Parallel Algorithms for PDE-Constrained Optimization

PDE-constrained optimization refers to the optimization of systems governed by partial differential equations (PDEs). The simulation problem is to solve the PDEs for the state variables (e.g. displacement, velocity, temperature, electric field, magnetic field, species concentration), given appropriate data (e.g. geometry, coefficients, boundary conditions, initial conditions, source functions). The optimization problem seeks to determine some of these data—the decision variables—given performance goals in the form of an objective function and possibly inequality or equality constraints on the behavior of the system. Since the behavior of the system is modeled by the PDEs, they appear as (usually equality) constraints in the optimization problem. We will refer to these PDE constraints as the state equations. Let u represent the state variables, d the decision variables, J the objective function, c the residual of the state equations, and h the residual of the inequality constraints. We

[1]  George Biros,et al.  DYNAMIC DATA-DRIVEN INVERSION FOR TERASCALE SIMULATIONS: REAL-TIME IDENTIFICATION OF AIRBORNE CONTAMINANTS , 2005, ACM/IEEE SC 2005 Conference (SC'05).

[2]  George Biros,et al.  Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part I: The Krylov-Schur Solver , 2005, SIAM J. Sci. Comput..

[3]  Karl Kunisch,et al.  A Comparison of a Moreau-Yosida-Based Active Set Strategy and Interior Point Methods for Constrained Optimal Control Problems , 2000, SIAM J. Optim..

[4]  Alfio Borzì,et al.  Multigrid methods for parabolic distributed optimal control problems , 2003 .

[5]  C. Vogel.  Computational Methods for Inverse Problems , 1987 .

[6]  Curtis R. Vogel,et al.  Iterative Methods for Total Variation Denoising , 1996, SIAM J. Sci. Comput..

[7]  Wolfgang Bangerth,et al.  Adaptive Finite Element Methods for the Identification of Distributed Parameters in Partial Differential Equations , 2002 .

[8]  Eldad Haber,et al.  A parallel method for large scale time domain electromagnetic inverse problems , 2008 .

[9]  Stefan Ulbrich,et al.  Superlinear Convergence of Affine-Scaling Interior-Point Newton Methods for Infinite-Dimensional Nonlinear Problems with Pointwise Bounds , 2000, SIAM J. Control. Optim..

[10]  D. Keyes,et al.  Jacobian-free Newton-Krylov methods: a survey of approaches and applications , 2004 .

[11]  William W. Symes,et al.  Velocity inversion by differential semblance optimization , 1991 .

[12]  Robert Michael Lewis,et al.  Practical Aspects Of Variable Reduction Formulations And Reduced Basis Algorithms In Multidisciplina , 1995 .

[13]  Larisa Beilina.  Adaptive hybrid FEM/FDM methods for inverse scattering problems , 2002 .

[14]  L. S. Hou,et al.  Numerical Approximation of Optimal Flow Control Problems by a Penalty Method: Error Estimates and Numerical Results , 1999, SIAM J. Sci. Comput..

[15]  Xiao-Chuan Cai,et al.  Parallel Full Space SQP Lagrange-Newton-Krylov-Schwarz Algorithms for PDE-Constrained Optimization Problems , 2005, SIAM J. Sci. Comput..

[16]  A. Tarantola.  Inversion of seismic reflection data in the acoustic approximation , 1984 .

[17]  O. Ghattas,et al.  Parallel Preconditioners for KKT Systems Arising in Optimal Control of Viscous Incompressible Flows , 2000 .

[18]  Matthias Heinkenschloss,et al.  DOMAIN DECOMPOSITION PRECONDITIONERS FOR LINEAR–QUADRATIC ELLIPTIC OPTIMAL CONTROL PROBLEMS , 2004 .

[19]  M. Heinkenschloss,et al.  A spatial domain decomposition method for parabolic optimal control problems , 2007 .

[20]  Stefan Volkwein,et al.  Mesh-Independence for an Augmented Lagrangian-SQP Method in Hilbert Spaces , 2000, SIAM J. Control. Optim..

[21]  Thomas Slawig,et al.  Adjoint gradients compared to gradients from algorithmic differentiation in instantaneous control of the Navier-Stokes equations , 2003, Optim. Methods Softw..

[22]  A. Walther,et al.  An optimal memory‐reduced procedure for calculating adjoints of the instationary Navier‐Stokes equations , 2006 .

[23]  M. Heinkenschloss,et al.  Domain decomposition methods for advection dominated linear-quadratic elliptic optimal control problems , 2005 .

[24]  C. Kelley,et al.  Quasi Newton methods and unconstrained optimal control problems , 1986, 1986 25th IEEE Conference on Decision and Control.

[25]  Tiankai Tu,et al.  High Resolution Forward And Inverse Earthquake Modeling on Terascale Computers , 2003, ACM/IEEE SC 2003 Conference (SC'03).

[26]  V. Schulz,et al.  Multigrid optimization in applications , 2000 .

[27]  Rolf Rannacher,et al.  A Priori Error Estimates for the Finite Element Discretization of Elliptic Parameter Identification Problems with Pointwise Measurements , 2005, SIAM J. Control. Optim..

[28]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[29]  Ekkehard W. Sachs,et al.  Multilevel Algorithms for Constrained Compact Fixed Point Problems , 1994, SIAM J. Sci. Comput..

[30]  Matthias Heinkenschloss,et al.  A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems , 2005 .

[31]  C. Bunks,et al.  Multiscale seismic waveform inversion , 1995 .

[32]  E. Haber,et al.  A multigrid method for distributed parameter estimation problems. , 2003 .

[33]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[34]  David E. Keyes,et al.  Using automatic differentiation for second-order matrix-free methods in PDE-constrained optimization , 2000 .

[35]  W. K. Anderson,et al.  Achieving High Sustained Performance in an Unstructured Mesh CFD Application , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[36]  Max Gunzburger,et al.  Perspectives in flow control and optimization , 1987 .

[37]  Hoang Nguyen,et al.  Neumann-Neumann Domain Decomposition Preconditioners for Linear-Quadratic Elliptic Optimal Control Problems , 2006, SIAM J. Sci. Comput..

[38]  Matthias Heinkenschloss,et al.  Preconditioners for Karush-Kuhn-Tucker Matrices Arising in the Optimal Control of Distributed Systems , 1998 .

[39]  Barbara Kaltenbacher.  V-cycle convergence of some multigrid methods for ill-posed problems , 2003, Math. Comput..

[40]  Roland Becker,et al.  A posteriori error estimation for finite element discretization of parameter identification problems , 2004, Numerische Mathematik.

[41]  A. Borzì.  MULTIGRID METHODS FOR OPTIMALITY SYSTEMS , 2004 .

[42]  J. King,et al.  Multilevel algorithms for ill-posed problems , 1992 .

[43]  Karl Kunisch,et al.  A multigrid approach to the optimal control of solid fuel ignition problems , 2000 .

[44]  George Biros,et al.  Parallel Lagrange-Newton-Krylov-Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange-Newton Solver and Its Application to Optimal Control of Steady Viscous Flows , 2005, SIAM J. Sci. Comput..

[45]  Michael Hintermüller,et al.  Globalization of SQP-Methods in Control of the Instationary Navier-Stokes Equations , 2002 .

[46]  Jacques-Louis Lions.  Some Aspects of the Optimal Control of Distributed Parameter Systems , 1987 .

[47]  M. Heinkenschloss,et al.  Global Convergence of Trust-Region Interior-Point Algorithms for Infinite-Dimensional Nonconvex Mini , 1999 .

[48]  S. Scott Collis,et al.  Analysis of the Streamline Upwind/Petrov Galerkin Method Applied to the Solution of Optimal Control Problems ∗ , 2002 .

[49]  Ekkehard W. Sachs,et al.  Block Preconditioners for KKT Systems in PDE—Governed Optimal Control Problems , 2001 .

[50]  S. Ulbrich.  Generalized SQP-Methods with ''Parareal'' Time-Domain Decomposition for Time-dependent PDE-constrained Optimization , 2007 .

[51]  Jorge Nocedal,et al.  A Reduced Hessian Method for Large-Scale Constrained Optimization , 1995, SIAM J. Optim..

[52]  Volker Schulz,et al.  Simultaneous Pseudo-Timestepping for PDE-Model Based Optimization Problems , 2004, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[53]  Marek Behr,et al.  The effect of stabilization in finite element methods for the optimal boundary control of the Oseen equations , 2004 .

[54]  Karl Kunisch,et al.  The Numerical Solution of the Steady State Solid Fuel Ignition Model and Its Optimal Control , 2000, SIAM J. Sci. Comput..

[55]  Barbara Kaltenbacher,et al.  Identification of nonlinear B–H curves based on magnetic field computations and multigrid methods for ill-posed problems , 2003, European Journal of Applied Mathematics.

[56]  P. Hansen.  Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion , 1987 .

[57]  George Biros,et al.  Parallel Multiscale Gauss-Newton-Krylov Methods for Inverse Wave Propagation , 2002, ACM/IEEE SC 2002 Conference (SC'02).

[58]  Stephen G. Nash,et al.  Model Problems for the Multigrid Optimization of Systems Governed by Differential Equations , 2005, SIAM J. Sci. Comput..

[59]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[60]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[61]  Mark F. Adams,et al.  Ultrascalable Implicit Finite Element Analyses in Solid Mechanics with over a Half a Billion Degrees of Freedom , 2004, Proceedings of the ACM/IEEE SC2004 Conference.

[62]  Eyal Arian,et al.  MULTIGRID ONE SHOT METHODS FOR OPTIMAL CONTROL PROBLEMS: INFINITE DIMENSIONAL CONTROL , 1994 .

[63]  E. Haber,et al.  Preconditioned all-at-once methods for large, sparse parameter estimation problems , 2001 .

[64]  E. Haber.  Quasi-Newton methods for large-scale electromagnetic inverse problems , 2005 .

[65]  S. Ta'asan,et al.  ANALYSIS OF THE HESSIAN FOR AERODYNAMIC OPTIMIZATION: INVISCID FLOW , 1996 .

[66]  Andreas Griewank,et al.  Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation , 1992 .

[67]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[68]  E. Haber,et al.  Inversion of 3D electromagnetic data in frequency and time domain using an inexact all-at-once approach , 2004 .