Adaptive wavelet methods for elliptic operator equations: Convergence rates

Abstract

Abstract. 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.

Adaptive wavelet algorithms for elliptic PDE's on product domains

“…the adaptive wavelet algorithms from [CDD01,CDD02,GHS05] are proven to produce a sequence of approximations that converge with this rate s, requiring a number of operations equivalent to their length, under the assumptions that one knows how to produce approximations for f of length N in O(N ) operations that converge with rate s, and that A can be sufficiently well approximated by computable sparse matrices. These assumptions are made to be able to control the cost of the successive approximate residual computations inside these iterative algorithms.…”

“…Other invertible systems can be put into this form by forming the normal equations A T Au = A T f . As shown in [Gan06], for mildly non-symmetric or indefinite equations, the algorithms from [CDD01,GHS05] can be applied directly to the original system, avoiding the squaring of the condition number.…”

An optimal adaptive wavelet method without coarsening of the iterands

“…Obviously, C D is defined as D C, and C D as C D and C D. The aforementioned convergence rates under the mild Besov regularity assumption concern best N -term approximations, whose computation, however, requires full knowledge of the solution u, which is only implicitly given. In [CDD01,CDD02], iterative methods for solving Au = f were developed which produce a sequence of approximations that converges with the same rate as is guaranteed for best N -term approximations, whereas their computation requires a number of operations that is equivalent to their support size. Together, both properties show that these methods are of optimal computational complexity.…”

“…Since, generally, each column of A contains infinitely many nonzero entries, clearly this matrix-vector product cannot be computed exactly, and has to be approximated. For sufficiently smooth wavelets that have sufficiently many vanishing moments and for both differential operators with piecewise sufficiently smooth coefficients, or singular integral operators on sufficiently smooth manifolds, the results from [Ste04,GS06a,GS06b] The results concerning optimal computational complexity of the iterative methods from [CDD01,CDD02] require the properties of APPLY and RHS mentioned above. Moreover, the methods apply under the condition that A is symmetric, positive definite (SPD), which, since A = Ψ, AΨ , is equivalent to v, Aw = Av, w , v, w ∈ H, and v, Av v 2 H , v ∈ H. For the case that A does not have both properties, the methods can be applied to the normal equations A * Au = A * f .…”

“…The principle behind the method from [CDD01] is to improve a given approximation w for u by realizing the saturation property: Let ·, · := A·, · and ||| · ||| := ·, · 1 2 . For any Λ ⊂ ∇, let P Λ denote the 2 -orthogonal projector onto 2 (Λ), i.e., P Λ replaces all coefficients outside Λ by zeros.…”