Wave Equation Simulation Using a Compressed Modeler

Victor Pereyra

Energy Resources Engineering, Stanford University, Stanford, USA.

DOI: 10.4236/ajcm.2013.33033                

Abstract

Repeated simulations of large scale wave propagation problems are prevalent in many fields. In oil exploration earth imaging problems, the use of full wave simulations is becoming routine and it is only hampered by the extreme computational resources required. In this contribution, we explore the feasibility of employing reduced-order modeling techniques in an attempt to significantly decrease the cost of these calculations. We consider the acoustic wave equation in two-dimensions for simplicity, but the extension to three-dimensions and to elastic or even anysotropic problems is clear. We use the proper orthogonal decomposition approach to model order reduction and describe two algorithms: the traditional one using the SVD of the matrix of snapshots and a more economical and flexible one using a progressive QR decomposition. We include also two a posteriori error estimation procedures and extensive testing and validation is presented that indicates the promise of the approach.

Keywords

Wave Equation; POD; Model Order Reduction; Earth Imaging

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	D. J. Lucia, P. S. Beran and W. A. Silva, “Reduced-Order Modeling: New Approaches for Computational Physics,” Progress in Aerospace Sciences, Vol. 40, No. 1-2, 2001, pp. 51-117. doi:10.1016/j.paerosci.2003.12.001
[2]	T. J. Klemas, “Full-Wave Algorithms for Model Order Reduction and Electromagnetic Analysis of Impedance and Scattering,” PhD Thesis, MIT, Electrical Engineering, 2005.
[3]	F. Troltzsch, S. Volkwein, L. X. Wang and R. V. N. Melnik, “Model Reduction Applied to Square Rectangular Martensitic Transformations Using Proper Orthogonal Decomposition,” Applied Numerical Mathematics, Vol. 57, No. 5-7, 2007, pp. 510-520. doi:10.1016/j.apnum.2006.07.004
[4]	D. Chapelle, A. Gariah and J. Sainte-Marie, “Galerkin Approximation with Proper Orthogonal Decomposition: New Error Estimates and Illustrative Examples,” ESAIM: Mathematical Modeling and Numerical Analysis, Vol. 46, Cambridge University Press, 2012, pp. 731-757.
[5]	S. Herkt, M. Hinze and R. Pinnau, “Convergence Analysis of Galerkin POD for Linear Second Order Evolution Equations,” Manuscript, 2011.
[6]	K. Grau, “Applications of the Proper Orthogonal Decomposition Method,” WN/CFD/07/97, 1997.
[7]	V. Pereyra and B. Kaelin, “Fast Wave Propagation by Model Order Reduction,” ETNA, Vol. 30, 2008, pp. 406-419.
[8]	S. Weiland, “Course Model Reduction,” Department of Electrical Engineering, Eindhoven University of Technology, 2005.
[9]	F. Troltzsch and S. Volkwein, “POD A-Posteriori Error Estimates for Linear-Quadratic Optimal Control Problems,” Computational Optimization and Applications, Vol. 44, No. 1, 2009, pp. 83-115. doi:10.1007/s10589-008-9224-3
[10]	M. Rathinam and L. R. Petzold, “A New Look at Proper Orthogonal Decomposition,” SIAM Journal on Numerical Analysis, Vol. 41, No. 5, 2003, pp. 1893-1925. doi:10.1137/S0036142901389049
[11]	R. Kosloff and D. Kosloff, “Absorbing Boundaries for Wave Propagation Problems,” Journal of Computational Physics, Vol. 63, No. 2, 1986, pp. 363-376. doi:10.1016/0021-9991(86)90199-3
[12]	H. B. Keller and V. Pereyra, “Symbolic Generation of Finite Difference Formulas,” Mathematics of Computation, Vol. 32, 1978, pp. 955-971. doi:10.1090/S0025-5718-1978-0494848-1
[13]	B. Biondi and G. J. Shan, “Prestack Imaging of Overturned Reflections by Reverse Time Migration,” SEG Annual Meeting, 2002.
[14]	V. Pereyra, “Iterated Deferred Corrections for Nonlinear Operator Equations,” Numerische Mathematik, Vol. 10, No. 4, 1967, pp. 316-323. doi:10.1007/BF02162030
[15]	V. Pereyra, J. W. Daniel and L. L. Schumaker, “Iterated Deferred Corrections for Initial Value Problems,” Acta Científica Venezolana, Vol. 19, 1968, pp. 128-135.
[16]	P. E. Zadunaisky, “A Method for the Estimation of Errors Propagated in the Numerical Solution of a System of Ordinary Differential Equations,” In: G. I. Kontopoulos, Ed., The Theory of Orbits in the Solar System and in Stellar Systems. Proceedings from Symposium No. 25 Held in Thessaloniki, International Astronomical Union, Academic Press, London, 1964, p. 281.
[17]	M. Rewienski and J. White, “A Trajectory Piecewise-Linear Approach to Model Order Reduction and Fast Simulation of Nonlinear Circuits and Micromachined Devices,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 22, No. 2, 2003, pp. 155-170. doi:10.1109/TCAD.2002.806601
[18]	M. A. Grepl and A. T. Patera, “A Posteriori Error Bounds for Reduced-Basis Approximations of Parametrized Parabolic Partial Differential Equations,” ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 39, No. 1, 2005, pp. 157-181. doi:10.1051/m2an:2005006
[19]	G. H. Golub and C. F. Van Loan, “Matrix Computations,” 3rd Edition, Johns Hopkins University Press, Baltimore, 1996.
[20]	P. C. Hansen, V. Pereyra and G. Scherer, “Least Squares Data Fitting with Applications,” Johns Hopkins University Press, Baltimore, 2012.
[21]	C. L. Wu, D. Bevc and V. Pereyra, “Model Order Reduction for Efficient Seismic Modeling,” Accepted for Publication in SEG 83rd Annual Meeting Extended Abstracts, Houston, 2013.