Welcome to the TMR Project "Wavelets in Numerical Simulation"

Research Topic

( Extract from TMR network Proposal "Wavelets and Multiscale Methods in Numerical Analysis and Simulation" )

During the past three decades, numerical simulation and scientific computing have become increasingly important in many areas of industrial and academic research. Physicists and engineers are now able to compute in a short time and with high accuracy the solutions to complex problems that typically arise from computational fluid dynamics, electromagnetism and elasticity. This dramatic evolution is due not only to the improvement in speed and memory space of computers, but also to substantial progress in developing efficient numerical analysis techniques. The success of numerical methods hinges, among other things, on finding an optimal compromise between the accuracy of the solution and the computational cost for a given problem. In this respect, essential breakthroughs have been accomplished recently by high order schemes, such as spectral or h-p discretization, multigrid methods and adaptive strategies using local mesh refinement techniques. Most of the principal developements of these methods occured during the 70's and 80's. At the present stage the understanding of these concepts is being consolidated primarily in connection with attempts to expand their scope of applicability. Recently, new concepts related to multiscale decompositions and wavelet bases have appeared, fueled by a strong input of theoretical analysis and approximation theory. Several principal features of such concepts raise strong expectations that, in the long run, a corresponding fully developed methodology will be capable of even further advancing the frontiers of numerical simulation. This may materialize through new types of algorithms as well as by complementing existing techniques with new ingredients. These promising perspectives are mainly based on the following facts, which will be discussed in further detail later. - A wide class of differential and integral operators, governing the above mentionened areas of applications, as well as their inverses, has sparse representation in wavelet bases. Moreover, the solutions of such equations are often very smooth except in isolated regions so that their wavelet representation involves only relatively few significant coefficients. The fact that under suitable circumstances wavelet expansions induce isomorphisms between the relevant functions spaces and the discrete realm provides extremely strong analysis tools that allow one to combine the sparsity representation of functions and operators with the adaptive extraction of the intrinsic complexity of the underlaying problem as well as the fast solution of the resulting discrete problems. - A corresponding dramatic reduction in storage and computational cost should then manifest itself through fast O(N) algorithms where N is the number of parameters needed to represent the solution with the desired accuracy. - Wavelet-based discretizations offer a versatile collection of tools that combine all the above important features with high order approximation schemes. - Wavelet discretizations are not only support for efficient numerical processing but are also powerful analysis tools for aquiring a better understanding of the various multiscale phenomena governing the problems arising in the above application areas. The understanding and development of these concepts is still far from having matured. So far some of the principles in the above mentioned perspectives have been realized only under rather constrained and special circumstances. Nevertheless,the experiences gained up to now strongly convince us that concentrated efforts to develop the new multiscale concepts to maturity is not only justified but necessary. The development of corresponding theoretical foundations, as well as the implementation of resulting algorithms for real life application in fluid dynamics, electromagnetism and elasticity, is the main objective of the project. Such developments necessarily have to go through several stages. Initial efforts have focused on only a few of the various aspects outlined above primarily under idealized assumptions. Several European academic groups have already contribued towards our goal and have acquired complementary experiences on different levels : - Numerical tools : each team has developed and implemented different types of wavelet and multiscale decomposition tools. - Type of research : some teams have worked primarily on the theoretical analysis of these methods, while others have concentrated on their implementation. - Applications : different problems arising from computational fluid dynamics, electromagnetism and elasticity have been considered by different teams. In spite of these promising perspectives, present implementations cannot yet compete with the best existing schemes that have been developed over the years. This will probably remain the case for quite some while. This is partly because new complex data structure have to be developed to exploit the full potential of wavelet based multiscale methods. So far, only simplified model problems have been treated by wavelet schemes. One reason is that the tools themselves are still quite young and are much more sophisticated than conventional discretizations, a natural consequence of their providing much more refined information than conven- tional discretizations. The process of identifying, on the one hand, the widest possible scope of problems and corresponding essential common principles that drive wavelet schemes in all these cases, and, on the other hand the concrete demands imposed on the tools by specific applications as well as creating a proper balance between both extremes, has only started. The diversity of different efforts indicated above has been essential in this regard. However, because of the complexity of the emerging methodology and the many facets of relevant know-how, our goal of developing the so far unexploited full potential in the application of wavelet-based techniques to solving more complex - and industrially realistic - problems inevitably requires to join the efforts and combine the complementary experiences of the the different teams. A project of the present type would provide an excellent platform for making significant progress towards our goal.