A Monte Carlo method for solving unsteady adjoint equations

Traditionally, solving the adjoint equation for unsteady problems involves solving a large, structured linear system. This paper presents a variation on this technique and uses a Monte Carlo linear solver. The Monte Carlo solver yields a forward-time algorithm for solving unsteady adjoint equations. When applied to computing the adjoint associated with Burgers' equation, the Monte Carlo approach is faster for a large class of problems while preserving sufficient accuracy.

[1]  Vikram Aggarwal,et al.  Improved Monte Carlo Linear Solvers Through Non-diagonal Splitting , 2003, ICCSA.

[2]  J. Westlake.  Handbook of Numerical Matrix Inversion and Solution of Linear Equations , 1968 .

[3]  Ivan Dimov,et al.  Random walk on distant mesh points Monte Carlo methods , 1993 .

[4]  Roger Temam,et al.  DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms , 2001, Journal of Fluid Mechanics.

[5]  T. Barth,et al.  Error estimation and adaptive discretization methods in computational fluid dynamics , 2003 .

[6]  Johan Hoffman,et al.  WEAK UNIQUENESS OF THE NAVIER-STOKES EQUATIONS AND ADAPTIVE TURBULENCE SIMULATION , 2005 .

[7]  Vassil N. Alexandrov,et al.  Relaxed Monte Carlo Linear Solver , 2001, International Conference on Computational Science.

[8]  Chih Jeng Kenneth Tan.  Solving Systems of Linear Equations with Relaxed Monte Carlo Method , 2004, The Journal of Supercomputing.

[9]  Andreas Griewank,et al.  Advantages of Binomial Checkpointing for Memory-reduced Adjoint Calculations , 2004 .

[10]  Arthur E. Bryson,et al.  Energy-state approximation in performance optimization of supersonic aircraft , 1969 .

[11]  Pat Hanrahan,et al.  Monte Carlo evaluation of non-linear scattering equations for subsurface reflection , 2000, SIGGRAPH.

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

[13]  E. S. Levinsky,et al.  Lifting-surface theory for V/STOL aircraft in transition and cruise. II , 1969 .

[14]  Andreas Griewank,et al.  A mathematical view of automatic differentiation , 2003, Acta Numerica.

[15]  Vassil Alexandrov.  Efficient parallel Monte Carlo methods for matrix computations , 1998 .

[16]  Giray Ökten,et al.  Solving Linear Equations by Monte Carlo Simulation , 2005, SIAM J. Sci. Comput..

[17]  L. J. Comrie,et al.  Mathematical Tables and Other Aids to Computation. , 1946 .

[18]  M. Giles,et al.  Adjoint Error Correction for Integral Outputs , 2003 .

[19]  Niles A. Pierce,et al.  An Introduction to the Adjoint Approach to Design , 2000 .

[20]  J. Hoffman.  ON DUALITY BASED A POSTERIORI ERROR ESTIMATION IN VARIOUS NORMS AND LINEAR FUNCTIONALS FOR LES , 2004 .

[21]  R. A. Leibler,et al.  Matrix inversion by a Monte Carlo method , 1950 .

[22]  Johan Hoffman,et al.  On Duality-Based A Posteriori Error Estimation in Various Norms and Linear Functionals for Large Eddy Simulation , 2004, SIAM J. Sci. Comput..

[23]  R. Temam,et al.  Feedback control for unsteady flow and its application to the stochastic Burgers equation , 1993, Journal of Fluid Mechanics.

[24]  Max Gunzburger,et al.  The Velocity Tracking Problem for Navier--Stokes Flows with Bounded Distributed Controls , 1999 .

[25]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[26]  Isabelle Charpentier,et al.  Checkpointing Schemes for Adjoint Codes: Application to the Meteorological Model Meso-NH , 2000, SIAM J. Sci. Comput..

[27]  Max Gunzburger,et al.  Analysis and approximation for linear feedback control for tracking the velocity in Navier–Stokes flows , 2000 .

[28]  N. Metropolis,et al.  The Monte Carlo method. , 1949 .

[29]  J. Hammersley.  SIMULATION AND THE MONTE CARLO METHOD , 1982 .

[30]  Antony Jameson,et al.  Aerodynamic design via control theory , 1988, J. Sci. Comput..