Lecture on August 17, 2015

2015-08-07

Topic: Adaptive wavelet methods: How to avoid the apply-routine?

Speaker: Prof. Rob Stevenson

  Korteweg-de Vries Institute (KdVI), Amsterdam, The Netherlands

Time: August 17 (Monday) 10:00-11:00

Location: Room 415, New Maths Building

Abstract:

We give an overview of adaptive wavelet methods for solving operator equations, as introduced by Cohen, Dahmen and DeVore in [CDD01, CDD02], and further developed in e.g. [XZ03, CDD03, GHS07, Ste14]. These methods were shown to converge to the solution with the best possible rate in linear computational complexity.

Compared to adaptive finite element methods, for which, in any case for elliptic problems, similar theoretical results were proven, in a quantitative sense the results obtained with wavelet schemes are sometimes somewhat disappointing. The approximate matrix-vector multiplication routine, known as the apply-routine, is easily identified as the computational bottleneck. In this talk, it will be shown how the application of this routine can be avoided by writing the PDE as a first order system least squares problem.

Promising applications include those to simultaneous space-time variational formulations of evolutionary PDEs as parabolic PDEs, or the instationary (Navier-) Stokes equations. Equipping the relevant function spaces with bases that consist of tensor products of temporal and spatial wavelets, the whole time evolution problem can be solved at a complexity of solving one instance of the corresponding stationary problem.

References

[CDD01] A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods for elliptic operator equations - Convergence rates. Math. Comp, 70:27-75, 2001.

[CDD02] A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods II - Beyond the elliptic case. Found. Comput. Math., 2(3):203-245, 2002.

[CDD03] A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet schemes for non-linear variational problems. SIAM J. Numer. Anal., 41:1785-1823, 2003.

[GHS07] T. Gantumur, H. Harbrecht, and R.P. Stevenson. An optimal adaptive

wavelet method without coarsening of the iterands. Math. Comp., 76:615-629, 2007.

[Ste14] R.P. Stevenson. Adaptive wavelet methods for linear and nonlinear least-squares problems. Found. Comput. Math., 14(2):237-283, 2014.

[XZ03] Y. Xu and Q. Zou. Adaptive wavelet methods for elliptic operator equations with nonlinear terms. Adv. Comput. Math., 19(1-3):99-146, 2003. Challenges in computational mathematics (Pohang, 2001).

Introduction of the speaker:

Since September 1st, Prof. Rob Stevenson has been head of the UvA research group that concentrates on this particular form of mathematics. More specifically, Prof. Stevenson works on partial differential equations, which are formulated in terms of derivatives of an unknown function with two or more variables. See more at: