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25
Pseudotransient continuation and differentialalgebraic equations
 SIAM J. Sci. Comp
, 2003
"... Abstract. Pseudotransient continuation is a practical technique for globalizing the computation of steadystate solutions of nonlinear differential equations. The technique employs adaptive timestepping to integrate an initial value problem derived from an underlying ODE or PDE boundary value prob ..."
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Cited by 22 (8 self)
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Abstract. Pseudotransient continuation is a practical technique for globalizing the computation of steadystate solutions of nonlinear differential equations. The technique employs adaptive timestepping to integrate an initial value problem derived from an underlying ODE or PDE boundary value problem until sufficient accuracy in the desired steadystate root is achieved to switch over to Newtonâ€™s method and gain a rapid asymptotic convergence. The existing theory for pseudotransient continuation includes a global convergence result for differential equations written in semidiscretized methodoflines form. However, many problems are better formulated or can only sensibly be formulated as differentialalgebraic equations (DAEs). These include systems in which some of the equations represent algebraic constraints, perhaps arising from the spatial discretization of a PDE constraint. Multirate systems, in particular, are often formulated as differentialalgebraic systems to suppress fast time scales (acoustics, gravity waves, Alfven waves, near equilibrium chemical oscillations, etc.) that are irrelevant on the dynamical time scales of interest. In this paper we present a global convergence result for pseudotransient continuation applied to DAEs of index 1, and we illustrate it with numerical experiments on model incompressible flow and reacting flow problems, in which a constraint is employed to step over acoustic waves.
Domain decomposition preconditioners for linearâ€“quadratic elliptic optimal control problems
, 2004
"... ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquad ..."
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Cited by 12 (4 self)
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ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the twolevel versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.
PARALLEL ALGORITHMS FOR FLUIDSTRUCTURE INTERACTION PROBLEMS IN HAEMODYNAMICS
"... Abstract. The increasing computational load required by most applications and the limits in hardware performances affecting scientific computing contributed in the last decades to the development of parallel software and architectures. In FluidStructure Interaction (FSI, in short) for haemodynamic ..."
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Cited by 8 (2 self)
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Abstract. The increasing computational load required by most applications and the limits in hardware performances affecting scientific computing contributed in the last decades to the development of parallel software and architectures. In FluidStructure Interaction (FSI, in short) for haemodynamic applications, parallelization and scalability are key issues (see [20]). In this work we introduce a class of parallel preconditioners for the FSI problem obtained by exploiting the blockstructure of the linear system. We stress the possibility of extending the approach to a general linear system with a blockstructure, then we provide a bound in the condition number of the preconditioned system in terms of the conditioning of the preconditioned diagonal blocks, finally we show that the construction and evaluation of the devised preconditioner is modular. The preconditioners are tested on a benchmark 3D geometry discretized in both a coarse and a fine mesh, as well as on two physiological aorta geometries. The simulations that we have performed show an advantage in using the block preconditioners introduced and confirm our theoretical results.
Parallel multilevel restricted Schwarz preconditioners with pollution removing for PDEconstrained optimization
 SIAM J. Sci. Comput
"... Abstract. We develop a class of Vcycle type multilevel restricted additive Schwarz (RAS) methods and study the numerical and parallel performance of the new fully coupled methods for solving large sparse Jacobian systems arising from the discretization of some optimization problems constrained by n ..."
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Cited by 4 (3 self)
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Abstract. We develop a class of Vcycle type multilevel restricted additive Schwarz (RAS) methods and study the numerical and parallel performance of the new fully coupled methods for solving large sparse Jacobian systems arising from the discretization of some optimization problems constrained by nonlinear partial differential equations. Straightforward extensions of the onelevel RAS to multilevel do not work due to the pollution effects of the coarse interpolation. We then introduce, in this paper, a pollution removing coarsetofine interpolation scheme for one of the components of the multicomponent linear system, and show numerically that the combination of the new interpolation scheme with the RAS smoothed multigrid method provides an effective family of techniques for solving rather difficult PDEconstrained optimization problems. Numerical examples involving the boundary control of incompressible NavierStokes flows are presented in detail. Key words. Schwarz preconditioners, domain decomposition, multilevel methods, parallel computing, partial differential equations constrained optimization, inexact Newton, flow control. 1. Introduction. There are two major families of Newton techniques for solving nonlinear optimization problems: reduced space methods, characterized by the partition of the problem into smaller ones at each Newton step, and full space ones. As
On Linear Monotone Iteration And Schwarz Methods For Nonlinear Elliptic PDEs
"... . The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each o ..."
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Cited by 4 (1 self)
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. The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology. Key words. domain decomposition, nonlinear elliptic PDE, Schwarz alternating method, monotone methods, subsolution, supersolution AMS subject classifications. 65N55, 65J15 1.
Using Automatic Differentiation for SecondOrder Matrixfree Methods in PDEconstrained Optimization
, 2000
"... Classical methods of constrained optimization are often based on the assumptions that projection onto the constraint manifold is routine but accessing secondderivative information is not. Both assumptions need revision for the application of optimization to systems constrained by partial differe ..."
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Cited by 2 (0 self)
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Classical methods of constrained optimization are often based on the assumptions that projection onto the constraint manifold is routine but accessing secondderivative information is not. Both assumptions need revision for the application of optimization to systems constrained by partial differential equations, in the contemporary limit of millions of state variables and in the parallel setting. Largescale PDE solvers are complex pieces of software that exploit detailed knowledge of architecture and application and cannot easily be modified to fit the interface requirements of a blackbox optimizer. Furthermore, in view of the expense of PDE analyses, optimization methods not using second derivatives may require too many iterations to be practical. For general problems, automatic differentiation is likely to be the most convenient means of exploiting second derivatives. We delineate a role for automatic differentiation in matrixfree optimization formulations involving Newto...
Using domain decomposition in the JacobiDavidson method
, 2000
"... The JacobiDavidson method is suitable for computing solutions of large ndimensional eigenvalue problems. It needs (approximate) solutions of specific ndimensional linear systems. Here we propose a strategy based on a nonoverlapping domain decomposition technique in order to reduce the wall clock ..."
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Cited by 2 (0 self)
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The JacobiDavidson method is suitable for computing solutions of large ndimensional eigenvalue problems. It needs (approximate) solutions of specific ndimensional linear systems. Here we propose a strategy based on a nonoverlapping domain decomposition technique in order to reduce the wall clock time and local memory requirements. For a model eigenvalue problem we derive optimal coupling parameters. Numerical experiments show the effect of this approach on the overall JacobiDavidson process. The implementation of the eventual process on a parallel computer is beyond the scope of this paper. 2000 Mathematics Subject Classification: 65F15, 65N25, 65N55. Keywords and Phrases: Eigenvalue problems, domain decomposition, JacobiDavidson, Schwarz method, nonoverlapping, iterative methods. Note: The first author's contribution was carried out partly under project MAS2, and sponsored by the Netherlands Organization for Scientific Research (NWO) under Grant No. 611302100. Note: This wo...
Parallel performance of some twolevel ASPIN algorithms, This Proceedings
"... Summary. In this paper we study the parallel performance of some nonlinear additive Schwarz preconditioned inexact Newton methods for solving large sparse system of nonlinear equations arising from the discretization of partial differential equations. The main idea of nonlinear preconditioning is to ..."
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Cited by 1 (1 self)
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Summary. In this paper we study the parallel performance of some nonlinear additive Schwarz preconditioned inexact Newton methods for solving large sparse system of nonlinear equations arising from the discretization of partial differential equations. The main idea of nonlinear preconditioning is to replace an illconditioned nonlinear system by an equivalent nonlinear system that has more balanced nonlinearities. In addition to balance the nonlinearities through nonlinear preconditioning, we also need to make sure that the multilayered iterative solver is scalable with respect to the number of processors. We focus on some twolevel nonlinear additive Schwarz preconditioners, and show numerically that these twolevel methods can reduce the nonlinearities and at the same time maintain the parallel scalability. Parallel numerical results for some high Reynolds number incompressible NavierStokes equations will be presented. 1
Improving Robustness and Parallel Scalability of Newton Method Through Nonlinear
"... Summary. Inexact Newton method with backtracking is one of the most popular techniques for solving large sparse nonlinear systems of equations. The method is easy to implement, and converges well for many practical problems. However, the method is not robust. More precisely speaking, the convergence ..."
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Cited by 1 (1 self)
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Summary. Inexact Newton method with backtracking is one of the most popular techniques for solving large sparse nonlinear systems of equations. The method is easy to implement, and converges well for many practical problems. However, the method is not robust. More precisely speaking, the convergence may stagnate for no obvious reason. In this paper, we extend the recent work of Tuminaro, Walker and Shadid [2002] on detecting the stagnation of Newton method using the angle between the Newton direction and the steepest descent direction. We also study a nonlinear additive Schwarz preconditioned inexact Newton method, and show that it is numerically more robust. Our discussion will be based on parallel numerical experiments on solving some high Reynolds numbers steadystate incompressible NavierStokes equations in the velocitypressure formulation. 1
PARALLEL FULLY COUPLED SCHWARZ PRECONDITIONERS FOR SADDLE POINT PROBLEMS
"... We study some parallel overlapping Schwarz preconditioners for solving Stokeslike problems arising from finite element discretization of incompressible flow problems. Most of the existing methods are based on the splitting of the velocity and pressure variables. With the splitting, fast solution met ..."
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We study some parallel overlapping Schwarz preconditioners for solving Stokeslike problems arising from finite element discretization of incompressible flow problems. Most of the existing methods are based on the splitting of the velocity and pressure variables. With the splitting, fast solution methods are often constructed using various fast Poisson solvers for one of the variables. More recently, several papers have investigated the socalled fully coupled approaches in which the variables are not separated. The fully coupled approach has some advantages over the variable splitting method when solving Stokeslike equations with many variables, where a good splitting may be hard to obtain. In this paper we systematically study the parallel scalability of several versions of the fully coupled Schwarz method for both symmetric and nonsymmetric Stokeslike problems. We show numerically that the performance of a twolevel method with a multiplicative iterative coarse solver is superior to the other variants of Schwarz preconditioners.