Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients
Loading...
Date
2019-02-22
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
MDPI
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.
Description
Subject(s)
Riesz basis, wavelet, spline, interval, differential equation, sparse matrix, Black–Scholes equation