Fast Wavelet Transforms and Numerical Algorithms

The purpose of this paper is to introduce a class of numerical algorithms designed for rapid application of dense matrices (or integral operators) to vectors. As is well-known, applying directly a dense N×N− matrix to a vector requires roughly N operations, and this simple fact is a cause of serious difficulties encountered in large-scale computations. For example, the main reason for the limited use of integral equations as a numerical tool in large-scale computations is that they normally lead to dense systems of linear algebraic equations, and the latter have to be solved, either directly or iteratively. Most iterative methods for the solution of systems of linear equations involve the application of the matrix of the system to a sequence of recursively generated vectors, which tends to be prohibitively expensive for large-scale problems. The situation is even worse if a direct solver for the linear system is used, since such solvers normally require O(N) operations. As a result, in most areas of computational mathematics dense matrices are simply avoided whenever possible. For example, finite difference and finite element methods can be viewed as devices for reducing a partial differential equation to a sparse linear system. In this case, the cost of sparsity is the inherently high condition number of the resulting matrices. For translation invariant operators, the problem of excessive cost of applying (or inverting) the dense matrices has been met by the Fast Fourier Transform (FFT ) and related algorithms (fast convolution schemes, etc.). These methods use algebraic properties of a matrix to apply it to a vector in order N log(N) operations. Such schemes are exact in exact arithmetic, and are fragile in the sense that they depend on the exact algebraic properties of the operator for their applicability. A more recent group of fast algorithms [1, 2, 5, 9] uses explicit analytical properties of specific operators to rapidly apply them