Buczkowski, Nicole E.; Foss, Mikil D.; Parks, Michael L.; Radu, Petronela
The paper presents a collection of results on continuous dependence for solutions to nonlocal problems under perturbations of data and system parameters. The integral operators appearing in the systems capture interactions via heterogeneous kernels that exhibit different types of weak singularities, space dependence, even regions of zero-interaction. The stability results showcase explicit bounds involving the measure of the domain and of the interaction collar size, nonlocal Poincaré constant, and other parameters. In the nonlinear setting, the bounds quantify in different Lp norms the sensitivity of solutions under different nonlinearity profiles. The results are validated by numerical simulations showcasing discontinuous solutions, varying horizons of interactions, and symmetric and heterogeneous kernels.
Krylov subspace recycling is a powerful tool when solving a long series of large, sparse linear systems that change only slowly over time. In PDE constrained shape optimization, these series appear naturally, as typically hundreds or thousands of optimization steps are needed with only small changes in the geometry. In this setting, however, applying Krylov subspace recycling can be a difficult task. As the geometry evolves, in general, so does the finite element mesh defined on or representing this geometry, including the numbers of nodes and elements and element connectivity. This is especially the case if re-meshing techniques are used. As a result, the number of algebraic degrees of freedom in the system changes, and in general the linear system matrices resulting from the finite element discretization change size from one optimization step to the next. Changes in the mesh connectivity also lead to structural changes in the matrices. In the case of re-meshing, even if the geometry changes only a little, the corresponding mesh might differ substantially from the previous one. Obviously, this prevents any straightforward mapping of the approximate invariant subspace of the linear system matrix (the focus of recycling in this work) from one optimization step to the next; similar problems arise for other selected subspaces. In this paper, we present an algorithm to map an approximate invariant subspace of the linear system matrix for the previous optimization step to an approximate invariant subspace of the linear system matrix for the current optimization step, for general meshes. This is achieved by exploiting the map from coefficient vectors to finite element functions on the mesh, combined with interpolation or approximation of functions on the finite element mesh. We demonstrate the effectiveness of our approach numerically with several proof of concept studies for a specific meshing technique.
The Computer Science Research Institute (CSRI) brings university faculty and students to Sandia for focused collaborative research on Department of Energy (DOE) computer and computational science problems. The institute provides an opportunity for university researchers to learn about problems in computer and computational science at DOE laboratories. Participants conduct leading-edge research, interact with scientists and engineers at the laboratories, and help transfer results of their research to programs at the labs. Some specific CSRI research interest areas are: scalable solvers, optimization, adaptivity and mesh refinement, graph-based, discrete, and combinatorial algorithms, uncertainty estimation, mesh generation, dynamic load-balancing, virus and other malicious-code defense, visualization, scalable cluster computers and heterogeneous computers, data-intensive computing, environments for scalable computing, parallel input/output, advanced architectures, and theoretical computer science. The CSRI Summer Program includes the organization of a weekly seminar series and the publication of this summer proceedings.
Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integro-differential equations with sparse, noisy, unstructured, and multifidelity data. PINNs incorporate all available information, including governing equations (reflecting physical laws), initial-boundary conditions, and observations of quantities of interest, into a loss function to be minimized, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal Laplace operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius δ goes to zero, and to the fractional Laplacian as δ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to δ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, δ and α, characterizing the kernel of the unified operator. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon, that is, different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order α(γ), which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as δ. More importantly, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.
We present a meshfree quadrature rule for compactly supported nonlocal integro-differential equations (IDEs) with radial kernels. We apply this rule to develop a meshfree discretization of a peridynamic solid mechanics model that requires no background mesh. Existing discretizations of peridynamic models have been shown to exhibit a lack of asymptotic compatibility to the corresponding linearly elastic local solution. By posing the quadrature rule as an equality constrained least squares problem, we obtain asymptotically compatible convergence by introducing polynomial reproduction constraints. Our approach naturally handles traction-free conditions, surface effects, and damage modeling for both static and dynamic problems. We demonstrate high-order convergence to the local theory by comparing to manufactured solutions and to cases with crack singularities for which an analytic solution is available. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff–Winkler experiments.
The Center for Computing Research (CCR) at Sandia National Laboratories organizes an active and productive summer program each year, in coordination with the Computer Science Research Institute (CSRI) and Cyber Engineering Research Institute (CERI). CERI focuses on open, exploratory research in cyber security in partnership with academia, industry, and government, and provides collaborators an accessible portal to Sandia's cybersecurity experts and facilities. Moreover, CERI provides an environment for visionary, threat-informed research on national cyber challenges. CSRI brings university faculty and students to Sandia National Laboratories for focused collaborative research on DOE computer and computational science problems. CSRI provides a mechanism by which university researchers learn about problems in computer and computational science at DOE Laboratories. Participants conduct leading - edge research, interact with scientists and engineers at the laboratories, and help transfer the results of their research to programs at the labs.