Computation of Optimal Weights for Solving the Atmospheric Source Term Estimation Problem

In case of a release of a hazardous material (e.g., a chemical or a biological agent) in the atmosphere, estimation of the source from concentration observations (provided by a network of sensors) is a challenging inverse problem known as the atmospheric source term estimation (STE) problem. This study emphasizes a method, known in the literature as the renormalization inversion technique, for addressing this problem. This method provides a solution that has been interpreted as a weighted minimal norm solution and can be computed in terms of a generalized inverse of the sensitivity matrix of the sensors. This inverse is constructed by using an appropriate diagonal weight matrix whose components fulfill the so-called renormalizing conditions. The main contribution of this paper is that it proposes a new compact algorithm (it requires less than 15 lines of MATLAB code) to obtain, in a fast and efficient way, those optimal weights. To show that the algorithm, based on the properties of the resolution matrix, matches the requirements of emergency situations, analysis of the computational complexity and memory requirements is included. Some numerical experiments are also reported to show the efficiency of the algorithm.

On weighted Strand’s iteration

In this paper we present a generalization of Strand’s iterative method for numerical approximation of the weighted minimal norm solution of a linear least squares problem. We prove convergence of the extended algorithm, and show that previous iterative algorithms proposed by L. Landweber, J. D. Riley and G. H. Golub are particular cases of it.

A general iterative solver for unbalanced inconsistent transportation problems

The transportation problem, as a particular case of a linear programme, has probably the highest relative frequency with which appears in applications. At least in its classical formulation, it involves demands and supplies. When, for practical reasons, the total demand cannot satisfy the total supply, the problem becomes unbalanced and inconsistent, and must be reformulated as e.g. finding a least squares solution of an inconsistent system of linear inequalities. A general iterative solver for this class of problems has been proposed by S. P. Han in his 1980 original paper. The drawback of Han’s algorithm consists in the fact that it uses in each iteration the computation of the Moore-Penrose pseudoinverse numerical solution of a subsystem of the initial one, which for bigger dimensions can cause serious computational troubles. In order to overcome these difficulties we propose in this paper a general projection-based minimal norm solution approximant to be used within Han-type algorithms for approximating least squares solutions of inconsistent systems of linear inequalities. Numerical experiments and comparisons on some inconsistent transport model problems are presented.

On a regularization technique for Kovarik-like approximate orthogonalization algorithms

In this paper we consider four versions of Kovarik’s iterative orthogonalization algorithm, for approximating the minimal norm solution of symmetric least squares problems. Although the theoretical convergence rate of these algorithms is at least linear, in practical applications we observed that a too big number of iterations can dramatically deteriorate the already obtained approximation. In this respect we analyse the above mentioned Kovarik-like methods according to the modifications they make on the “machine zero” eigenvalues of the problem’s (symmetric) matrix. We establish a theoretical almost optimal formula for the number of iterations necessary to obtain an enough accurate approximation, as well as to avoid the above mentioned troubles. Experiments on collocation discretization of a Fredholm first kind integral equation illustrate the efficiency of our considerations.