Václav Finěk

Cubic spline wavelets with complementary boundary conditions

Abstract We propose a new construction of a stable cubic spline-wavelet basis on the interval satisfying complementary boundary conditions of the second order. It means that the primal wavelet basis is adapted to homogeneous Dirichlet boundary conditions of the second order, while the dual wavelet basis preserves the full degree of polynomial exactness. We present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator.

Wavelet basis of cubic splines on the hypercube satisfying homogeneous boundary conditions

In this paper, we propose a construction of a new cubic spline-wavelet basis on the hypercube satisfying homogeneous Dirichlet boundary conditions. Wavelets have two vanishing moments. Stiffness matrices arising from discretization of elliptic problems using a constructed wavelet basis have uniformly bounded condition numbers and we show that these condition numbers are small. We present quantitative properties of the constructed basis and we provide a numerical example to show the efficiency of the Galerkin method using the constructed basis.

Approximate multiplication in adaptive wavelet methods

Cohen, Dahmen and DeVore designed in [Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75] and [Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245] a general concept for solving operator equations. Its essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l2-problem, finding the convergent iteration process for the l2-problem and finally using its finite dimensional approximation which works with an inexact right-hand side and approximate matrix-vector multiplication. In our contribution, we pay attention to approximate matrix-vector multiplication which is enabled by an off-diagonal decay of entries of the wavelet stiffness matrices. We propose a more efficient technique which better utilizes actual decay of matrix and vector entries and we also prove that this multiplication algorithm is asymptotically optimal in the sense that storage and number of floating point operations, needed to resolve the problem with desired accuracy, remain proportional to the problem size when the resolution of the discretization is refined.

On the exact values of coefficients of coiflets

In 1989, R. Coifman suggested the design of orthonormal wavelet systems with vanishing moments for both scaling and wavelet functions. They were first constructed by I. Daubechies [15, 16], and she named them coiflets. In this paper, we propose a system of necessary conditions which is redundant free and simpler than the known system due to the elimination of some quadratic conditions, thus the construction of coiflets is simplified and enables us to find the exact values of the scaling coefficients of coiflets up to length 8 and two further with length 12. Furthermore for scaling coefficients of coiflets up to length 14 we obtain two quadratic equations, which can be transformed into a polynomial of degree 4 for which there is an algebraic formula to solve them.

Hermite cubic spline multi-wavelets on the cube

In 2000, W. Dahmen et al. proposed a construction of Hermite cubic spline multi-wavelets adapted to the interval [0, 1]. Later, several more simple constructions of wavelet bases based on Hermite cubic splines were proposed. We focus here on wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of non-zero elements in each column is bounded by a constant. Then, a matrix-vector multiplication in adaptive wavelet methods can be performed exactly with linear complexity for any second order differential equation with constant coefficients. In this contribution, we shortly review these constructions, use an anisotropic tensor product to obtain bases on the cube [0, 1]3, and compare their condition numbers.

Optimized Construction of Biorthogonal Spline‐Wavelets

The paper is concerned with the construction of wavelet bases on the interval derived from B‐splines. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Inner wavelets are translated and dilated versions of well‐known wavelets designed by Cohen, Daubechies, Feauveau [5] while the construction of boundary wavelets is along the lines of [6]. The disadvantage of popular bases from [6] is their bad condition which cause problems in practical applications. Some modifications which lead to better conditioned bases were proposed in [1, 7, 8, 9, 10]. In this contribution, we further improve the condition of spline‐wavelet bases on the interval. Quantitative properties of these bases are presented.

Quadratic spline wavelets with short support satisfying homogeneous boundary conditions

In the paper, we construct a new quadratic spline-wavelet basis on the interval and a unit square satisfying homogeneous Dirichlet boundary conditions of the first order. Wavelets have one vanishing moment and the shortest support among known quadratic spline wavelets adapted to the same type of boundary conditions. Stiffness matrices arising from a discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers and the condition numbers are small. We present quantitative properties of the constructed basis. We provide numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis requires smaller number of iterations than these methods with other quadratic spline wavelet bases. Moreover, due to the short support of the wavelets one iteration requires smaller number of floating point operations.

Wavelets based on Hermite cubic splines

In 2000, W. Dahmen et al. designed biorthogonal multi-wavelets adapted to the interval [0,1] on the basis of Hermite cubic splines. In recent years, several more simple constructions of wavelet bases based on Hermite cubic splines were proposed. We focus here on wavelet bases with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in any column is bounded by a constant. Then, a matrix-vector multiplication in adaptive wavelet methods can be performed exactly with linear complexity for any second order differential equation with constant coefficients. In this contribution, we shortly review these constructions and propose a new wavelet which leads to improved Riesz constants. Wavelets have four vanishing wavelet moments.

Approximation properties of wavelets and relations among scaling moments II

A new orthonormality condition for scaling functions is derived. This condition shows a close connection between orthonormality and relations among discrete scaling moments. This new condition in connection with certain approximation properties of scaling functions enables to prove new relations among discrete scaling moments and consequently the same relations for continuous scaling moments.

A note on preconditioning for singularly perturbed problems discretized by wavelets

In the modeling of boundary value problems for singularly perturbed differential equations, one can observe boundary and interior layers whose width can be arbitrary small. To solve such types of problems a lot of methods were proposed. In this contribution, we focus on methods based on wavelets. Using wavelet discretization of singularly perturbed two-point boundary value problems, one can observe that the condition numbers of the arising stiffness matrices are growing with decreasing parameter e. We show here that a matrix splitting can stabilize the condition numbers of the stiffness matrices for small values of parameter e. Numerical examples are given.