An existence theory for a minimum energy problem

In this paper we deal with existence theory and develop it for the simple case of the minimum energy problem, as described by Pironneau (1984). We shall treat this problem for the differential inequality by introducing the penalized differential equation and then taking limits of the equations resulting from the penalized approximation.     

Method of penalization for the state equation for an elliptical optimal control problem

We solve by finite difference method an optimal control problem of a system governed by a linear elliptic equation with pointwise control constraints and non-local state constraints. A discrete optimal control problem is approximated by a minimization problem with penalized state equation. We derive the error estimates for the distance between the exact and regularized solutions. We also prove the rate of convergence of block Gauss–Seidel iterative solution method for the penalized problem. We present and analyze the results of the numerical experiments.     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.     

Penalty approximation of a Sobolev optimal control problem with state constraints

In this paper, we study the approximation by the penalty method of a control problem governed by a pseudo-parabolic equation with a noncoercive control functional and with control and state constraints. The existence of solutions to the penalized problems is established. In addition, the convergence of the penalized problems to the solution, the Lagrange multipliers, and the minimum value of the original problem is studied. The results apply to Sobolev and parabolic equations as well.This work was partially supported by the National Science Foundation, Grant No. MCS-79-02037. The author would like to thank Professor A. B. Schwarzkopf for his helpful comments on this paper.     

Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results.     

Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results. Received: February 16, 2005     

Obstacle problem for nonlinear integro-differential equations arising in option pricing

Abstract  We study the obstacle problem for a class of nonlinear integro-partial differential equations of second order, possibly degenerate, which includes the equation modeling American options in a jump-diffusion market with large investor. The viscosity solutions setting reveals appropriate, because of a monotonicity property with respect to the integral term. The same property allows to approximate the problem by penalization and to obtain the existence and uniqueness of solutions via a comparison principle. We also give uniform estimates of the solutions of the penalized problems which allow to prove further regularity. Keywords: Integro-differential equations, Obstacle problem, Viscosity solutions, American options Mathematics Subject Classification (2000): 45K05, 35K85, 49L25, 91B24     

Analysis and discretization of the volume penalized Laplace operator with Neumann boundary conditions

We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains.     

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non‐linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed‐point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

A new class of complementarity functions for symmetric cone complementarity problems

A popular approach to solving the complementarity problem is to reformulate it as an equivalent equation system via a complementarity function. In this paper, we propose a new class of functions, which contains the penalized natural residual function and the penalized Fischer–Burmeister function for symmetric cone complementarity problems. We show that this class of functions is indeed a class of complementarity functions. We finally prove that the merit function of this new class of complementarity functions is coercive.     

A Penalization Approach for Tomographic Reconstruction of Binary Axially Symmetric Objects

We propose a variational method for tomographic reconstruction of blurred and noised binary images based on a penalization process of a minimization problem settled in the space of bounded variation functions. We prove existence and/or uniqueness results and derive a penalized optimality system. Numerical simulations are provided to demonstrate the relevance of the approach.     

Variational approach and optimal control of a PEM fuel cell

The purpose of this paper is to propose and study a mathematical model and a boundary control problem associated to the miscible displacement of hydrogen through the porous anode of a PEM fuel cell. Throughout the paper, we study certain variational problems with a priori regularity properties of the weak solutions. We obtain the existence of less regular solutions and then we prove the desired regularity of these solutions. We consider a control problem that permits to determine the boundary distribution of the pressure which provides an optimal configuration for the temperature and for the concentration, as well. Since the solution of the problem is not unique, the control variable does not appear explicitly in the definition of our cost functional. To overcome this difficulty, we introduce a family of penalized control problems which approximates our boundary control problem. The necessary conditions of optimality are derived by passing to the limit in the penalized optimality conditions.     

A Penalty Approach to Optimal Control of Allen-Cahn Variational Inequalities: MPEC-View

A scalar Allen-Cahn-MPEC problem is considered and a penalization technique is applied to show the existence of an optimal control. We show that the stationary points of the penalized problems converge to some stationary points of the limit problem, which however are weaker than C-stationarity conditions.     

Estimation of elastic coefficients for a multiple beam structure

In this paper, we consider the estimation of spatially dependent elastic parameters in a static distributed model of a simple structure composed of two beams at a fixed angle to one another. We formulate the potential energy functional of the system and obtain existence of optimal estimators to a regularized-output least-squares estimation problem. We discuss regularity and approximation results for the basic problem and penalized problem in which nonconforming elements are used to model the junction of the beams. Numerical examples are presented and generalizations to multiple-beam systems are discussed.This work was supported in part by AFOSR Grant 91-0017.     

Inverse problems for multi-valued quasi variational inequalities and noncoercive variational inequalities with noisy data

ABSTRACT

We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models. We study the inverse problem by posing optimization problems using the output least-squares and the modified output least-squares. Using regularization, penalization, and smoothing, we obtain a single-valued parameter-to-selection map and study its differentiability. We consider optimization problems using the output least-squares and the modified output least-squares for the regularized, penalized and smoothened variational inequality. We give existence results, convergence analysis, and optimality conditions. We provide applications and numerical examples to justify the proposed framework.     

Power Penalty Method for a Linear Complementarity Problem Arising from American Option Valuation

In this paper, we present a power penalty function approach to the linear complementarity problem arising from pricing American options. The problem is first reformulated as a variational inequality problem; the resulting variational inequality problem is then transformed into a nonlinear parabolic partial differential equation (PDE) by adding a power penalty term. It is shown that the solution to the penalized equation converges to that of the variational inequality problem with an arbitrary order. This arbitrary-order convergence rate allows us to achieve the required accuracy of the solution with a small penalty parameter. A numerical scheme for solving the penalized nonlinear PDE is also proposed. Numerical results are given to illustrate the theoretical findings and to show the effectiveness and usefulness of the method. This work was partially supported by a research grant from the University of Western Australia and the Research Grant Council of Hong Kong, Grants PolyU BQ475 and PolyU BQ493.     

带注资的二维复合泊松模型的最优分红

研究建立两类理赔关系的二维复合泊松模型的最优分红与注资问题,目标为最大化分红减注资的折现. 该问题由随机控制问题刻画, 通过解相应的哈密尔顿-雅克比-贝尔曼(HJB)方程,得到了最优分红策略,并在指数理赔时明确地解决该问题.     

Finite element approximation of a prestressed shell model

This work deals with the finite element approximation of a prestressed shell model formulated in Cartesian coordinates system. The considered constrained variational problem is not necessarily positive. Moreover, because of the constraint, it cannot be discretized by conforming finite element methods. A penalized version of the model and its discretization are then proposed. We prove existence and uniqueness results of solutions for the continuous and discrete problems, and we derive optimal a priori error estimates. Numerical tests that validate and illustrate our approach are given.     

Optimal control of solutions of the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation

An optimal control problem for a second-order Sobolev type equation with a relatively polynomially bounded operator pencil is considered. We prove the existence and uniqueness of a strong solution of the Showalter-Sidorov problem for this equation. Necessary and sufficient conditions for the existence and uniqueness of an optimal control of such solutions are obtained. We study the Showalter-Sidorov-Dirichlet problem for the Boussinesq-Love equation.     

Minimax Control of Parabolic Systems with Dirichlet Boundary Conditions and State Constraints

In this paper we formulate and study a minimax control problem for a class of parabolic systems with controlled Dirichlet boundary conditions and uncertain distributed perturbations under pointwise control and state constraints. We prove an existence theorem for minimax solutions and develop effective penalized procedures to approximate state constraints. Based on a careful variational analysis, we establish convergence results and optimality conditions for approximating problems that allow us to characterize suboptimal solutions to the original minimax problem with hard constraints. Then passing to the limit in approximations, we prove necessary optimality conditions for the minimax problem considered under proper constraint qualification conditions. Accepted 7 June 1996