Marta D'Elia

Fractional diffusion on bounded domains

Abstract The mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. This paper discusses the application of nonlocal diffusion theory to specify well-posed fractional diffusion equations on bounded domains.

OPTIMAL DISTRIBUTED CONTROL OF NONLOCAL STEADY DIFFUSION PROBLEMS

A control problem constrained by a nonlocal steady diffusion equation that arises in several applications is studied. The control is the right-hand side forcing function and the objective of control is a standard matching functional. A recently developed nonlocal vector calculus is exploited to define a weak formulation of the state system. When sufficient conditions on certain kernel functions and the volume constraints hold, the existence and uniqueness of the optimal state and control is demonstrated and an optimality system is derived. We demonstrate the convergence, as the nonlocal interactions vanish, of the optimal nonlocal state to the optimal state of a local PDE-constrained control problem. We also define continuous and discontinuous Galerkin finite element discretizations of the optimality system for which we derive a priori error estimates. Numerical examples are provided illustrating these convergence results and also illustrating the differences between optimal controls and states obtained f...

A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions

We develop and analyze an optimization-based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions. The approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. When some assumptions on the kernel functions hold, we prove that the resulting optimization problem is well-posed and discuss its implementation using Sandias agile software components toolkit. The latter provides the groundwork for the development of engineering analysis tools, while numerical results for nonlocal diffusion in three-dimensions illustrate key properties of the optimization-based coupling method.

Coarse-Grid Sampling Interpolatory Methods for Approximating Gaussian Random Fields

Random fields can be approximated using grid-based discretizations of their covariance functions followed by, e.g., an eigendecomposition (i.e., a Karhunen--Loeve expansion) or a Cholesky factorization of the resulting covariance matrix. In this paper, we consider Gaussian random fields and we analyze the efficiency gains obtained by using low-rank approximations based on constructing a coarse grid covariance matrix, followed by either an eigendecomposition or a Cholesky factorization of that matrix, followed by interpolation from the coarse grid onto the fine grid. The result is coarser sampling and smaller decomposition or factorization problems than that for full-rank approximations. With one-dimensional experiments we examine the relative merits, with respect to accuracy achieved for the same computational complexity, of the different low-rank approaches. We find that interpolation from the coarse grid combined with the Cholesky factorization of the coarse grid covariance matrix yields the most effici...

Embedded ensemble propagation for improving performance, portability, and scalability of uncertainty quantification on emerging computational architectures

Quantifying simulation uncertainties is a critical component of rigorous predictive simulation. A key component of this is forward propagation of uncertainties in simulation input data to output quantities of interest. Typical approaches involve repeated sampling of the simulation over the uncertain input data and can require numerous samples when accurately propagating uncertainties from large numbers of sources. Often simulation processes from sample to sample are similar, and much of the data generated from each sample evaluation could be reused. We explore a new method for implementing sampling methods that simultaneously propagates groups of samples together in an embedded fashion, which we call embedded ensemble propagation. We show how this approach takes advantage of properties of modern computer architectures to improve performance by enabling reuse between samples, reducing memory bandwidth requirements, improving memory access patterns, improving opportunities for fine-grained parallelization, ...

Nonlocal Convection-Diffusion Problems on Bounded Domains and Finite-Range Jump Processes

Abstract A nonlocal convection-diffusion model is introduced for the master equation of Markov jump processes in bounded domains. With minimal assumptions on the model parameters, the nonlocal steady and unsteady state master equations are shown to be well-posed in a weak sense. Then the nonlocal operator is shown to be the generator of finite-range nonsymmetric jump processes and, when certain conditions on the model parameters hold, the generators of finite and infinite activity Lévy and Lévy-type jump processes are shown to be special instances of the nonlocal operator.

Optimization-based mesh correction with volume and convexity constraints

We consider the problem of finding a mesh such that 1) it is the closest, with respect to a suitable metric, to a given source mesh having the same connectivity, and 2) the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh. This volume correction problem arises in important simulation contexts, such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi-Lagrangian transport schemes. In this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective, while the prescribed cell volumes, mesh validity and/or cell convexity specify the constraints. We solve this problem numerically using a sequential quadratic programming (SQP) method whose performance scales with the mesh size. To achieve scalable performance we develop a specialized multigrid-based preconditioner for optimality systems that arise in the application of the SQP method to the volume correction problem. Numerical examples illustrate the importance of volume correction, and showcase the accuracy, robustness and scalability of our approach.