Penalization techniques in L ∞ optimization problems with unbounded horizon

In this work we present a numerical procedure for the ergodic optimal minimax control problem. Restricting the development to the case with relaxed controls and using a perturbation of the instantaneous cost function, we obtain discrete solutions U _ε ^k that converge to the optimal relaxed cost U when the relation ship between the parameters of discretization k and penalization ε is an appropriate one. This paper aims to be a tribute to Prof. Roberto L.V. González who died after we finished this work. This paper was supported by grant PIP 5379 CONICET.     

Fast Computational Procedure for Solving Multi-Item Single-Machine Lot Scheduling Optimization Problems

In this paper, we deal with the numerical solution of the optimal scheduling problem in a multi-item single machine. We develop a method of discretization and a computational procedure which allows us to compute the solution in a short time and with a precision of order k, where k is the discretization size. In our method, the nodes of the triangulation mesh are joined by segments of trajectories of the original system. This special feature allows us to obtain precision of order k, which is in general impossible to achieve by usual methods. Also, we develop a highly efficient algorithm which converges in a finite number of steps.     

Two-grid methods for –P1 mixed finite element approximation of general elliptic optimal control problems with low regularity

In this paper, we present a two-grid mixed finite element scheme for distributed optimal control governed by general elliptic equations. –P₁ mixed finite elements are used for the discretization of the state and co-state variables, whereas piecewise constant function is used to approximate the control variable. We first use a new approach to obtain the superclose property between the centroid interpolation and the numerical solution of the optimal control u with order h² under the low regularity. Based on the superclose property, we derive the optimal a priori error estimates. Then, using a postprocessing projection operator, we get a second-order superconvergent result for the control u. Next, we construct a two-grid mixed finite element scheme and analyze a priori error estimates. In the two-grid scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, a numerical example is presented to verify the theoretical results.     

Optimal buffer size for a stochastic processing network in heavy traffic

We consider a one-dimensional stochastic control problem that arises from queueing network applications. The state process corresponding to the queue-length process is given by a stochastic differential equation which reflects at the origin. The controller can choose the drift coefficient which represents the service rate and the buffer size b>0. When the queue length reaches b, the new customers are rejected and this incurs a penalty. There are three types of costs involved: A “control cost” related to the dynamically controlled service rate, a “congestion cost” which depends on the queue length and a “rejection penalty” for the rejection of the customers. We consider the problem of minimizing long-term average cost, which is also known as the ergodic cost criterion. We obtain an optimal drift rate (i.e. an optimal service rate) as well as the optimal buffer size b ^*>0. When the buffer size b>0 is fixed and where there is no congestion cost, this problem is similar to the work in Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005). Our method is quite different from that of (Ata, Harrison and Shepp (Ann. Appl. Probab. 15, 1145–1160, 2005)). To obtain a solution to the corresponding Hamilton–Jacobi–Bellman (HJB) equation, we analyze a family of ordinary differential equations. We make use of some specific characteristics of this family of solutions to obtain the optimal buffer size b ^*>0. A.P. Weerasinghe’s research supported by US Army Research Office grant W911NF0510032.     

Stable enrichment and local preconditioning in the particle-partition of unity method

This paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods. In particular we focus on the particle-partition of unity method (PPUM) yet the presented results hold for any partition of unity based enrichment scheme. The goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h-version independent of the regularity of the solution. Hence, we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme. To this end we enrich our PPUM function space in an enrichment zone hierarchically near the singularities of the solution. This initial enrichment however can lead to a severe ill-conditioning and can compromise the stability of the discretization. To overcome the ill-conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well-conditioned basis independent of the employed initial enrichment. The construction of this preconditioner is of linear complexity with respect to the number of discretization points. We obtain optimal error bounds for an enriched PPUM discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super-convergence. The results of our numerical experiments clearly show that our enriched PPUM with local preconditioning recovers the optimal convergence rate of O(h ^p ) of the uniform h-version globally. For the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of O(h ^p+δ ) with d 3 \frac12{\delta\geq\frac{1}{2}} for p = 1. The convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation.     

Risk sensitive control of pure jump processes on a general state space

We study stochastic control problem for pure jump processes on a general state space with risk sensitive discounted and ergodic cost criteria. For the discounted cost criterion we prove the existence and Hamilton–Jacobi–Bellman characterization of optimal α-discounted control for bounded cost function. For the ergodic cost criterion we assume a Lyapunov type stability assumption and a small cost condition. Under these assumptions we show the existence of the optimal risk-sensitive ergodic control.     

Numerical solution of an optimal investment problem with proportional transaction costs

This paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite-horizon optimal investment problem with proportional transaction costs. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable. The proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain, a characteristics method for time discretization to deal with the large gradients of the solution, a Newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle (free boundary) feature. Moreover, piecewise linear Lagrange finite elements for spatial discretization are considered. Numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in Dai and Yi (2009) [6].     

Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

We present an optimal error estimate of the numerical velocity, pressure, and angular velocity for the fully discrete penalty finite element method of the micropolar equations when the parameters ?, Δ t, and h are sufficiently small. In order to obtain this estimate, we present the time discretization of the penalty micropolar equation that is based on the backward Euler scheme; the spatial discretization of the time discretized penalty micropolar equation is based on a finite elements space pair (X _h , M _h ) that satisfies some approximations properties.     

Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical Schemes

Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretization parameters approach zero. We also prove convergence of an Eulerian scheme, under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation. As a by-product, we obtain existence and uniqueness of the solution. All the results of convergence are proved with respect to the Wasserstein distance. We also show that L ¹ spaces are not natural for such equations, since we lose uniqueness of the solution.     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

Numerical solution of a variational problem with L∞ functionals

We consider the problem which consists in finding an optimal Lipschitz extension to the domain Ω of functions that verify the restriction u = g on ∂Ω. This work deals with the numerical approximations of the problem in dimension two. Using a discretization procedure based on finite differences method we obtain a large scale non smooth convex minimization problem, which is solved via Variable Metric Hibrid Proximal Point Method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Bellman Equations in Quadratic Ergodic Control with Controller Constraints

We consider the Bellman equation related to the quadratic ergodic control problem for stochastic differential systems with controller constraints. We solve this equation rigidly in C ² -class, and give the minimal value and the optimal control. Accepted 9 January 1997     

Numerical analysis of incompressible wormhole propagation with mass-preserving characteristic mixed finite element procedure

In this article, we construct a new combined characteristic mixed finite element procedure to simulate the incompressible wormhole propagation. In this procedure, we use the classical mixed finite element method to solve the pressure equation and a modified mass-preserving characteristic finite element method for the solute transport equation, and solve the porosity function straightly by the given concentration. This combined method not only keeps mass balance globally but also preserves maximum principle for the porosity. We considered the corresponding convergence and derive the optimal L²-norm error estimate. Finally, we present some numerical examples to confirm theoretical analysis.

     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

Search for maximum points of a vector criterion based on decomposition properties

We address the problem of optimal reconstruction of the values of a linear operator on ℝ ^d or ℤ ^d from approximate values of other operators. Each operator acts as the multiplication of the Fourier transform by a certain function. As an application, we present explicit expressions for optimal methods of reconstructing the solution of the heat equation (for continuous and difference models) at a given instant of time from inaccurate measurements of this solution at other time instants.     

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L ¹-norm and with order 1/2 w.r.t. the L ²-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.     

On a backward problem for fractional diffusion equation with Riemann-Liouville derivative

In the present paper, we study the initial inverse problem (backward problem) for a two-dimensional fractional differential equation with Riemann-Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well-posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L² and H^τ norms is considered and illustrated by numerical example.     

Stability of central finite difference schemes for the Heston PDE

This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semidiscrete systems with nonnormal matrices A. By employing the logarithmic spectral norm we prove practical, rigorous stability bounds. Our theoretical stability results are illustrated by ample numerical experiments. We also apply the analysis to obtain useful stability bounds for time discretization methods.     

A comparison of asymptotic analytical formulae with finite-difference approximations for pricing zero coupon bond

In this paper we solve numerically a degenerate parabolic equation with dynamical boundary condition for pricing zero coupon bond and compare numerical solution with asymptotic analytical solution. First, we discuss an approximate analytical solution of the model and its order of accuracy. Then, starting from the divergent form of the equation we implement the finite-volume method of Song Wang (IMA J Numer Anal 24:699–720, 2004) to discretize the differential problem. We show that the system matrix of the discretization scheme is a M-matrix, so that the discretization is monotone. This provides the non-negativity of the price with respect to time if the initial distribution is nonnegative. Numerical experiments demonstrate second order of convergence for difference scheme when the node is not too close to the point of degeneration.     

Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.