We discuss complexity issues in time dependent adjoint evaluation. We address the question of storage complexity and redundant calculation of intermediate states in adjoint calculations for time dependent flows. Parallel in time solutions are introduced in reverse time integration together with reduced order modelling for the recovery of intermediate forward states between checkpoints.

The approach is illustrated on an identification problem from partial macroscopic variables fields observations and also in the context of shape sensitivity evaluation in fluids for the pressure and viscous drag coefficients.

DOI: https://doi.org/10.5802/smai-jcm.2

Classification: 65Y00, 65Y05, 68W10, 35Q93, 90C52

Keywords: LBM, discrete adjoint, meta model, uncertainty, contour identification, shape optimization, parallel time reversal.

@article{SMAI-JCM_2015__1__5_0, author = {Bijan Mohammadi}, title = {Parallel reverse time integration and reduced order models}, journal = {The SMAI journal of computational mathematics}, pages = {5--28}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {1}, year = {2015}, doi = {10.5802/smai-jcm.2}, mrnumber = {3620368}, zbl = {1416.65391}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.2/} }

Bijan Mohammadi. Parallel reverse time integration and reduced order models. The SMAI journal of computational mathematics, Volume 1 (2015) , pp. 5-28. doi : 10.5802/smai-jcm.2. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.2/

[1] Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., Volume 30 (1998), pp. 139-165 | Article | MR 1609626

[2] A penalization method to take into account obstacles in viscous flows, Numerische Mathematik, Volume 81 (1999), pp. 497-520 | Article | MR 1675200 | Zbl 0921.76168

[3] A multiple shooting algorithm for direct solution of optimal control problems, Proceedings of the 9th IFAC World Congress, Budapest Univ., 1984, pp. 22-31

[4] Reverse accumulation and implicit functions, Optimization Methods and Software, Volume 9 (1998), pp. 307-322 | Article | MR 1637708 | Zbl 0922.65013

[5] The finite element method for elliptic problems, North-Holland, Amsterdam, 1978

[6] A Finite element method for the simulation of Rayleigh-Taylor instability, Lecture Notes in Mathematics, Volume 771 (1979), pp. 145-159 | Article | Zbl 0438.76044

[7] Multifuid incompressible fows by a finite element method, Lecture Notes in Physics, Volume 11 (1991), pp. 158-163

[8] Analysis of the parareal time-parallel time integration method, SIAM J. Sci. Comput., Volume 29 (2007), pp. 556-578 | Article | MR 2306258 | Zbl 1141.65064

[9] Automatic mesh generation. Applications to finite element method, Wiley, London, 1991 | Zbl 0808.65122

[10] A fictitious domain method for external incompressible viscous flows modeled by Navier-Stokes equations, Comput. Meth. Appl. Mech. Eng., Volume 112 (1994), pp. 133-148 | Article | MR 1263284 | Zbl 0845.76069

[11] Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optimization Methods and Software, Volume 1 (1992), pp. 35-54 | Article

[12] Computational derivatives, Springer, New York, 2001

[13] Enabling user-driven checkpointing strategies in reverse mode AD, ECCOMAS CFD conference (Wesseling, ed.), Springer, 2006, pp. 153-162

[14] Tapenade user’s guide, INRIA Technical report, INRIA, 2004, pp. 1-31

[15] An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., Volume 171 (2001), pp. 132-150 | Article | MR 1843643 | Zbl 1057.76039

[16] Fluid flow simulation and optimisation with lattice Boltzmann methods on high performance computers (2010) (Ph. D. Thesis)

[17] Theory of the Lattice Boltzmann Method: Dispersion, Dissipation, Isotropy, Galilean Invariance, and Stability, NASA/CR-2000-210103, ICASE Report No. 2000-17,, ICASE, 2000, pp. 1-45

[18] The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Num. Anal., Volume 31 (1994), pp. 1001-1025 | Article | MR 1286215

[19] A parareal in time discretization of PDE’s, C.R. Acad. Sci. Paris, Serie I., Volume 332 (2001), pp. 1-8 | Zbl 0984.65085

[20] Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Letters, Volume 61 (1988), pp. 2332-2335 | Article

[21] Global optimization, level set dynamics, incomplete sensitivity and regularity control, Int. J. Comp. Fluid. Dynamics, Volume 21 (2008), pp. 61-68 | Article | MR 2352061 | Zbl 1184.76833

[22] Reduced sampling and incomplete sensitivity for low-complexity robust parametric optimization, Int. J. Num. Meth. Fluids, Volume 73 (2013), pp. 307-323 | Article | MR 3104453

[23] Uncertainty Quantification by geometric characterization of sensitivity spaces, Compt. Meth. Appl. Mech. Eng., Volume 280 (2014), pp. 197-221 | Article | MR 3255538 | Zbl 1425.65058

[24] Value at Risk for confidence level quantifications in robust engineering optimization, Optimal Control: Applications and Methods, Volume 35 (2014), pp. 179-190 | Article | MR 3191357 | Zbl 1290.49067

[25] Applied shape optimization for fluids (2nd Edition), Oxford Univ. Press, Oxford, 2009

[26] Plate rigidity inversion in southern California using interseismic GPS velocity field, Geophys. J. Int., Volume 187 (2011), pp. 783-796 | Article

[27] Shape Optimization in Fluid Mechanics, Annu. Rev. of Fluid Mech., Volume 36 (2004), pp. 255-279 | Article | MR 2062314 | Zbl 1076.76020

[28] Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., Volume 79 (1998), pp. 12-49 | Article | MR 965860 | Zbl 0659.65132

[29] The fluid dynamics of heart valves: experimental, theoretical and computational methods, Annu. Rev. Fluid Mech., Volume 14 (1981), pp. 235-259 | Article

[30] A parametric level-set approach for topology optimization of flow domains, Struct. Multidisc. Optim., Volume 41 (2010), pp. 117-131 | Article | MR 2577727 | Zbl 1274.76183

[31] On optimal shapes for Stokes flow, J. Fluid Mech., Volume 70 (1973), pp. 331-340

[32] Optimal shape design for elliptic systems, Springer, Berlin, 1984 | Article

[33] Direct and revers modes of AD for inverse problems, SIAM workshop on computational differentiation, SIAM, 1996, pp. 253-282

[34] Sensitivity estimates for nonlinear mathematical models, Mathematical Modelling and Computational Experiments, Volume 1 (1993), pp. 407-414 | MR 1335161

[35] Derivative based global sensitivity measures and their link with global sensitivity indices, Mathematics and Computers in Simulation, Volume 79 (2009), pp. 3009-3017 | Article | MR 2541313 | Zbl 1167.62005

[36] Adjoint lattice Boltzmann equation for parameter identification, Computers and Fluids, Volume 35 (2006), pp. 805-813 | Article | Zbl 1177.76330