Our Analysis and Applied Mathematics research group is our largest group as so many of our faculty work on applications of mathematics. Faculty working on mathematical analysis study functional and harmonic analysis and analysis of partial differential equations, along with applications in other areas and disciplines like geometry, probability, number theory and numerical analysis.

## Analysis and applied mathematics research faculty

# Vrushali A. Bokil

#### Interim Dean – College of Science, Professor

Professor Bokil's general research interests are in applied mathematics, scientific computing, numerical analysis and mathematical biology. Her primary research interests are in the numerical solution of wave propagation problems. Specifically, she has conducted research on the numerical solution of Maxwell's equations using a variety of finite difference and finite element methods. Bokil is also working on several problems in mathematical ecology which involve the construction and analyses of deterministic and stochastic models for applications in population dynamics, epidemiology and spatial ecology.

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# Elaine Cozzi

#### Associate Professor

Professor Cozzi investigates problems on existence, uniqueness, and regularity of solutions to partial differential equations arising in fluid mechanics.

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# Radu Dascaliuc

#### Associate Professor

Dascaliuc's research focuses on qualitative properties and long-time behavior of nonlinear PDE. In particular, he is interested in dynamics of the incompressible Navier-Stokes and Euler equations, especially in relation to the turbulence theory. The main goals are to derive the tenets of the heuristic theory of turbulence directly from the physical equations of motion and to understand connections between turbulence in fluids and regularity theory of the 3D Navier-Stokes and Euler equations.

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# Patrick De Leenheer

#### Professor

My research is on solitons, spinning things, donuts, and something random (sometimes on computers). My primary interest is applications of complex analysis to the study solutions to completely integrable Hamiltonian systems in analytical mechanics, nonlinear wave mechanics, and mathematical physics. I am also interested in probability (random waves, soliton gasses, and Markov processes), and geometry (Riemann surfaces, translation surfaces, zero curvature connections, and complex algebraic geometry).

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# Adel Faridani

#### Professor

Faridani conducts research in numerical analysis, investigating problems arising in signal processing and tomography. In signal processing he is interested in uniform and non-uniform sampling of bandlimited functions in one and several variables. His research in tomography comprises questions of optimal sampling and resolution; error estimates for reconstruction algorithms in two and three dimensions; and theory and implementation of local tomography.

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# David V. Finch

#### Professor

Finch works on inverse problems, particularly those arising in medical imaging. He has worked on x-ray computed tomography and vector tomography. He is currently working on problems arising in the medical imaging technique called thermoacoustic tomography and other forms of hybrid imaging.

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# Nathan L. Gibson

#### Professor

Gibson's primary research interests are computational electromagnetics, uncertainty quantification, and inverse problems. Research topics coinciding with primary interests include: wave propagation modeling, homogenization, optimization and regularization techniques, high performance and parallel computing, parameter identification and sensitivity analysis, finite element and finite difference methods. Current work involves finite element and finite difference methods for time-domain electromagnetics in dispersive and random media.

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# Robert L. Higdon

#### Professor

Higdon has worked on open boundary conditions for wave propagation problems and on issues related to the well-posedness of hyperbolic initial-boundary value problems and the stability of their numerical approximations. He is presently working on some mathematical and computational problems related to the numerical modeling of ocean circulation. This modeling involves the solution of a system of partial differential equations that describes fluid flow, as adapted to that case. In this area, Higdon first worked on some problems with stability and efficiency that can arise from the multiple time scales that are contained in the system. He is presently working on a project to develop, analyze, and test some procedures for using discontinuous Galerkin numerical methods in multi-layer models of ocean circulation.

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# John W. Lee

#### Emeritus Professor

Lee has made contributions to the existence, uniqueness, and continuous dependence theory for solutions to nonlinear boundary value problems. This work, which continues, is joint with R.B. Guenther of Oregon State and A. Granas of the University of Montreal. Lee has also worked on the numerical calculation of solutions to such problems. He has helped develop the extension of Sturm oscillation theory and the properties of Sturm-Liouville eigenvalue problems to higher order equations. This work is closely related to the branch of approximation theory which deals with Tchebycheff systems and with positive operator theory. It led to related work on best quadrature formulas.

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# Nicholas Marshall

#### Assistant Professor

I am interested in problems that involve interplay between analysis, geometry, and probability (especially such problems motivated by applications to data science).

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# Malgorzata Peszynska

#### Professor

Malgo Peszynska's research interests are in applied mathematics and computational modeling of real life phenomena. Originally trained in pure mathematics, she came to applied mathematics projects through her interest in parallel and high performance computing and variational theory for finite element methods. Since her PhD she has worked on models of flow and transport using mathematical and numerical analysis as well as computer simulations to understand these phenomena better across the various time and spatial scales. She is involved in a variety of interdisciplinary projects with academic, national lab, and industry collaborators from hydrology, oceanography, statistics, environmental, petroleum, civil and coastal engineering, physics, and materials science. Her research projects were funded by NSF, DOE-NETL and by DOE; her current projects include the 2015-2020 NSF-DMS 1522734 "Phase transitions in porous media across multiple scales", and 2019-2022 NSF-DMS 1912938 on "Modeling with Constraints and Phase Transitions in Porous Media". She believes in "paying it forward" and is engaged in external and university service via editorial work and conference organization: she served as the President of Pacific Northwest SIAM Section, Chair (2011-12) and Program Officer (2009-10) of SIAM Activity Group on Geosciences and in other functions.

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# Petri Pohjanpelto

#### Professor

Pohjanpelto works on the theory and applications of generalized symmetries of differential equations. He has studied the structure of symmetries of the electromagnetic field and applied symmetries in the construction of conservation laws and classification of group invariant solutions. He has also used variational bicomplexes to study the correspondence of generalized symmetries of equations in physical and potential formulation.

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# Ralph E. Showalter

#### Professor

My research interests include singular or degenerate nonlinear evolution equations and partial differential equations, variational inequalities, free-boundary problems, and related applications to initial-boundary-value problems of mechanics and diffusion. Current work is focused on the development of multiphysics mathematical models of coupled fluid-solid dynamics, deformable porous media, and upscaled models of transport and flow in heterogeneous media.

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# Enrique A. Thomann

#### Professor

Thomann's research is primarily in problems in partial differential equations arising from fluid mechanics. He also collaborates with colleagues in other departments, as well as in the Mathematics Department, in the development of mathematical models to problems in Ecology, Oceanography, Hydrology and management of Natural Resources.