Variational Inequality Approach to Stochastic Nash Equilibrium Problems with an Application to Cournot Oligopoly

In this note we investigate stochastic Nash equilibrium problems by means of monotone variational inequalities in probabilistic Lebesgue spaces. We apply our approach to a class of oligopolistic market equilibrium problems where the data are known through their probability distributions.Comment: 19 pages, 2 table

Identification Of A Parameter in Fourth-Order Partial Differential Equations By An Equation Error Approach

The objective of this short note is to employ an equation error approach to identify a variable parameter in fourth-order partial differential equations. Existence and convergence results are given for the optimization problem emerging from the equation error formulation. Finite element based numerical experiments show the effectiveness of the proposed framework

Identification of Flexural Rigidity in a Kirchhoff Plates Model Using a Convex Objective and Continuous Newton Method

This work provides a detailed theoretical and numerical study of the inverse problem of identifying flexural rigidity in Kirchhoff plate models. From a mathematical standpoint, this inverse problem requires estimating a variable coefficient in a fourth-order boundary value problem.This inverse problem and related estimation problems associated with general plates and shellmodels have been investigated by numerous researchers through an optimization framework using the output least-squares (OLSs) formulation. OLS yields a nonconvex framework and hence it is suitable for investigating only the local behavior of the solution. In this work, we propose a new convex framework for the inverse problem of identifying a variable parameter in a fourth-order inverse problem. Existence results, optimality conditions, and discretization issues are discussed in detail. The discrete inverse problem is solved by using a continuous Newton method. Numerical results show the feasibility of the proposed framework

Efficiency and Vulnerability Analysis for Congested Networks with Random Data

In this note, we combine two theories that have been proposed in the last decade: the theory of vulnerability and efficiency of a congested network, and the theory of stochastic variational inequalities. As a result, we propose a model that describes the performance and vulnerability of a congested network with random traffic demands and where the travel time can be affected by uncertainty. As an application, we investigate in detail the famous Braess’ network

Equation error approach for elliptic inverse problems with an application to the identification of Lamé parameters

The method of equation error can be posed and analyzed in an abstract setting that encompasses a variety of elliptic inverse problems, in which a coefficient in an elliptic partial differential equation is to be estimated from a measurement of the solution to a boundary value problem. Stability in the presence of measurement error is obtained by regularization, and since the abstract setting admits the use of total variation regularization, rapidly varying or even discontinuous coefficients can be estimated. The proposed method effectively identifies Lamé\u27 parameters in the system of linear isotropic elasticity

An equation error approach for the elasticity imaging inverse problem for predicting tumor location

The primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body. This nonlinear inverse problem not only represents an important and interesting application, it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility. Due to the locking effect, classical finite element methods are not effective for incompressible elasticity equations. Therefore, special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we propose an extension of the equation error approach. We focus on two cases, namely when the material parameter is sufficiently smooth and when it is may be discontinuous. For the latter case, we extend the total variation regularization method to the elasticity imaging inverse problem. We give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem. We give sufficient details on the discrete formulas as well as on the implementation issues. Numerical examples for smooth and discontinuous coefficients are given