How can wavelets be applied to PDE?

I would like to learn how wavelet methods can be applied to PDE, but unfortunately I do not know a good resource to learn about this topic.

It seems that many introductions to wavelets focus on interpolation theory, e.g., assembling a signal by a superposition of preferably few wavelets. Applications to PDEs are sometimes mentioned, without going deeper into that subject. I am interested in good summary articles for people who have seen a WFT but do not have any more knowledge on that topic. A good summary would be interesing as well, of course, if you think that can be done.

I am particularly interested in getting an impression which kind of questions typically appear. For example, I know that finite elements are typically applied to a PDE on a bounded domain with Lipschitz boundary, which are the typical questions in choosing the ansatz space (conforming, non-conforming, geometry and combinatorics), how convergence theory is established (actually the Galerkin theory should not be so different for Wavelets), and I have some intuition which mathematical things are feasible in implementations. Such a bird's eye perspective on Wavelets for PDE would be very helpful for me.

-

Wavelets have nice multi-resolution approximation properties, but are not especially popular for solving PDEs. The most commonly cited reasons are difficulty imposing boundary conditions, treatment of unaligned anisotropy, evaluation of nonlinear terms, and efficiency.

Wavelets were first to obtain strong convergence results for fully adaptive methods (see Cohen, Dahmen, and DeVore 2001 and 2002). However, this crucial theory was quickly followed by Binev, Dahmen, and DeVore (2004) who proved a similar result for adaptive finite element methods which are more popular for traditional PDE problems in moderate dimensions. Wavelet bases are popular for higher dimensional problems such as sparse tensor methods for stochastic PDEs Schwab and Gittelson (2011) and this discussion.

Differential operators have bounded condition number when expressed in wavelet bases and preconditioned with Jacobi (thus Krylov methods converge in a constant number of iterations independent of resolution). This is related to the hierarchical multigrid methods of Yserentant (1984), Bank, Dupont, and Yserentant (1988), and others. Note that multiplicative multigrid methods have superior convergence properties to additive methods. A standard multigrid V-cycle is essentially equivalent to standard symmetric Gauss-Seidel in the wavelet basis with the usual ordering. Note that this is rarely the best way to implement, especially in parallel.

Calederon-Zygmund operators and pseudo-differential operators are sparse in wavelet bases. Thus, many problems for which $\mathcal H$-matrices are useful with compact bases can be treated elegantly using wavelet bases.

Differential operators are relatively more expensive to evaluate in wavelet bases and it can be difficult to establish desired conservation properties. Some authors (e.g. Vasilyev, Paolucci, and Sen 1995) resort to collocation methods and use finite difference stencils to evaluate derivatives and nonlinear terms. If the wavelet expansion is blocked (usually good for computational efficiency), these methods become very similar to block-structured AMR.

I suggest Beylkin and Keiser (1997) as a practical introduction to solving PDEs with wavelets. The MADNESS code is based on these methods. It has support for immersed boundaries (see Reuter, Hill, and Harrison 2011), but has no efficient way to represent boundary layers in complicated geometry. The software is often used for chemistry problems in which geometry is not a concern.

For general numerical analysis of wavelets, I suggest Cohen's 2003 book. It presents an analysis framework in which the continuum solution is manipulated up until you want to evaluate it to a given accuracy, at which point the wavelet basis is evaluated as necessary.

-