavelet Frame Methods for Operator Equations: Sparse Grids, Vector-Valued Spaces and Applications to Nonlinear Inverse Problems. This project has been concerned with (nonlinear) elliptic and parabolic operator equations on nontrivial domains as well as with related inverse parameter identification problems. One crucial step has been the design of an efficient forward solver. We employed a spatially adaptive wavelet Rothe scheme. The resulting elliptic subproblems have been solved by adaptive wavelet Galerkin schemes based on generalized tensor wavelets that realize dimension-independent approximation rates. In this chapter, we present the construction of these new tensor bases and discuss some numerical experiments.

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic partial differential equation (PDE). We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on $\sigma_j$ with $j\in\mathbb{N}$, and that the PDE operator may depend nonaffinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [A. Cohen, R. DeVore, and Ch. Schwab, Found. Comput. Math., 10 (2010), pp. 615--646; Anal. Appl., 9 (2011), pp. 1--37], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. We establish sparsity of generalized polynomial chaos (gpc) expansions of both state and control in terms of the stochastic coordinate sequence $\sigma = (\sigma_j)_{j\geq 1}$ of the random inputs, and we prove convergence rates of best $N$-term truncations of these expansions. Such truncations are the key for subsequent computations since they do not assume that the stochastic input data has a finite expansion. In a follow-up paper [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation], we explain two methods for how such best $N$-term truncations can be computed practically: by greedy-type algorithms as in [Ch. Schwab and C. J. Gittelson, Acta Numer., 20 (2011), pp. 291--467; C. J. Gittelson, Report 2011-12, Seminar for Applied Mathematics, ETH Zürich, Zürich, Switzerland, 2011], or by multilevel Monte-Carlo methods as in [F. Y. Kuo, Ch. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374]. The sparsity result allows, in conjunction with adaptive wavelet Galerkin schemes, for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [W. Dahmen and A. Kunoth, SIAM J. Control Optim., 43 (2005), pp. 1640--1675; M. D. Gunzburger and A. Kunoth, SIAM J. Control. Optim., 49 (2011), pp. 1150--1170; A. Kunoth, Numer. Algorithms, 39 (2005), pp. 199--220]; see [A. Kunoth and Ch. Schwab, Sparse adaptive tensor Galerkin approximations of stochastic PDE-constrained control problems, in preparation].

Adaptive wavelet-Galerkin methods for limited angle tomography

This paper studied incomplete data problems of computed tomography that frequently occur in medical or industrial imaging, for example, when the high-density region of objects can only be penetrated by X-rays at a limited angular range. When projection data are available only in an angular range, the incomplete data problem can be attributed to the limited angle problem, which is a severely ill-posed inverse problem. In this paper, a numerical method for the treatment of inverse problems based on an adaptive wavelet-Galerkin method is introduced and investigated. The paper focuses especially on how to avoid inverse crimes in numerical simulations. The method used here combines numerical simplicity and characteristics of adapting to the unknown smoothness of a reconstructed image, which leads to significant reduction in the computational cost. The reconstruction strategy has a comparable performance with a significant reduction in computational time.

Adaptive Wavelet Galerkin Solution of Some Elastostatics Problems on Irregularly Spaced Nodes

In this paper the second generation wavelets are applied as a basis in finite element method. The wavelet basis is constructed over typical nonequispaced nodes and on boundaries. In addition the wavelet bases are tailored to the Poisson's operator. The wavelet basis is lifted to enforce operator orthogonality, this eliminates coupling between coarse and detail parts of the stiffness matrix. The scale decoupled stiffness matrix permits optimal O(N) computation. The Lagrangian second order wavelets are chosen for demonstration purposes, and a Poisson equation is solved. The potential application of this method in simulating a heterogenous material is outlined.

Poisson inverse problems

In this paper we focus on nonparametric estimators in inverse problems for Poisson processes involving the use of wavelet decompositions. Adopting an adaptive wavelet Galerkin discretization, we find that our method combines the well-known theoretical advantages of wavelet--vaguelette decompositions for inverse problems in terms of optimally adapting to the unknown smoothness of the solution, together with the remarkably simple closed-form expressions of Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann. Statist. 19 (1991) 1347--1369] to the context of log-intensity functions approximated by wavelet series with the use of the Kullback--Leibler distance between two point processes, we also present an asymptotic analysis of convergence rates that justifies our approach. In order to shed some light on the theoretical results obtained and to examine the accuracy of our estimates in finite samples, we illustrate our method by the analysis of some simulated examples.

2020 Bスプラインウェーブレットガラーキン法を用いたアダプティブ解析に関する研究(S04-3 FEM/メッシュフリー法/粒子法とその関連技術(3),S04 FEM/メッシュフリー法/粒子法とその関連技術)

An adaptive B-spline wavelet Galerkin method is adopted to the solid and structural analyses. The B-spline scaling function and wavelets used as the basis functions have simple forms and are relatively easy to integrate and differentiate. The basis functions have the so called multiresolution properties. High gradient region of stresses and strains can be refined by the superposition of the different length scale basis functions. As the wavelet Galerkin method is discretized based on structured cells, there are no re-meshing processes for the refinement of the solution. Due to control the spatial resolution of the wavelet Galerkin solution based on the posteriori estimate of its discretization error, an adaptive analysis is adopted. In this paper, adaptive B-spline wavelet Galerkin analyses to the stress concentration problems are performed.

Implementation Strategy for the Numerical Efficiency Improvement of the Multiscale Interpolation Wavelet-Galerkin Method

The multiscale wavelet-Galerkin method implemented in an adaptive manner has an advantage of obtaining accurate solutions with a substantially reduced number of interpolation points. The method is becoming popular, but its numerical efficiency still needs improvement. The objectives of this investigation are to present a new numerical scheme to improve the performance of the multiscale adaptive wavelet-Galerkin method and to give detailed implementation procedure. Specifically, the subdomain technique suitable for multiscale methods is developed and implemented. When the standard wavelet-Galerkin method is implemented without domain subdivision, the interaction between very long scale wavelets and very short scale wavelets leads to a poorly-sparse system matrix, which considerably worsens numerical efficiency for large-sized problems. The performance of the developed strategy is checked in terms of numerical costs such as the CPU time and memory size. Since the detailed implementation procedure including preprocessing and stiffness matrix construction is given, researchers having experiences in standard finite element implementation may be able to extend the multiscale method further or utilize some features of the multiscale method in their own applications.

Adaptive multigrid solution strategy for the dynamic simulation of petroleum mixture processes

The paper presents the extension of our work on the adaptive steady-state simulation of refinery processes to dynamic problems. The results show that the adaptive algorithm is capable of following the changes in the composition by adapting the composition representation during the course of simulation. The method is based on a continuous model formulation. In principle, each model given in a discrete (or pseudocomponent) formulation can be rewritten by means of continuous distribution functions. This continuous formulation does not yet reduce the model size but comprises the full detailed model. A model reduction is accomplished by applying a discretization scheme to the continuous formulation. Thus, the continuous reformulation allows the usage of modern discretization techniques including grid adaptation (adaptive wavelet-Galerkin discretization) and multigrid methods. In the iterative algorithm suggested, the approximate solution of the dynamic process model is obtained on the full composition grid. In each iteration loop the current iterate is corrected by a correction term, which is determined on a coarse grid for the composition representation. According to the multigrid framework the obtained intermediate solution is prolongated to the full composition detail by means of a linearization of the model equations. Within these iterations, the discretization grid is adapted by exploitation of the particular properties of the employed wavelet basis functions.

Remesh-free shape optimization using the wavelet-Galerkin method

The objective of this investigation is to develop a new remesh-free shape optimization method based on the adaptive wavelet-Galerkin analysis. To avoid cumbersome remeshing processes, we embed the original analysis design domain inside a simple fictitious domain and use the predetermined analysis grid points. For efficient adaptive analysis, we utilize the multiscale multiresolution characteristics of the hat interpolation wavelets that are associated with predetermined grid points. Furthermore, we approximate the design boundary curve by oblique lines that do not necessarily pass through the analysis grid points and develop an accurate stress evaluation technique based on the proposed boundary approximation. The validity of the proposed remesh-free shape optimization formulation based on the wavelet-Galerkin method is verified through benchmark shape optimization problems.

Adaptive Wavelet Galerkin Methods for Linear Inverse Problems

We introduce and analyze numerical methods for the treatment of inverse problems, based on an adaptive wavelet Galerkin discretization. These methods combine the theoretical advantages of the wavelet-vaguelette decomposition (WVD) in terms of optimally adapting to the unknown smoothness of the solution, together with the numerical simplicity of Galerkin methods. In a first step, we simply combine a thresholding algorithm on the data with a Galerkin inversion on a fixed linear space. In a second step, a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed; this leads to a significant reduction of the computational cost.

An adaptive wavelet-Galerkin method for an elastic-plastic-damage constitutive model: 1D problem

We present a wavelet-Galerkin method for the analysis of elastic-plastic-damage systems. Ju's model (J.W. Ju, Int. J. Solids Struct. 25 (1989) 803) is explicitly considered and numerically tested but the method applies equally well to a broad range of damage-mechanics models. A three-step elastic-predictor, plastic-corrector, damage-corrector strategy is the core of the proposed approach. We use a displacement approach with wavelet interpolating functions instead of classical polynomials. This allows to clearly locate those sites in the system where damage is taking place, thanks to the high-pass filtering action operated by the wavelets. This opens the way to inexpensive self-adaptive strategies that are introduced and preliminary tested herein.

Adaptive Wavelet-Galerkin Method for Structural Ananlysis

The object of the present study is to present an adaptive wavelet-Galerkin method for the analysis of thin-walled box beam. Due to good localization properties of wavelets, wavelet methods emerge as alternative efficient solution methods to finite element methods. Most structural applications of wavelets thus far are limited in fixed-scale, non-adaptive frameworks, but this is not an appropriate use of wavelets. On the other hand, the present work appears the first attempt of an adaptive wavelet-based Galerkin method in structural problems. To handle boundary conditions, a fictitous domain method with penalty terms is employed. The limitation of the fictitious domain method is also addressed.