Wavelets as a Tool for Numerical Simulation
by Silvia Bertoluzza
Wavelets were first introduced in the mid-eighties as a tool for signal and image processing, a field where they almost immediately showed their effectiveness. The good properties of localization in space and frequency of these functions, together with the existence of a fast O(n) algorithm (the 'Fast Wavelet Transform') make wavelets an efficient tool for tasks such as image compression, pattern recognition and signal denoising.
Recently, several researchers, many of them European, began to study the applicability of such functions as a tool for numerical simulation and scientific computing. As a result, new techniques for the numerical solution of partial differential and integral equations, based on the use of multiscale decomposition and wavelet bases have been proposed. These techniques have benefited from a strong input from theoretical analysis and approximation theory. There is strong expectation that, in the long run, a corresponding fully developed methodology will further advance the frontiers of numerical simulation, allowing the design of efficient algorithms which should combine high accuracy (attained through the use of high order adaptive schemes) with low computational cost. This goal may be attained through new types of algorithms as well as by complementing existing techniques with new ingredients. These promising perspectives are mainly based on the following considerations:
- A wide class of differential and integral operators, found in equations used in fields such as computational fluid dynamics, electromagnetism and elasticity, as well as their inverses have sparse representations in wavelet bases, and the number of parameters needed to represent the solution accurately is usually small.
- Optimal O(n) preconditioners exist for elliptic operators of any order, when they are discretized in wavelet bases. These are based on the property of the stability of such bases with respect to Sobolev norms: the norm in a Sobolev space Hs of a linear combination of wavelets is equivalent to a weighted Euclidean norm of the coefficients. This holds for all Sobolev exponents in a suitable range (depending on the wavelet basis).
- Furthermore, the previous norm equivalence result makes it possible to read the local regularity of a function on its wavelet coefficients. This is at the basis of the design of simple adaptivity strategies that have been successfully applied in different fields.
We would like to stress that all these features are a consequence of the good space-frequency localization properties of wavelet bases.
Several researchers are currently studying ways to make this new and promising tool applicable to complex real life problems by facing open questions - like the treating of a complex geometry, or the approximation of non linear operators that need to be solved if wavelets are to be proved effective in real life scientific computing.
With similar objectives, an EC Training, Mobility and Research Network is expected to start in Spring 98. The Network, with the title 'Wavelets and Multiscale Methods in Numerical Analysis and Simulation' will be coordinated by the 'Istituto di Analisi Numerica' of the Italian National Research Council (IAN-CNR, Pavia, Italy) and will include nine teams working in six countries. Researchers from the following institutions will participate in the project: IGPM-RWTH Aachen, Univ. Chemnitz, Univ. Karlsruhe, IST Lisbon, IRPHE-CNRS Marseilles, LAN Paris VI, Univ. Paris XIII, Politecnico Turin, Univ. Bologna, Univ. Messina, Univ. Valencia, ETH-Zurich.
Silvia Bertoluzza - IAN-CNR
Tel: +39 382 505687