A3.: Efficient Sampling in Stochastic Simulation of Inelastic Solids via Compressive Sensing Techniques

Polynomial Chaos Expansion (PCE) is an established technique to accelerate stochastic computations, however, it is limited to small dimensions and linear or smoothly non-linear problems. For improvement, i.e. reducing the number of polynomials, e.g. Sparse PCE has been suggested [1]. However, limitations with regard to the dimension of the stochastic space are still remaining, in particular when non-linearities have to be considered. There is a point, at which brute Monte Carlo Simulation (MCS) performs better [2].

One obstacle in the computation of PCE is, that a larger number of snapshots computed with a high-fidelity model is needed to compute the coefficients of the PCE via least-squares (L2) minimization. Here comes the idea for this project into the game. Compressive Sensing (CS) have been successfully used for signal reconstruction using sparse data based on L1 projection, i.e. optimization based on underdetermined systems [3]. That has been recognized recently in computational mechanics too [Cueto]. Goal of this project to investigate more efficient techniques for SPCE construction thus enabling a space for broadening the efficient use of this model order reduction technique for stochastic computations in non-linear solid mechanics.

[1] Géraud Blatman, Bruno Sudret, An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probabilistic Engineering Mechanics, Volume 25, Issue 2, 2010, Pages 183-197, ISSN 0266-8920, https://doi.org/10.1016/j.probengmech.2009.10.003.

[2] Esther dos Santos Oliveira, Udo Nackenhorst, Sparse polynomial chaos expansion for high-dimensional nonlinear damage mechanics, Probabilistic Engineering Mechanics, Volume 75, 2024, doi.org/10.1016/j.probengmech.2023.103556.

[3] Emmanuel J. Candes and Michael B. Wakin, An Introduction into Compressive Sampling, IEEE Signal Processing Magazine, March 2008.

[4] Ibañez, R., Abisset-Chavanne, E., Cueto, E.et al. Some applications of compressed sensing in computational mechanics: model order reduction, manifold learning, data-driven applications and nonlinear dimensionality reduction.Comput Mech, 64,1259–1271 (2019). doi.org/10.1007/s00466-019-01703-5.

Supervision: Prof. Dr.-Ing. Udo Nackenhorst (LUH), NN (ENS)