Optimal Shape Problems for Eigenvalues

ABSTRACT

We consider the first Dirichlet eigenvalue for nonhomogeneous membranes. For given volume we want to find the domain which minimizes this eigenvalue. The problem is formulated as a variational free boundary problem. The optimal domain is characterized as the support of the first eigenfunction. We prove enough regularity for the eigenfunction to conclude that the optimal domain has finite parameter. Finally an overdetermined boundary value problem on the regular part of the free boundary is given.     

Conjugate gradient techniques for the optimal control evolution dam problem

This paper considers the numerical simulation of optimal control evolution dam problem by using conjugate gradient method.The paper considers the free boundary value problem related to time dependent fluid flow in a homogeneous earth rectangular dam.The dam is taken to be sufficiently long that the flow is considered to be two dimensional.On the left and right walls of the dam there is a reservoir of fluid at a level dependent on time.This problem can be transformed into a variational inequality on a fixed domain.The numerical techniques we use are based on a linear finite element method to approximate the state equations and a conjugate gradient algorithm to solve the discrete optimal control problem.This algorithm is based on Armijo's rule in the unconstrained optimization theory.The convergence of the discrete optimal solutions to the continuous optimal solutions,and the convergence of the conjugate gradient algorithm are proved.A numerical example is given to determine the location of the minimum surface     

Solution with an internal transition layer for a singularly perturbed second-order equation with an advancing and a retarded argument

We consider a singularly perturbed boundary value problem for a differential equation with a retarded and a deviating argument. By using the method of boundary functions and the sewing method, we find not only a continuous but also a smooth solution of the problem. We prove the existence of a solution with an internal transition layer. A graphical numerical example is presented.     

A multi-item mixture inventory model involving random lead time and demand with budget constraint and surprise function

This study deals with a multi-item mixture inventory model in which both demand and lead time are random. A budget constraint is also added to this model. The optimization problem with budget constraint is then transformed into a multi-objective optimization problem with the help of fuzzy chance-constrained programming technique and surprise function. In our studies, we relax the assumption about the demand, lead time and demand during lead time that follows a known distribution and then apply the minimax distribution free procedure to solve the problem. We develop an algorithm procedure to find the optimal order quantity and optimal value of the safety factor. Finally, the model is illustrated by a numerical example.     

A NUMERICAL METHOD FOR DETERMINING THE OPTIMAL EXERCISE PRICE TO AMERICAN OPTIONS

American Options can be exercised prior to the date of expiration,the valuation of American options then constitutes a free boundary value problem.How to determine the free boundary,i.e. the optimal exercise price,is a key problem.In this paper,a nonlinear equation is given.The free boundary can be obtained by solving the nonlinear equation and the numerical results are better.     

The Stochastic Control Model for Use Conversion of Land

In this paper, the authors investigate the optimal conversion rate at which land use is irreversibly converted from biodiversity conservation to agricultural production. This problem is formulated as a stochastic control model, then transformed into a HJB equation involving free boundary. Since the state equation has singularity, it is difficult to directly derive the boundary value condition for the HJB equation. They provide a new method to overcome the difficulty via constructing another auxi...     

More on the Problem and the Solution of the Optimal Shape of a Javelin

The analogy between the optimal javelin problem and the problem of determining the optimal shape of the free rotating rod has been established and employed to determine the optimal shape of the javelin via Pontryagin's maximum principle. Five distinct variational principles are constructed for boundary value problem describing optimal shape of the javelin. The first integral for this nonlinear system is found. An a priori estimate of the cross-sectional area is obtained. The optimal shape of the javelin or free rotating rod is determined by numerical integration.     

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.     

Minimizing the ruin probability allowing investments in two assets: a two-dimensional problem

We consider in this paper that the reserve of an insurance company follows the classical model, in which the aggregate claim amount follows a compound Poisson process. Our goal is to minimize the ruin probability of the company assuming that the management can invest dynamically part of the reserve in an asset that has a positive fixed return. However, due to transaction costs, the sale price of the asset at the time when the company needs cash to cover claims is lower than the original price. This is a singular two-dimensional stochastic control problem which cannot be reduced to a one-dimensional problem. The associated Hamilton–Jacobi–Bellman (HJB) equation is a variational inequality involving a first order integro-differential operator and a gradient constraint. We characterize the optimal value function as the unique viscosity solution of the associated HJB equation. For exponential claim distributions, we show that the optimal value function is induced by a two-region stationary strategy (“action” and “inaction” regions) and we find an implicit formula for the free boundary between these two regions. We also study the optimal strategy for small and large initial capital and show some numerical examples.     

An application of the exponential cubic splines to numerical solution of a self-adjoint perturbation problem

We present a class of the second order optimal splines difference schemes derived from expomential cubic splines for self-adjoint singularly perturbed 2-point boundary value problem. We prove an optimal error estimate and give illustrative numerical example.     

A problem-solving approach to product design using decision tree induction based on intuitionistic fuzzy

Customer complaint problem is a product design used to understand customer requirements. Furthermore, product design corresponding to customer requirement does not feel adequately solved for a cause of problem. The cause of the problem affecting product design is solved to prevent customer complaint from reoccurring. However, the problems by customer may have observation uncertainty and fuzzy. Fuzzy concept considers not only the degree of membership to an accept set, but also the degree of non-membership to a rejection set. Therefore, we present a new approach for problem solving using decision tree induction based on intuitionistic fuzzy sets in this paper. Under this approach, we first develop the problem formulation for the symptoms and causes of the problem based on intuitionistic fuzzy sets. Next, we identify the cause of the problem using intuitionistic fuzzy decision tree by the problem formulation. We then provide the approach to find the optimal cause of the problem for the consideration of product design. A numerical example is used to illustrate the approach applied for product design.     

Exercisability Randomization of the American Option

Abstract

The valuation of American options is an optimal stopping time problem which typically leads to a free boundary problem. We introduce here the randomization of the exercisability of the option. This method considerably simplifies the problematic by transforming the free boundary problem into an evolution equation. This evolution equation can be transformed in a way that decomposes the value of the randomized option into a European option and the present value of continuously paid benefits. This yields a new binomial approximation for American options. We prove that the method is accurate and numerical results illustrate that it is computationally efficient.     

A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem

This paper is aimed at studying finite element discretization for a class of quadratic boundary optimal control problems governed by nonlinear elliptic equations. We derive a posteriori error estimates for the coupled state and control approximation. Such estimates can be used to construct a reliable adaptive finite element approximation for the boundary optimal control problem. Finally, we present a numerical example to confirm our theoretical results.     

关于某类带小周期参数的方程的数值解法

In this paper,we propose a numerical method for the point boundary value problem with small periodic structure.we give error estimates and numerical example.     

On the treatment of free boundary problems with the heat equation via optimal control

In this paper we consider a free boundary problem of general type for the heat equation in one space dimension. We formulate this problem as an optimal control problem and derive necessary conditions for a solution of it. In order to compute a solution of the control problem, we apply the projection-gradient-method. Simple numerical examples illustrate the results.     

Valuing continuous-installment options

Installment options are path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing European continuous-installment options written on dividend-paying assets in the standard Black–Scholes–Merton framework. The valuation of installment options can be formulated as a free boundary problem, due to the flexibility of continuing or stopping to pay installments. On the basis of a PDE for the initial premium, we derive an integral representation for the initial premium, being expressed as a difference of the corresponding European vanilla value and the expected present value of installment payments along the optimal stopping boundary. Applying the Laplace transform approach to this PDE, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping boundary in a closed form. Abelian theorems of Laplace transforms enable us to characterize asymptotic behaviors of the stopping boundary close and at infinite time to expiry. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the optimal stopping boundary.     

A fast high-order finite difference algorithm for pricing American options

We describe an improvement of Han and Wu’s algorithm [H. Han, X.Wu, A fast numerical method for the Black–Scholes equation of American options, SIAM J. Numer. Anal. 41 (6) (2003) 2081–2095] for American options. A high-order optimal compact scheme is used to discretise the transformed Black–Scholes PDE under a singularity separating framework. A more accurate free boundary location based on the smooth pasting condition and the use of a non-uniform grid with a modified tridiagonal solver lead to an efficient implementation of the free boundary value problem. Extensive numerical experiments show that the new finite difference algorithm converges rapidly and numerical solutions with good accuracy are obtained. Comparisons with some recently proposed methods for the American options problem are carried out to show the advantage of our numerical method.     

On approximation in the fictitious domain method for a bidimensional transport equation

In this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D₂. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D₂. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Optimal investment, consumption and retirement choice problem with disutility and subsistence consumption constraints

In this paper we consider a general optimal consumption-portfolio selection problem of an infinitely-lived agent whose consumption rate process is subject to subsistence constraints before retirement. That is, her consumption rate should be greater than or equal to some positive constant before retirement. We integrate three optimal decisions which are the optimal consumption, the optimal investment choice and the optimal stopping problem in which the agent chooses her retirement time in one model. We obtain the explicit forms of optimal policies using a martingale method and a variational inequality arising from the dual function of the optimal stopping problem. We treat the optimal retirement time as the first hitting time when her wealth exceeds a certain wealth level which will be determined by a free boundary value problem and duality approaches. We also derive closed forms of the optimal wealth processes before and after retirement. Some numerical examples are presented for the case of constant relative risk aversion (CRRA) utility class.     

An adaptive algorithm for the Thomas–Fermi equation

A free boundary value problem is introduced to approximate the original Thomas–Fermi equation. The unknown truncated free boundary is determined iteratively. We transform the free boundary value problem to a nonlinear boundary value problem defined on [0,1]. We present an adaptive algorithm to solve the problem by means of the moving mesh finite element method. Comparison of our numerical results with those obtained by other approaches shows high accuracy of our method.