Multifidelity methods + nonlocal models: a marriage made in computational heaven

Max Gunzburger, Florida State University
11/8, 2023 at 4:10PM-5PM in 939 Evans (for in-person talks) and

We are given a discretized partial differential equation (PDE) system for which for a desired accuracy, a single simulation run is costly to obtain so that uncertainty quantification and other many query applications become very expensive endeavors. We also have available surrogate models (such as coarser grids, reduced-order models, interpolants, etc.) which are less costly and less accurate. Multifidelity approaches leverage the use of surrogate models to obtain the desired accuracy at significantly lower cost. Instead of PDEs, we also consider nonlocal models which are a type of integral equations which have been shown to produce better fidelity (when compared to PDEs) in several application settings such as anomalous diffusion, fracture in solid mechanics, etc. By "nonlocal" we mean that points interact with each other even when separated by a finite distance, in contrast to PDEs for which derivatives are defined within infinitesimal neighborhoods. However, nonlocality induces discretizations that have reduced sparsity (and therefore increased computational costs) compared to that for related PDEs. Thus, multifidelity methods are especially of interest for nonlocal modeling. In this talk, we briefly review the two components (multifidelity methods and nonlocal models) and consider their use in tandem. We illustrate the efficacy of the two components and of their combination using toy problems and also phase-field applications.