Monte Carlo simulation via a numerical algorithm for solving a nonlinear inverse problem

This paper is intended to provide a numerical algorithm involving the combined use of the finite differences scheme and Monte Carlo method for estimating the diffusion coefficient in a one-dimensional nonlinear parabolic inverse problem. In the present study, the functional form of the diffusion coefficient is unknown a priori. The unknown diffusion coefficient is approximated by the polynomial form and the present numerical algorithm is employed to find the solution. To modify the values of estimated coefficients of this polynomial form, we introduce a random search algorithm in Monte Carlo method for global optimization. A numerical test is performed in order to show the efficiency and accuracy of the present work.     

A new matrix inverse

We compute the inverse of a specific infinite-dimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type.

     

Computing strategies for achieving acceptability: A Monte Carlo approach

We consider a trader who wants to direct his or her portfolio towards a set of acceptable wealths given by a convex risk measure. We propose a Monte Carlo algorithm, whose inputs are the joint law of stock prices and the convex risk measure, and whose outputs are the numerical values of initial capital requirement and the functional form of a trading strategy for achieving acceptability. We also prove optimality of the capital obtained. Explicit theoretical evaluations of hedging strategies are extremely difficult, and we avoid the problem by resorting to such computational methods. The main idea is to utilize the finite Vapnik–C?ervonenkis dimension of a class of possible strategies.     

A real delivery problem dealt with Monte Carlo Techniques

In this paper we use Monte Carlo Techniques to deal with a real world delivery problem of a food company in Valencia (Spain). The problem is modeled as a set of 11 instances of the well known Vehicle Routing Problem, VRP, with additional time constraints. Given that VRP is a NP-hard problem, a heuristic algorithm, based on Monte Carlo techniques, is implemented. The solution proposed by this heuristic algorithm reaches distance and money savings of about 20% and 5% respectively. This work has been partially supported by thePlan de Incentivo a la Investigación/98 of the Universidad Politécnica de Valencia, under the project “Técnicas Monte Carlo aplicadas a Problemas de Rutas de Vehículos”.     

A new inverse problem for the diffusion operator

In this work, we have estimated nodal points and nodal lengths for the diffusion operator. Furthermore, by using these new spectral parameters, we have shown that the potential function of the diffusion operator can be established uniquely. An analogous inverse problem was solved for the Sturm–Liouville problem in recent years.     

Adaptive Monte Carlo methods for matrix equations with applications

This paper discusses empirical studies with both the adaptive correlated sequential sampling method and the adaptive importance sampling method which can be used in solving matrix and integral equations. Both methods achieve geometric convergence (provided the number of random walks per stage is large enough) in the sense: e_ν≤cλ^ν, where e_ν is the error at stage ν, λ∈(0,1) is a constant, c>0 is also a constant. Thus, both methods converge much faster than the conventional Monte Carlo method. Our extensive numerical test results show that the adaptive importance sampling method converges faster than the adaptive correlated sequential sampling method, even with many fewer random walks per stage for the same problem. The methods can be applied to problems involving large scale matrix equations with non-sparse coefficient matrices. We also provide an application of the adaptive importance sampling method to the numerical solution of integral equations, where the integral equations are converted into matrix equations (with order up to 8192×8192) after discretization. By using Niederreiter’s sequence, instead of a pseudo-random sequence when generating the nodal point set used in discretizing the phase space Γ, we find that the average absolute errors or relative errors at nodal points can be reduced by a factor of more than one hundred.     

A Perturbation approach for an inverse quadratic programming problem

We consider an inverse quadratic programming (QP) problem in which the parameters in both the objective function and the constraint set of a given QP problem need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a linear complementarity constrained minimization problem with a positive semidefinite cone constraint. With the help of duality theory, we reformulate this problem as a linear complementarity constrained semismoothly differentiable (SC¹) optimization problem with fewer variables than the original one. We propose a perturbation approach to solve the reformulated problem and demonstrate its global convergence. An inexact Newton method is constructed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. As the objective function of the problem is a SC¹ function involving the projection operator onto the cone of positively semi-definite symmetric matrices, the analysis requires an implicit function theorem for semismooth functions as well as properties of the projection operator in the symmetric-matrix space. Since an approximate proximal point is required in the inexact Newton method, we also give a Newton method to obtain it. Finally we report our numerical results showing that the proposed approach is quite effective.     

A new iterative method for solving split feasibility problem

In this paper, we construct a new iterative algorithm and show that the newly introduced iterative algorithm converges faster than a number of existing iterative algorithms for contractive-like mappings. We present a numerical example followed by graphs to validate our claim. We prove strong and weak convergence results for approximating fixed points of generalized $\alpha$-nonexpansive mappings. Again we reconfirm our results by an example and table. Further, we utilize our proposed algorithm to solve split feasibility problem.     

A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix

Based on a quadratical convergence method, a family of iterative methods to compute the approximate inverse of square matrix are presented. The theoretical proofs and numerical experiments show that these iterative methods are very effective. And, more importantly, these methods can be used to compute the inner inverse and their convergence proofs are given by fundamental matrix tools.     

The inverse problem for Krein orthogonal matrix functions

In the mid-fifties, in a seminal paper, M. G. Krein introduced continuous analogs of Szeg? orthogonal polynomials on the unit circle and established their main properties. In this paper, we generalize these results and subsequent results that he obtained jointly with Langer to the case of matrix-valued functions. Our main theorems are much more involved than their scalar counterparts. They contain new conditions based on Jordan chains and root functions. The proofs require new techniques based on recent results in the theory of continuous analogs of resultant and Bezout matrices and solutions of certain equations in entire matrix functions.     

On iterative and Monte Carlo solutions of the Boltzmann equation

The two most commonly used techniques for solving the Boltzmann equation, with given boundary conditions, are first iterative equations (typically the BGK equation) and Monte Carlo methods. The present work examines the accuracy of two different iterative solutions compared with that of an advanced Monte Carlo solution for a one-dimensional shock wave in a hard sphere gas. It is found that by comparison with the Monte Carlo solution the BGK model is not as satisfactory as the other first iterative solution (Holway's) and that the BGK solution may be improved by using directional temperatures rather than a mean temperature.     

A Jacobi matrix inverse eigenvalue problem with mixed data

In this paper, the inverse eigenvalue problem of reconstructing a Jacobi matrix from part of its eigenvalues and its leading principal submatrix is considered. The necessary and sufficient conditions for the existence and uniqueness of the solution are derived. Furthermore, a numerical algorithm and some numerical examples are given.     

A new iterative approach to fractal models

Mandelbrot is best appreciated for his broad attempt to describe irregular shapes in nature. He founded fractal geometry in 1975. Subsequently the whole fractal theory developed using one-step feedback systems. In 2002, an attempt was made to study and analyze fractal objects using two-step feedback systems. Researchers used superior iteration methods to implement two-step feedback systems. This was the beginning of a new iterative approach in the study of fractal models, and it seems promising to extend fractal theory. The purpose of this paper is to present a review of literature in fractal analysis using this new iterative approach and explore its potential applications.     

A new reconstruction method for a parabolic inverse source problem

In this paper, we focus on the detection of the shape and location of a discontinuous source term from the knowledge of boundary measurements. We propose a non-iterative reconstruction algorithm based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The inverse source problem is formulated as a topology optimization one. A topological sensitivity analysis is derived from an energy-like cost function. The unknown shape of the term source support is reconstructed using a level-set curve of the topological gradient. The efficiency of our algorithm is illustrated by some numerical simulations.     

A proximal iterative approach to a non-convex optimization problem

We consider a variable Krasnosel’skii-Mann algorithm for approximating critical points of a prox-regular function or equivalently for finding fixed-points of its proximal mapping prox_λf. The novelty of our approach is that the latter is not non-expansive any longer. We prove that the sequence generated by such algorithm (via the formula x_k+1=(1−α_k)x_k+α_kprox_λkfx_k, where (α_k) is a sequence in (0,1)), is an approximate fixed-point of the proximal mapping and converges provided that the function under consideration satisfies a local metric regularity condition.     

一个来自振动理论反问题的矩阵方程

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＲｎ×ｍｄｅｎｏｔｅｔｈｅｒｅａｌｎ×ｍｍａｔｒｉｘｓｐａｃｅ ,Ｒｎ×ｍｒ ｉｔｓｓｕｂｓｅｔｗｈｏｓｅｅｌｅｍｅｎｔｓｈａｖｅｒａｎｋｒ ,ＯＲｎ×ｎｔｈｅｓｅｔｏｆａｌｌｎ×ｎｏｒｔｈｏｇｏｎａｌｍａｔｒｉｃｅｓ,ＳＲｎ×ｎ(ＳＲｎ×ｎ≥ ,ＳＲｎ×ｎ>)ｔｈｅｓｅｔｏｆａｌｌｎ×ｎｒｅａｌｓｙｍｍｅｔｒｉｃ (ｓｙｍｍｅｔｒｉｃｐｏｓｉｔｉｖｅｓｅｍｉｄｅｆｉｎｉｔｅ ,ｐｏｓｉｔｉｖｅｄｅｆｉｎｉｔｅ)ｍａｔｒｉｃｅｓ.ＴｈｅｎｏｔａｔｉｏｎＡ>0 (≥ 0 ,<0 ,≤ 0 )ｍ…     

A stability analysis of the jacobi matrix inverse eigenvalue problem

In this paper, we give a perturbation bound for the solution of the Jacobi matrix inverse eigenvalue problem.China State Major Key Project for Basic Researches.