Research

References, abstracts, and links (when applicable) for my recent publications and preprint arcticles.

Convergence of Steady-State Nonlocal Nonlinear Diffusion Systems in the Limit of the Vanishing Horizon

Mikil Foss, Hayley Olson, and Petronela Radu. 2022 Preprint.

Nonlinear nonlocal diffusion models arise when the diffusivity coefficient is state dependent and the operators governing the equations are chosen to have integral, rather than differential, formulations. In this paper we will show that nonlocal solutions to such systems are close approximations to classical solutions, as the horizon of interaction shrinks to zero. The convergence of the nonlocal operators is used to show the convergence of solutions to boundary value problems with Dirichlet-type data. Due to the lack of compactness in the nonlocal framework, the results require bounds on the Lipschitz nonlinearity, as well as on the solutions to the classical system.

Towards a Unified Theory of Fractional and Nonlocal Vector Calculus (Journal) (arXiv)

Marta D'Elia, Mamikon Gulian, Hayley Olson, and George Em Karniadakis. Fractional Calculus and Applied Analysis. 24 (2021) pp. 1301-1355 .

Nonlocal and fractional models capture effects that classical (or standard) partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus based on nonlocal and fractional derivatives to derive models of, e.g., subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. At the moment, this fragmented literature suffers from a lack of rigorous comparison and unified notation, hindering the development of nonlocal modeling. The ultimate goal of this work is to provide a new theory and to "connect all the dots" by defining a universal form of nonlocal vector calculus operators under a theory that includes, as a special case, several well-known proposals for fractional vector calculus in the limit of infinite interactions. We show that this formulation enjoys a form of Green's identity, enabling a unified variational theory for the resulting nonlocal exterior-value problems, and is consistent with several independent results in the fractional calculus literature. The proposed unified vector calculus has the potential to go beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators and providing useful analogues of standard tools from classical vector calculus.

Comparison of Tempered and Truncated Fractional Models (PDF) (Full Proceedings)

Hayley Olson, Marta D'Elia, Mikil Foss, Mamikon Gulian, and Petronela Radu. Computer Science Research Institute Summer Proceedings 2021. Technical Report SAND2022-0653R, Sandia National Laboratories (2021) pp. 112-119

Tempered fractional operators are able to model effects that classical partial differential equations cannot capture, such as the super- and sub-diffusive effects that are present in hydrology and geophysics models. However, tempered fractional operators are computationally intensive due to the infinite range of interaction for the integral operator. We analyze a truncated variation of the fractional operators, which are less computationally intensive, in an effort to use them in place of the more complex tempered variation. In particular, we train parameters of the truncated operator using neural networks in order to optimize the difference of the actions of the two operators.

Linking numbers of Klein links (Journal)

Steven Beres, Vesta Coufal, Kate Kearney, Ryan Lattanzi, and Hayley Olson. The College Mathematics Journal. 52, no.2 (2021) pp. 106-114.

We compute the linking numbers of links embedded on a punctured Klein bottle (called Klein links). We give a complete formula for the linking number of any pair of components of a Klein link, and work out a detailed example of a calculation.

The tempered fractional Laplacian as a special case of the nonlocal Laplace operator (PDF) (Full Proceedings)

Hayley Olson, Marta D'Elia, and Mamikon Gulian. Computer Science Research Institute Summer Proceedings 2020. Technical Report SAND2020-12580R, Sandia National Laboratories (2020) pp. 111-126.

Tempered fractional operators provide an improved predictive capability for modeling anomalous effects that cannot be captured by standard partial differential equations. These effects include subdiffusion and superdiffusion, which often occur in, e.g., geoscience and hydrology. Tempered operators can be used in such models of heavy-tailed behavior while circumventing consequences of standard fractional models, such as divergent moments. In the first part of this work, we investigate the relationship between tempered and truncated fractional operators and the unified nonlocal diffusion operator, building upon the recently developed unified nonlocal calculus. In the second part of this work, with the purpose of finding a computationally cheap alternative to tempered fractional operators, we investigate the relationship between the (computationally expensive) tempered fractional Laplacian and the (computationally cheaper) truncated fractional Laplacian. Our main result shows the equivalence between truncated and tempered fractional energies and represents the first step towards the approximation of expensive fractional models with cheaper, but equivalent, alternatives

Towards a unified nonlocal vector calculus (PDF) (Full Proceedings)

Hayley Olson, Marta D'Elia, and Mamikon Gulian. Computer Science Research Institute Summer Proceedings 2019. Technical Report SAND2020-9969R, Sandia National Laboratories (2020) pp. 11-19.

In this work we provide the groundwork for a unified theory of nonlocal operators. Specifically, in the context of nonlocal diffusion, we prove the equivalence, for certain kernel functions, of weighted and unweighted operators. After studying general properties of the “equivalence” kernel, we show that the equivalence holds for fractional-type operators. We also make preliminary steps towards a unified wellposedness theory that holds for broad class of nonlocal operators by leveraging the well-established theory for unweighted operators, and the generalized operator definition that arises from our equivalence result.

Hyperbolic tangle surgeries on augmented links (Journal)

John Harnois, Hayley Olson, and Rolland Trapp. Algebraic and Geometric Topology. 18, no.3 (2018) pp. 1573-1602.

Changes to gluing patterns for fully augmented links are shown to result in generalized fully augmented links. The first changes considered result in 4-tangle surgeries on hyperbolic fully augmented links that produce hyperbolic generalized fully augmented links. These surgeries motivate the definition of nested links, and a characterization of hyperbolic nested links is given. Finally, the geometry of nested links is compared to that of fully augmented links.

A classification of Klein links as torus links (Journal)

Steven Beres, Vesta Coufal, Kaia Havacek, M. Kate Kearney, Ryan Lattanzi, Hayley Olson, Joel Pereira, and Bryan Strub. Involve. 11, no.4 (2018) pp. 609-624.

We classify Klein links. In particular, we calculate the number and types of components in a K p , q Klein link. We completely determine which Klein links are equivalent to a torus link, and which are not.

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