UvA-DARE ( Digital Academic Repository ) Adaptive wavelet schemes for parabolic problems : sparse matrices and numerical results (original) (raw)
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SIAM Journal on Control and Optimization, 2011
An adaptive algorithm based on wavelets is proposed for the efficient numerical solution of a control problem governed by a linear parabolic evolution equation. First, the constraints are represented by means of a full weak space-time formulation as a linear system in 2 in wavelet coordinates, following a recent approach by Schwab and Stevenson. Second, a quadratic cost functional involving a tracking-type term for the state and a regularization term for the distributed control is also formulated in terms of 2 sequence norms of wavelet coordinates. This functional serves as a representer for a functional involving different Sobolev norms with possibly non-integral smoothness parameter. Standard techniques from optimization are then used to derive the first order necessary conditions as a coupled system in 2 -coordinates.
Adaptive Wavelet Methods II—Beyond the Elliptic Case
Foundations of Computational Mathematics, 2002
This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet-based method developed in for symmetric positive definite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in ) that transform the original (continuous) problem into an equivalent infinite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve Date A. Cohen, W. Dahmen, and R. DeVore the resulting infinite system of equations. This second step requires a significant modification of the ideas from . The main departure from [17] is to develop an iterative scheme that directly applies to the infinite-dimensional problem rather than finite subproblems derived from the infinite problem. This rests on an adaptive application of the infinite-dimensional operator to finite vectors representing elements from finite-dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding wavelet-best N -term approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces, like the LBB condition, no longer arise.
Simultaneous space–time adaptive wavelet solution of nonlinear parabolic differential equations
Journal of Computational Physics, 2006
Dynamically adaptive numerical methods have been developed to efficiently solve differential equations whose solutions are intermittent in both space and time. These methods combine an adjustable time step with a spatial grid that adapts to spatial intermittency at a fixed time. The same time step is used for all spatial locations and all scales: this approach clearly does not fully exploit space-time intermittency. We propose an adaptive wavelet collocation method for solving highly intermittent problems (e.g. turbulence) on a simultaneous space-time computational domain which naturally adapts both the space and time resolution to match the solution. Besides generating a near optimal grid for the full space-time solution, this approach also allows the global time integration error to be controlled. The efficiency and accuracy of the method is demonstrated by applying it to several highly intermittent (1D + t)-dimensional and (2D + t)-dimensional test problems. In particular, we found that the space-time method uses roughly 18 times fewer space-time grid points and is roughly 4 times faster than a dynamically adaptive explicit time marching method, while achieving similar global accuracy.
Adaptive Wavelet Schemes for Nonlinear Variational Problems
SIAM Journal on Numerical Analysis, 2003
We develop and analyze wavelet based adaptive schemes for nonlinear variational problems. We derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal. Our approach is based on a new paradigm that has been put forward recently for a class of linear problems. The original problem is transformed first into an equivalent one which is well posed in the Euclidean metric 2 . Then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step. This iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances. The main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions. We prove asymptotically optimal complexity for adaptive realizations of first order iterations and of Newton's method.
jiip, 2012
In this paper, we combine concepts from two different mathematical research topics: adaptive wavelet techniques for well-posed problems and regularization theory for nonlinear inverse problems with sparsity constraints. We are concerned with identifying certain parameters in a parabolic reaction-diffusion equation from measured data. Analytical properties of the related parameter-to-state operator are summarized, which justify the application of an iterated soft shrinkage algorithm for minimizing a Tikhonov functional with sparsity constraints. The forward problem is treated by means of a new adaptive wavelet algorithm which is based on tensor wavelets. In its general form, the underlying PDE describes gene concentrations in embryos at an early state of development. We implemented an algorithm for the related nonlinear parameter identification problem and numerical results are presented for a simplified test equation.
Wavelet Method for Numerical Solution of Parabolic Equations
Journal of Computational Engineering, 2014
We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.
Adaptive wavelet methods for elliptic operator equations: Convergence rates
Mathematics of Computation, 2000
This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called N -term approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N −s ) in the energy norm, whenever such a rate is possible by N -term approximation. The range of s > 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N . The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization.
Advances in Computational Mathematics, 2020
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 wellknown 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 from [1] and an AWGM method as described in [13]. HT-AWGM performs a sequence of Galerkin solves based upon a truncated preconditioned conjugate gradient (PCG) algorithm from [33] in combination with a tensor-based preconditioner from [3]. 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.