Décomposition de domaine pour le contrôle de systèmes gouvernés par des équations d'évolution

We present a non overlapping iterative domain decomposition method with “coupled” Robin transmission conditions. We prove its convergence on an optimal control problem for the wave equation. The linear part of the “feed-back” law associated to the local optimal control problems set on subdomains is independent of the iterative process. The method can be applied, at least formally, to the optimal control of systems governed by evolution equations.     

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.     

Vector domain decomposition schemes for parabolic equations

A new class of domain decomposition schemes for finding approximate solutions of timedependent problems for partial differential equations is proposed and studied. A boundary value problem for a second-order parabolic equation is used as a model problem. The general approach to the construction of domain decomposition schemes is based on partition of unity. Specifically, a vector problem is set up for solving problems in individual subdomains. Stability conditions for vector regionally additive schemes of first- and second-order accuracy are obtained.     

Parallel algorithm for the solutions of PDEs in linux clustered workstations

In this paper we propose parallel algorithm for the solution of partial differential equations over a rectangular domain using the Crank–Nicholson method by cooperation with the DuFort–Frankel method and apply it on a model problem, namely, the heat conduction equation. One of the well known parallel techniques in solving partial differential equations in cluster computing environment is the domain decomposition technique. Using this technique, the whole domain is decomposed into subdomains, each of them has its own boundaries that are called the interface points. Parallelization is realized by approximating interface values using the unconditionally stable DuFort–Frankel explicit scheme, and these values serve as Neumann boundary conditions for the Crank–Nicholson implicit scheme in the subdomains. The numerical results show that our algorithm is more accurate than the algorithm based on the forward explicit method to approximate the values of the interface points, especially, when we use a small number of time steps. Moreover, these numerical results show that increasing the number of processors which are used in the cluster, yields an increase in the algorithm speedup.     

Domain decomposition method and elastic multi-structures: The stiffened plate problem

Summary Domain decomposition methods allow faster solution of partial differential equations in many cases. The efficiency of these methods mainly depends on the variables and operators chosen for the coupling between the subdomains; it is the preconditioning problem. In the modeling of multistructures, the partial differential equations have some specific properties that must be taken into account in a domain decomposition method. Different kinds of elliptic problems modeling stiffened plates in linearized elasticity are compared. One of them is remarkable as far as domain decomposition is concerned, since it is possible to associate particularly efficient preconditioner. A theoretical estimate for the conditioning is given, which is confirmed by several numerical experiments.     

Conditions d'interface pour les méthodes de décomposition de domaine pour le système d'Oseen en dimensions 2 et 3

We study domain decomposition methods for two- and three-dimensional Oseen equations. We propose pseudo-differential interface conditions which lead to the convergence of an additive Schwarz type method in a number of steps equal to the number of subdomains. These conditions are approximated by partial differential conditions for which convergence is proved. These results generalize previous works on scalar equations.     

Optimal Control of Interface Problems with hp-finite Elements

We investigate the optimal control of elliptic partial differential equations with jumping coefficients. As discretization, we use interface concentrated finite elements on subdomains with smooth data. In order to apply convergence results, we prove higher regularity of the optimal solution using the concept of quasi-monotone coefficients and a domain that is injective modulo polynomials of degree 1 at each vertex. Numerical results are presented for a semi-linear control problem with a non-local radiation operator, which models the production process of silicon carbide single crystals.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

Equations of attainable set dynamics,part 2: Partial differential equations

We derive three partial differential equations describing the attainable set dynamics from the local integral funnel equation. They can be considered as new partial differential equations for optimal control. The Bellman equation is a special case of one of them. Three examples are given.     

Studies of an interface relaxation domain decomposition technique using finite elements on a parallel computer

Studies are presented for an interface relaxation domain decomposition technique using finite elements on an iPSC/2 D5 Hypercube Concurrent computer. The general type of problem to be solved is one governed by a partial differential equation. The application of the approach, however, will be extended to a free boundary value problem by appropriate modification of the numerical scheme. Using the domain decomposition technique, the computation domain is subdivided into several subdomains. In addition, on the interfaces between two adjacent subdomains are imposed a continuity condition on one side and an equilibrium condition on the other side. Successive overrelaxation iterative processes are then carried out in all subdomains with a relaxation process imposed on the interfaces. With this domain decomposition technique, the problem can be solved parallelly until convergence is reached both in the interiors and on the interfaces of all subdomains. Moreover, the formulation includes a simple domain decomposer that automatically divides a finite element mesh into a list of subdomains to guarantee load balancing. Furthermore, it is shown, through numerical experiments performed on an example problem of free surface seepage through a porous dam, how the values of the relaxation parameters, the choice of imposed boundary conditions, and the number of subdomains (i.e., the number of processors used) affect the solution convergence in this parallel computing environment. © 1993 John Wiley & Sons, Inc.     

Existence and uniqueness of optimal solutions for multirate partial differential algebraic equations

The numerical simulation of electric circuits including multirate signals can be done by a model based on partial differential algebraic equations. In the case of frequency modulated signals, a local frequency function appears as a degree of freedom in the model. Thus the determination of a solution with a minimum amount of variation is feasible, which allows for resolving on relatively coarse grids. We prove the existence and uniqueness of the optimal solutions in the case of initial-boundary value problems as well as biperiodic boundary value problems. The minimisation problems are also investigated and interpreted in the context of optimal control. Furthermore, we construct a method of characteristics for the computation of optimal solutions in biperiodic problems. Numerical simulations of test examples are presented.     

A characteristic nonoverlapping domain decomposition method for multidimensional convection‐diffusion equations

We develop a quasi‐two‐level, coarse‐mesh‐free characteristic nonoverlapping domain decomposition method for unsteady‐state convection‐diffusion partial differential equations in multidimensional spaces. The development of the domain decomposition method is carried out by utilizing an additive Schwarz domain decomposition preconditioner, by using an Eulerian‐Lagrangian method for convection‐diffusion equations and by delicately choosing appropriate interface conditions that fully respect and utilize the hyperbolic nature of the governing equations. Numerical experiments are presented to illustrate the method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Optimization of Schwarz waveform relaxation over short time windows

Schwarz waveform relaxation algorithms (SWR) are naturally parallel solvers for evolution partial differential equations. They are based on a decomposition of the spatial domain into subdomains, and a partition of the time interval of interest into time windows. On each time window, an iteration, during which subproblems are solved in space-time subdomains, is then used to obtain better and better approximations of the overall solution. The information exchange between subdomains in space-time is performed through classical or optimized transmission conditions (TCs). We analyze in this paper the optimization problem when the time windows are short. We use as our model problem the optimized SWR algorithm with Robin TCs applied to the heat equation. After a general convergence analysis using energy estimates, we prove that in one spatial dimension, the optimized Robin parameter scales like the inverse of the length of the time window, which is fundamentally different from the known scaling on general bounded time windows, which is like the inverse of the square root of the time window length. We illustrate our analysis with a numerical experiment.     

Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

The explicit implicit domain decomposition methods are noniterative types of methods for nonoverlapping domain decomposition but due to the use of the explicit step for the interface prediction, the methods suffer from inaccuracy of the usual explicit scheme. In this article a specific type of first‐ and second‐order splitting up method, of additive type, for the dependent variables is initially considered to solve the two‐ or three‐dimensional parabolic problem over nonoverlapping subdomains. We have also considered the parallel explicit splitting up algorithm to define (predict) the interface boundary conditions with respect to each spatial variable and for each nonoverlapping subdomains. The parallel second‐order splitting up algorithm is then considered to solve the subproblems defined over each subdomain; the correction step will then be considered for the predicted interface nodal points using the most recent solution values over the subdomains. Finally several model problems will be considered to test the efficiency of the presented algorithm. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

条件平均场随机微分方程的最优控制问题

作者研究了一个条件平均场随机微分方程的最优控制问题.这种方程和某些部分信息下的随机最优控制问题有关,并且可以看做是平均场随机微分方程的推广.作者以庞特里雅金最大值原理的形式给出最优控制满足的必要和充分条件.此外,文中给出一个线性二次最优控制问题来说明理论结果的应用.     

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.     

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.     

Distributed optimal control of the viscous Burgers equation via a Legendre pseudo‐spectral approach

This paper presents a computational technique based on the pseudo‐spectral method for the solution of distributed optimal control problem for the viscous Burgers equation. By using pseudo‐spectral method, the problem is converted to a classical optimal control problem governed by a system of ordinary differential equations, which can be solved by well‐developed direct or indirect methods. For solving the resulting optimal control problem, we present an indirect method by deriving and numerically solving the first‐order optimality conditions. Numerical tests involving both unconstrained and constrained control problems are considered. Copyright © 2015 John Wiley & Sons, Ltd.     

Solving the 2-D elliptic Monge-Ampère equation by a Kansa’s method

In this paper, a Kansa’s method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa’s method is a meshfree method which applying the combination of some radial basis functions (such as Hardy’s MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.