Researchers in the Numerical Analysis group work on various theoretical issues for applications ranging from signal processing, inverse problems and more specifically computed tomography. This process uses large-scale high-resolution modeling of ocean circulation to models of flow and transport in porous media, and models for wave propagation in liner and nonlinear materials at multiple spatial and temporal scales. Faculty in this group are interested in both the theoretical and implementation issues. The former include development of algorithms, error and stability analysis; the latter concern high-performance computing techniques such as parallel algorithms.
Numerical analysis 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.
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.
Nathan L. Gibson
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.
Robert L. Higdon
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.
I am interested in problems that involve interplay between analysis, geometry, and probability (especially such problems motivated by applications to data science).
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.