Tensorial Reduced Order Models: Development, Analysis, and Applications

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Large-scale numerical simulation of dynamical systems is ubiquitous in all areas of computational science and engineering. With many computational tasks in biomedical or Earth sciences involving billions of degrees of freedom, model order reduction is necessary to scale the problem of interest down to a tractable size that fits onto the available computing platforms. Model order reduction techniques allow to construct a reduced order model (ROM) that retains the key features of a high-fidelity computational model while being much cheaper to simulate. Conventionally, a ROM represents a specific instance of the system and needs to be recomputed from scratch should the system experience significant changes to its properties. Thus, the question of constructing a ROM that captures the dependence of the system on its parameters arises. This is the main objective of the so-called parametric model reduction. Its main challenge is to develop efficient ROMs that can accurately predict solutions of parametrized high-fidelity models for parameter values that lie outside of the "training" set. The project will include training of graduate students.This project aims at addressing the above-mentioned challenges by developing a ROM that extends the ideas of conventional projection-based model order reduction to parametric systems using the concepts and tools of modern numerical multi-linear algebra. The main techniques utilized are tensor decompositions, low-rank tensor approximation and completion. In particular, low-rank tensor approximations such as canonical polyadic, Tucker (a.k.a. high order SVD, HOSVD) and tensor train are employed in place of truncated SVD, a key conventional dimension-reduction technique. The resulting reduced model is referred to as tensorial ROM (TROM). The three main objectives of the project are: (1) developing a two-stage (training/evaluation) TROM for non-linear dynamical systems, including a tensor version of the Discrete Empirical Interpolation Method; (2) developing novel low-rank tensor completion methods for use with TROM to ease the burden of the training stage by working with a sparse sampling of the parameter space; (3) integrating TROM into inverse modeling workflows, in particular, parameter estimation of the phase-field model for a multi-component lipid membrane modeled by surface Cahn-Hilliard equations, and quantitative imaging with waves for medical imaging and geophysical monitoring.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.