We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in d â‰¥ 1â€¦ (More)

The fractional Laplacian in Rd, which we write as (âˆ’âˆ†)Î±/2 with Î± âˆˆ (0, 2), has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in theseâ€¦ (More)

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one toâ€¦ (More)

Given a weighted undirected graph G = (V,E,d) with d : E â†’ Q+ and a positive integer K, the Distance Geometry Problem (DGP) asks to find an embedding x : V â†’ R of G such that for each edge {i, j} weâ€¦ (More)

The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods.â€¦ (More)

We design and analyze solution techniques for a linear-quadratic optimal control problem involving the integral fractional Laplacian. We derive existence and uniqueness results, first orderâ€¦ (More)

Parallel implementations of linear iterative solvers generally alternate between phases of data exchange and phases of local computation. Increasingly large problem sizes on more heterogeneousâ€¦ (More)

Computing at the exascale level is expected to be affected by a significantly higher rate of faults, due to increased component counts as well as power considerations. Therefore, current dayâ€¦ (More)