Parameter estimation using minimum norm differential approximation

We present mathematical results which can be used to compute the parameters of a system described by differential equations, using the method of minimum norm differential approximation. The algorithm is described and several examples are given in both the ordinary and partial differential equations cases. The approximating subspaces used in this algorithm are those spanned by certain B-splines of degree 3 in the O.D.E. case and by tensor products of B-splines in the P.D.E. case. Singular value decomposition is used in two distinct ways in the algorithm. The method described can be used on any type of differential operator with constant coefficients, i.e., elliptic, hyperbolic, parabolic, although only in the case of elliptic operators can error bounds between the data function and a generalized solution of the D.E. with the approximated parameters be estimated.     

Minimum norm solutions of single stiff linear analytic differential equations

The $\text{l}$₂-norm of the infinite vector of the terms of the Taylor series of an analytic function is used to measure the “unsmoothness” of the function. The sets of solutions to the scalar differential equations y′(t) = λy(t) + f(t) and y′(t) = q(t)y(t) + f(t) are analyzed with respect to this norm. A number of results on the particular solution with minimum norm are given.     

Minimum norm solutions for cooperative games

A solution f for cooperative games is a minimum norm solution, if the space of games has a norm such that f(v) minimizes the distance (induced by the norm) between the game v and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.      

Minimum norm problems over transportation polytopes

We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? $\text{R}$^m, v?$\text{R}$ⁿ, find an (m,n) matrix X ? 0 with prescribed row sums Σⁿ_j=1X_ij = u_i (i = 1,…,m) and prescribed column sums Σ^m_i=1X_ij = v_j (j = 1,…,n) which fits the relations X_ij = A_ij + λ_iW_ij + W_ij + W_ijμ_j for all i,j and some λ?$\text{R}$^m, μ?$\text{R}$ⁿ. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2].     

On generalized matrix approximation problem in the spectral norm

In this paper theoretical results regarding a generalized minimum rank matrix approximation problem in the spectral norm are presented. An alternative solution expression for the generalized matrix approximation problem is obtained. This alternative expression provides a simple characterization of the achievable minimum rank, which is shown to be the same as the optimal objective value of the classical problem considered by Eckart–Young–Schmidt–Mirsky, as long as the generalized problem is feasible. In addition, this paper provides a result on a constrained version of the matrix approximation problem, establishing that the later problem is solvable via singular value decomposition.     

On vector valued function approximation using a peak norm

A peak norm is defined for L_p spaces of E-valued Bochner integrable functions, where E is a Banach space, and best approximations from a sun to elements of the space are characterized. Applications are given to some families of simultaneous best approximation problems.     

Minimum norm solution to the positive semidefinite linear complementarity problem

In this article, we present an algorithm to compute the minimum norm solution of the positive semidefinite linear complementarity problem. We show that its solution can be obtained using the alternative theorems and a convenient characterization of the solution set of a convex quadratic programming problem. This problem reduces to an unconstrained minimization problem with once differentiable convex objective function. We propose an extension of Newton's method for solving the unconstrained optimization problem. Computational results show that convergence to high accuracy often occurs in just a few iterations.     

Coconvex approximation in the uniform norm: the final frontier

Summary The paper deals with approximation of a continuous function, on a finite interval, which changes convexity finitely many times, by algebraic polynomials which are coconvex with it. We give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian-Totik moduli of smoothness.     

Quasioptimality of skeleton approximation of a matrix in the Chebyshev norm

For a given matrix, considered is the rank-r skeleton approximation which uses r columns and r rows of the given matrix. It is demonstrated that if the minor residing on the intersection of the chosen columns and rows has the maximal modulus among all minors of order r, the considered approximation is quasioptimal in Chebyshev norm.     

Weighted inequalities and applications to best local approximation in Luxemburg norm

In this paper we establish a result about uniformly equivalent norms and the convergence of best approximant pairs on the unitary ball for a family of weighted Luxemburg norms with normalized weight functions depending on ε, when ε→ 0. It is introduced a general concept of Pade approximant and we study its relation with the best local quasi-rational approximant. We characterize the limit of the error for polynomial approximation. We also obtain a new condition over a weight function in order to obtain inequalities in Lp norm, which play an important role in problems of weighted best local Lp approximation in several variables.     

Optimal Hankel norm approximation for the Pritchard-Salamon class of infinite-dimensional systems

The optimal Hankel norm approximation problem is solved under the assumptions that the system (A, B, C) is an exponentially stable, regular Pritchard-Salamon infinite-dimensional system. An explicit parameterization of all solutions is obtained in terms of the system parametersA, B, C.     

Sub-optimal Hankel norm approximation for the analytic class of infinite-dimensional systems

The sub-optimal Hankel norm approximation problem is solved under the assumptions that the system is given in terms of a triple of operators (–A, B, C), where–A is the infinitesimal generator of an exponentially stable, analytic semigroup on the Hilbert spaceZ,B L ( ^m ,Z where –1<0,C L is obtained in terms of the system parameters–A, B, C. (Z, ^p ), and the system is approximately controllable. An explicit parameterization of all solutions     

Monotonicity and best approximation in Orlicz-Sobolev spaces with the Luxemburg norm

Criteria for strict monotonicity, upper (lower) locally uniform monotonicity and uniform monotonicity of Orlicz-Sobolev spaces with the Luxemburg norm are given. Some applications to best approximation are presented.     

Seminorm and full norm order of linear approximation from shift-invariant spaces

In this paper, we continue our research on characterizing the order of linear approximation schemes from shift-invariant spaces, which was started in [4]. Our extension of earlier results applies to various aspects: Firstly, in the definition of the operators we allow more general functions, viz. distributions of finite order. Secondly, we consider the non-stationary case, where the operators may depend on the scaling parameter. Thirdly, we bound the error for derivatives as well, i.e., we can bound the error of simultaneous approximation. Finally, we derive a characterization of the full norm approximation order in addition to the usual seminorm approximation order. Our results are applied to the following examples: Thin-plate splines, multiquadrics, and four-directional box splines.     

On numerical approximation of differential polynomials

The formula for numerical approximation of differential polynomials of special form is obtained. The absolute error of approximation is shown. It is shown that the approximation considered is sharp on polynomials. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.