Catálogo de publicaciones - libros

Skew-Hamiltonian and Hamiltonian eigenvalue problems arise from a number of applications, particularly in systems and control theory. The preservation of the underlying matrix structures often plays an important role in these applications and may lead to more accurate and more efficient computational methods. We will discuss the relation of structured and unstructured condition numbers for these problems as well as algorithms exploiting the given matrix structures. Applications of Hamiltonian and skew-Hamiltonian eigenproblems are briefly described.

The aim of the present paper is to review the basic ideas of the so called (AS) and to show that they can be used to solve any problem concerning the construction of spline curves subject to (i.e. piecewise defined) constraints.

In particular, we will use AS to solve a planar parametric interpolation problem with free knots.

This paper addresses a new approach in solving the problem of shape preserving spline interpolation. Based on the formulation of the latter problem as a differential multipoint boundary value problem for hyperbolic and biharmonic tension splines we consider its finite-difference approximation. The resulting system of linear equations can be efficiently solved either by direct (Gaussian elimination) and iterative methods (successive over-relaxation (SOR) method and finite-difference schemes in fractional steps). We consider the basic computational aspects and illustrate the main advantages of this original approach.

We discuss a dimensionless formulation of the Black-Scholes equation for the value of a European call option. We observe that, for some values of the parameters, this may be a singularly perturbed problem. We demonstrate numerically that, in such a case, a standard numerical method on a uniform mesh does not produce robust numerical solutions. We then construct a new numerical method, on an appropriately fitted piecewise-uniform mesh, which generates numerical approximations that converge parameter-uniformly in the maximum norm to the exact solution.

In recent years the locally Sobolev functions got quite popular in works on applications of partial differential equations. However, the properties of those spaces have not been systematically studied and proved in the literature, resulting in many particular proofs by reduction to classical Sobolev spaces.

Following some hints of general theory scattered through classical literature, as well as some proofs of special cases, we systematically present the main results regarding the properties of W and W spaces, their duality, reflexivity, imbeddings, density, weak topologies, etc., with particular emphasis on applications in partial differential equations of mathematical physics.

We consider optimal design of stationary diffusion problems for two-phase materials. Such problems usually have no solution. A relaxation consists in introducing the notion of composite materials, as fine mixtures of different phases, mathematically described by the homogenisation theory. The problem can be written as an optimisation problem over (), the set of all possible composite materials with given local proportion . Tartar and Murat (1985) described the set (), for some vector e, and used this result to replace the optimisation over the complicated set () by a much simpler one. Analogous characterisation holds even for the case of mixing more than two materials (possibly anisotropic), where the set () is not effectively known (Tartar, 1995).

We address the question of describing the set {(, : ∊ ()} (for given and ), which is important for optimal design problems with multiple state equations (different right-hand sides). In other words, we are interested in describing two columns of matrices in (). In two dimensions we describe this set in appropriate coordinates and give some geometric interpretation. For the three-dimensional case we consider the set { : ∊ (), = }, for a fixed t, and show how it can be reduced to a two-dimensional one, albeit through tedious computations.

Tension spline is a function that, for given partition < < … < , on each interval [, ] satisfies differential equation ( − ρ) = 0, where ρ's are prescribed nonnegative real numbers. In the literature, tension splines are used in collocation methods applied to two-points singularly perturbed boundary value problems with Dirichlet boundary conditions.

In this paper, we adapt collocation method to solve a time dependent reaction-diffusion problem of the form with Dirichlet boundary conditions. We tested our method on the time-uniform mesh with × elements. Numerical results show -uniformly convergence of the method.

The singular value decomposition (SVD) of a general matrix is the fundamental theoretical and computational tool in numerical linear algebra. The most efficient way to compute the SVD is to reduce the matrix to bidiagonal form in a finite number of orthogonal (unitary) transformations, and then to compute the bidiagonal SVD. This paper gives detailed error analysis and proposes modifications of recently proposed one-sided bidiagonalization procedure, suitable for parallel computing. It also demonstrates its application in solving two common problems in linear algebra.

We develop a technique to calculate with weighted splines of arbitrary order, i.e. with splines from the kernel of the operator ωD, with piecewisely constant, based on knot insertion type algorithm. The algorithm is a generalization of de Boor algorithm for polynomial splines, and it inserts the evaluation point in the knot sequence with maximal multiplicity. To achieve this, we use a general form of knot insertion matrices, and an Oslo type algorithm for calculating integrals of B-splines in reduced Chebyshev systems. We use the fact that the space of weighted splines is a subspace of the polynomial spline space. The complexity of proposed algorithm can be reduced to the computationally reasonable size. Now we can calculate weighted splines, and the splines associated with their reduced system, in a stable and efficient manner.

Numerical procedures for the solution of an inverse problem of determining unknown source parameter in a parabolic equation are considered. Two different numerical procedures are studied and their comparison analysis is presented. One of these procedures is obtained by introducing transformation of an unknown function, while the other is based on trace functional formulation of the problem.