Abstract
We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefficients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efficiency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility.
Highlights
Wavelets are a powerful and useful tool for analyzing signals, the detection of singularities, data compression and the numerical solution of partial differential and integral equations
Vanishing wavelet moments ensure the so-called compression property of wavelets. This means that a function f that is smooth, except at some isolated singularities, typically has a sparse representation in a wavelet basis, i.e., only a small number of wavelet coefficients carry most of the information on f
The basis is adapted to homogeneous Dirichlet boundary conditions, and wavelets have eight vanishing moments
Summary
Wavelets are a powerful and useful tool for analyzing signals, the detection of singularities, data compression and the numerical solution of partial differential and integral equations. Vanishing wavelet moments ensure the so-called compression property of wavelets This means that a function f that is smooth, except at some isolated singularities, typically has a sparse representation in a wavelet basis, i.e., only a small number of wavelet coefficients carry most of the information on f. Certain differential and integral operators have sparse or quasi-sparse representation in a wavelet basis This compression property of wavelets leads to the design of many multiscale wavelet-based methods for the solution of differential equations. Our aim is to construct a wavelet basis such that the discretization matrix corresponding to (1) is sparse if the coefficients pk,l , qk and p0 are piecewise polynomial functions of degree at most n on the uniform grid, where n = 6 for pk,l , n = 5 for qk and n = 4 for p0. For the basis that will be constructed in this paper, the discretization matrix is sparse for the Black–Scholes equation with volatilities σi that are piecewise quadratic
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