Baasansuren Jadamba

A New Convex Inversion Framework for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

This work presents a thorough theoretical and numerical analysis of the elasticity imaging inverse problem of tumor identification in the soft tissue of the human body. Beyond the obvious merits of its applications, this problem also presents significant mathematical challenges. The near incompressibility inherent in the model of linear elasticity in the body gives rise to the “locking effect” and necessitates a unique treatment of both the direct and inverse problems. A general optimization framework for the identification of parameters in saddle point problems is presented along with a new modified output least-squares (MOLS) objective functional. The MOLS functional is shown to be convex, thus overcoming the nonconvexity of the classical output least-squares (OLS) functional, and the new framework is shown to be capable of accommodating both smooth and discontinuous parameters. Generalized derivative formulas for the coefficient-to-solution map are also given along with a complete convergence analysis....

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.

Generalized solutions of quasi variational inequalities

This paper deals with multivalued quasi variational inequalities with pseudo-monotone and monotone maps. The primary objective of this work is to show that the notion of generalized solutions can be employed to investigate multivalued pseudo-monotone quasi variational inequalities. It is a well-known fact that a quasi variational inequality can conveniently be posed as a fixed point problem through the so-called variational selection. For pseudo-monotone maps, the associated variational selection is a nonconvex map, and the fixed point theorems can only be applied under restrictive assumptions on the data of quasi variational inequalities. On the other hand, the generalized solutions are defined by posing a minimization problem which can be solved by a variant of classical Weierstrass theorem. It turns out that far less restrictive assumptions on the data are needed in this case. To emphasis on the strong difference between a classical solution and a generalized solution, we also give a new existence theorem for quasi variational inequalities with monotone maps. The main existence result is proved under a milder coercivity condition. We also relax a few other conditions from the monotone map. Due to its flexibility, it seems that the notion of generalized solutions can be employed to study quasi variational inequalities for other classes of maps as well.

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.

Proximal Methods for the Elastography Inverse Problem of Tumor Identification Using an Equation Error Approach

In this chapter, we study a nonlinear inverse problem in linear elasticity relating to tumor identification by an equation error formulation. This approach leads to a variational inequality as a necessary and sufficient optimality condition. We give complete convergence analysis for the proposed equation error method. Since the considered problem is highly ill-posed, we develop a stable computational framework by employing a variety of proximal point methods and compare their performance with the more commonly used Tikhonov regularization.

A New Energy Inversion for Parameter Identification in Saddle Point Problems with an Application to the Elasticity Imaging Inverse Problem of Predicting Tumor Location

The primary objective of this work is a detailed theoretical and computational study of the elasticity imaging inverse problem for tumor identification within the human body. Apart from this inverse problems important and interesting application, it also poses noteworthy mathematical challenges since the underlying mathematical model is a system of elasticity involving incompressibility. This gives rise to the “locking” effect and special treatment is necessary for both the direct and inverse problems. To study the inverse problem in an optimization framework, we introduce a general computational scheme for handling parameter identification in saddle point problems along with the introduction and analysis of a new energy output least-squares objective functionals. We also present a treatment of the identification of discontinuous elasticity coefficients using the total variation regularization method. General formulas for the computation of the coefficient-to-solution map and a complete convergence analysis are given for the continuous problem as well as for its discrete analogue. Discrete formulas and implementation issues are discussed in detail and numerical examples for smooth and discontinuous coefficients are given.

A first-order adjoint and a second-order hybrid method for an energy output least-squares elastography inverse problem of identifying tumor location

In this paper we investigate the elastography inverse problem of identifying cancerous tumors within the human body. From a mathematical standpoint, the elastography inverse problem consists of identifying the variable Lamé parameter μ in a system of linear elasticity where the underlying object exhibits nearly incompressible behavior. This problem is subsequently posed as an optimization problem using an energy output least-squares (EOLS) functional, but the nonlinearity that arises makes the computation of the EOLS functional’s derivatives challenging. We employ an adjoint method for the computation of the gradient, something shown to be an efficient method in recent studies, and also give a parallelizable hybrid method for the computation of the EOLS functional’s second derivative. Detailed discrete formulas and nontrivial computational examples are provided to show the feasibility of both the adjoint and hybrid approaches. Furthermore, all results are given in the framework of a general saddle point problem allowing easy adaptation to numerous other inverse problems.MSC:35R30, 65N30.

On the Modeling of Some Environmental Games with Uncertain Data

In this note, we deal with a class of environmental games, where the data are affected by uncertainty and are given through their probability distributions. We perform our investigation in the framework of stochastic variational inequalities in Lebesgue spaces.

On Stochastic Variational Inequalities with Mean Value Constraints

In this note, we consider a class of variational inequalities on probabilistic Lebesgue spaces, where the constraints are satisfied on average, and provide an approximation procedure for the solutions. As an application, we investigate the Nash–Cournot oligopoly problem with uncertain data and compare the solutions obtained when the constraints are satisfied on average with the ones obtained when the constraints are satisfied almost surely.

Regularization for state constrained optimal control problems by half spaces based decoupling

Abstract In this paper, we study an abstract constrained optimization problem which subsumes a common model for the optimal control of linear partial differential equations. Our emphasis is the case when the ordering cone for the optimization problem has an empty interior. To circumvent this difficulty, we propose a new regularization approach which is based on the decoupling of the ordering cone by using certain half-spaces. The approach results in a family of regular optimization problems having a simpler structure than the original optimization problem. Existence theorems, convergence analysis, and optimality conditions are given for the regularized problems. We present two numerical examples and the results are quite encouraging, showing the potential of the proposed approach.