A model-order reduction method based on wavelets and POD to solve nonlinear transient and steady-state continuation problems. (English) Zbl 1480.65310
Summary: We introduce a wavelet-based model-order reduction method (MOR) that provides an alternative subspace to Proper Orthogonal Decomposition (POD). We thus compare the wavelet and POD-based approaches for reducing high-dimensional nonlinear transient and steady-state continuation problems. We employ a global regularized Gauss-Newton (GN) algorithm for solving zero-residual problems on a reduced subspace. We rediscover that this latter is just a generalization of the Petrov-Galerkin method (PG) which retains GN’s fast convergence rate. Numerical results included herein indicate that wavelet-based method is competitive with POD, for small rank systems \((\approx 100)\) and compression ratios below 25% while POD achieves up to 90%. Full-order-model (FOM) results demonstrate that the proposed PGGN algorithm outperforms the standard PG method.
MSC:
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 65T60 | Numerical methods for wavelets |
References:
| [1] | Paul-Dubois-Taine, A.; Amsallem, D., An adaptive and efficient greedy procedure for the optimal training of parametric reduced-order models, Int. J. Numer. Methods Eng., 102, 1262-1292 (2015) · Zbl 1352.65217 |
| [2] | Nguyen, N.; Peraire, J., An efficient reduced-order modeling approach for non-linear parametrized partial differential equations, Int. J. Numer. Methods in Eng., 76, 27-55 (2008) · Zbl 1162.65407 |
| [3] | Carlberg, K., Model order reduction with oblique projections for large scale wave propagation, Int. J. Numer. Methods Eng., 102, 1192-1210 (2015) · Zbl 1352.65136 |
| [4] | Carlberg, K.; Bou-Mosleh, C.; Farhat, C., Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. J. Numer. Methods Eng., 86, 155-181 (2011) · Zbl 1235.74351 |
| [5] | Willcox, K.; Peraire, J., Balanced model reduction via the proper orthogonal decomposition, AIAA J., 40, 11, 2323-2330 (2002) |
| [6] | Chaturantabut, S.; Sorensen, D., Application of POD and DEIM on dimension reduction of non-linear miscible viscous fingering in porous media, Math. Comput. Model. Dyn. Syst., 17, 4, 337-353 (2011) · Zbl 1302.76127 |
| [7] | Walton, S.; Hassan, O.; Morgan, K., Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions, Appl. Math. Model., 37, 20-21, 8930-8945 (2013) · Zbl 1426.76576 |
| [8] | Amsallem, D.; Zahr, M. J.; Farhat, C., Nonlinear model order reduction based on local reduced-order bases, Int. J. Numer. Methods Eng., 92, 891-916 (2012) · Zbl 1352.65212 |
| [9] | Kerfriden, P.; Gosselet, P.; Adhikari, S.; Bordas, S., Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems, Comput. Methods Appl. Mech. Eng., 200, 850-866 (2011) · Zbl 1225.74092 |
| [10] | Ojo, S.; Grivet-Talocia, S.; Paggi, M., Model order reduction applied to heat conduction in photovoltaic modules, Compos. Struct., 119, 477-486 (2015) |
| [11] | Iuliano, E.; Quagliarella, D., Proper orthogonal decomposition, surrogate modelling and evolutionary optimization in aerodynamic design, Comput. Fluids, 84, 327-350 (2013) · Zbl 1290.76051 |
| [12] | Pereyra, V., Model order reduction with oblique projections for large scale wave propagation, J. Comput. Appl Math., 295, 103-114 (2016) · Zbl 1329.86018 |
| [13] | Chinesta, F.; Ammar, A.; Leygue, A.; Keunings, R., An overview of the proper generalized decomposition with applications in computational rheology, J. Non-Newton. Fluid Mech, 166, 11, 578-592 (2011) · Zbl 1359.76219 |
| [14] | Argaez, M.; Ceberio, M.; Florez, H.; Mendez, O., A model order reduction method for solving high-dimensional problems, Proceedings of the NAFIPS (2016), IEEE: IEEE El Paso, TX |
| [15] | Florez, H.; Argaez, M., Applications and comparison of model-order reduction methods based on wavelets and POD, Proceedings of the NAFIPS (2016), IEEE: IEEE El Paso, TX |
| [16] | Daubechies, I., Ten lectures on wavelets, Proceedings of the CBMS-NSF conference series in applied mathematics (1992), SIAM Ed.: SIAM Ed. Philadelphia, PA · Zbl 0776.42018 |
| [17] | Teolis, A., Computational Signal Processing with Wavelets (1998), Birkhauser · Zbl 0928.94002 |
| [18] | Mallat, S., A Wavelet Tour of Signal Processing (1999), London Academic Press · Zbl 0998.94510 |
| [19] | Le, T.-H.; Caracoglia, L., Reduced-order wavelet-Galerkin solution for the coupled, nonlinear stochastic response of slender buildings in transient winds, J. Sound Vib., 344, 179-208 (2015) |
| [20] | Goyal, K.; Mehra, M., Fast diffusion wavelet method for partial differential equations, Appl. Math. Model., 40, 78, 5000-5025 (2016) · Zbl 1459.65209 |
| [21] | Alsmadi, O. M.; Abo-Hammour, Z. S.; Al-Smadi, A. M., Artificial neural network for discrete model order reduction with substructure preservation, Appl. Math. Model., 35, 9, 4620-4629 (2011) · Zbl 1225.93031 |
| [22] | Mahadevan, N.; Hoo, K. A., Wavelet-based model reduction of distributed parameter systems, Chem. Eng. Sci., 55, 4271-4290 (2000) |
| [23] | Gunawan, F. E., Levenberg marquardt iterative regularization for the pulse-type impact-force reconstruction, J. Sound Vib., 331, 25, 5424-5434 (2012) |
| [24] | Marquardt, D. W., An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math., 11, 2, 431-441 (1963) · Zbl 0112.10505 |
| [25] | Kunisch, K.; Volkwein, S., Control of the burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl., 102, 345-371 (1999) · Zbl 0949.93039 |
| [26] | Barragy, E.; Carey, G. F., A partitioning scheme and iterative solution for sparse bordered systems, Comput. Methods Appl. Mech. Eng., 70, 321-327 (1998) · Zbl 0636.65024 |
| [27] | Glowinski, R.; Keller, H. B.; Reinhart, L., Continuation-conjugate gradient methods for the least squares solution of nonlinear boundary value problems, SIAM J. Sci. Stat. Comput., 6, 4, 793-832 (1985) · Zbl 0589.65075 |
| [28] | Chan, T. F.; Saad, Y., Iterative methods for solving bordered systems with applications to continuation methods, SIAM J. Sci. Stat. Comput., 6, 2, 438-451 (1985) · Zbl 0581.65039 |
| [29] | Anderson, A.; Tannehill, J. C.; Pletcher, R. H., Computational Fluid Mechanics and Heat Transfer (1984), Hemisphere Publishing Corporation: Hemisphere Publishing Corporation USA · Zbl 0569.76001 |
| [30] | C. Farhat, M.J. Zahr, MORTestbed User Guide, Institute for Computational and Mathematical Engineering, Stanford University, 2015.; C. Farhat, M.J. Zahr, MORTestbed User Guide, Institute for Computational and Mathematical Engineering, Stanford University, 2015. |
| [31] | Florez, H., Applications of model-order reduction to thermo-poroelasticity, Proceedings of the 51st US Rock Mechanics / Geomechanics Symposium (2017) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
