In our contribution, we study different Riesz wavelet bases in Sobolev spaces based on cubic splines satisfying homogeneous Dirichlet boundary conditions of the second order. These bases are consequently applied to the numerical solution of the biharmonic problem and their quantitative properties are compared.
For a class of anisotropic integrodifferential operators arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations u = f on [0,1]n with possibly large n. Under certain conditions on , the scheme is of essentially optimal and dimension independent complexity (h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements...
This paper examines the pricing of two-asset European options under the Merton model represented by a nonstationary integro-differential equation with two state variables. For its numerical solution, the wavelet-Galerkin method combined with the Crank-Nicolson scheme is used. A drawback of most classical methods is the full structure of discretization matrices. In comparison, the wavelet method enables the approximation of discretization matrices with sparse matrices. Sparsity is essential for the...
Wavelets (see [2, 3, 4]) are a recent mathematical tool that is applied in signal processing, numerical mathematics and statistics. The wavelet transform allows to follow data in the frequency as well as time domain, to compute efficiently the wavelet coefficients using fast algorithm, to separate approximations from details. Due to these properties, the wavelet transform is suitable for analyzing and forecasting in time series. In this paper, Box-Jenkins models (see [1, 5]) combined with wavelets...
In this paper the theory of wavelets on the integers is developed. For this, one needs to first find analogs of translations and dyadic dilations which appear in the classical theory. Translations in l2(Z) are defined in the obvious way, taking advantage of the additive group structure of the integers. Dyadic dilations, on the other hand, pose a greater problem. In the classical theory of wavelets on the real line, translation T and dyadic dilation T obey the commutativity relation DT^2 = TD. We...
We give conditions such that the least degree solution of a Bézout identity is nonnegative on the interval [-1,1].