Math/ICES Center of Numerical Analysis Seminar (Fall 2016)

Time and Location: Friday, 1:00-2:00PM, POB 6.304 Special time and locations are indicated in red.

If you are interested in meeting a speaker, please contact Kui Ren (

Here are the links to the past seminars: Spring 2016 Fall 2015 Spring 2015 Fall 2014 Spring 2014 Spring 2013 Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009


Spekers and Hosts

Title and Abstract



Patrick Bardsley

ICES, UT Austin

Kirchhoff migration without phases

Scatterers in a homogeneous medium can be imaged using the Kirchhoff migration functional. This imaging method generally requires full waveform measurements of the scattered field, and thus intensity (i.e., phaseless) measurements are insufficient to image with. However, if the scattered field is small compared to the probing field, we can solve a simple least-squares problem to recover the projection (on a known subspace) of the full waveform scattered field from intensity data. For high frequencies, this projection gives a Kirchhoff image asymptotically identical to the Kirchoff image obtained from full waveform data. This imaging method also works when the illuminating wavefields are stochastic and we measure autocorrelations at receivers. Since correlations are robust with respect to additive noise, we expect this imaging method to work well in situations with low signal-to-noise ratio.


Per-Gunnar Martinsson

University of Colorado, Boulder

Randomized methods for accelerating matrix factorization algorithms

The talk will describe accelerated algorithms for computing full or partial matrix factorizations such as the eigenvalue decomposition, the QR factorization, etc. The key technical novelty is the use of randomized projections to reduce the effective dimensionality of intermediate steps in the computation. The resulting algorithms execute faster on modern hardware than traditional algorithms, and are particularly well suited for processing very large data sets.

The algorithms described are supported by a rigorous mathematical analysis that exploits recent work in random matrix theory. The talk will briefly review some representative theoretical results.


Hongkai Zhao

UC Irvine

Why is high frequency Helmholtz equation difficult to solve

The Helmholtz equation is a linear partial differential equation modeling time harmonic wave propagation. It is notoriously difficult to solve numerically when the frequency is high. I will first present our recent study (joint with Engquist) on the intrinsic complexity of the Helmholtz operator in the high frequency limit. The study is based on the analysis of approximate separable representations of the Green’s functions. Computationally, being able to approximate a Green’s function by a sum with few separable terms is equivalent to the existence of low rank approximation of the discretized systems which can be explored for matrix compression and fast solution techniques. We prove both lower and upper bounds for the number of terms needed for a separable approximation of the Green’s function of the Helmholtz operator. Sharpness and implications of these bounds will be shown for computation setups that are commonly used in practice. I will also make comparisons with other types of differential operators such as coercive elliptic differential operators with rough coefficients and hyperbolic differential operators.


Mark A. Anastasio

Washington University in St. Louis

Image Reconstruction Methods for Photoacoustic Computed Tomography in Heterogeneous Media with Application to Experimental Data

Photoacoustic computed tomography (PACT) is an emerging biomedical imaging modality that has great potential for a wide range of preclinical and clinical imaging applications. It can be viewed as a hybrid imaging modality in the sense that it utilizes an optical contrast mechanism combined with ultrasonic detection principles, thereby combining the advantages of optical and ultrasonic imaging while circumventing their primary limitations. The goal of PACT is to reconstruct the distribution of an object's absorbed optical energy density from measurements of pressure wavefields that are induced via the thermoacoustic effect. This corresponds to an inverse source problem. In this talk, we review our recent advancements in image reconstruction approaches for PACT. Such advancements include physics-based models of the measurement process and associated inversion methods for reconstructing images from limited data sets in acoustically heterogeneous media. Applications of PACT to transcranial brain imaging and breast cancer detection will be presented.


Sarah Vallelian

SAMSI and North Carolina State University

Computationally efficient Markov chain Monte Carlo methods for hierarchical Bayesian inverse problems

In Bayesian inverse problems, the posterior distribution can be used to quantify uncertainty about the reconstructed solution. In practice, approximating the posterior requires Markov chain Monte Carlo (MCMC) algorithms, but these can be computationally expensive. We present a computationally efficient MCMC sampling scheme for ill-posed Bayesian inverse problems.


Junshan Lin

Auburn University

Scattering and Field Enhancement of Subwavelength Slits

Subwavelength apertures and holes on surfaces of metals induce strong electromagnetic field enhancement and extraordinary optical transmission. This remarkable phenomenon can lead to potentially significant applications in biological and chemical sensing, spectroscopy, and other novel optical devices.  In this talk, I will present mathematical studies of the enhancement mechanism for the scattering of narrow slits perforated in a slab of perfect conductor. Both the single slit and an array of slits will be discussed. It is demonstrated that the enhancement of electromagnetic fields for a single slit can be induced either by scattering resonances or certain non-resonant effect in the quasi-static regime. We derive the asymptotic expansions of resonances and quantitatively analyze the field enhancement at resonant frequencies. The field enhancement at non-resonant frequencies in the quasi-static regime is also investigated, and it is shown that the fast transition of the magnetic field in the slit induces strong electric field enhancement. For a periodic array of slits, we show that additional enhancement mechanism arises at the Wood’s anomaly.


Yingda Cheng

Michigan State University

A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations

In this talk, we present a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG schemes for hyperbolic problems and is proven to be $L^2$ stable and convergent. A major advantage of the scheme lies in its low computational and storage cost due to the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. We also discuss extension of the scheme to adaptive sparse grid methods.




Lise-Marie Imbert-Gerard

Courant Institute

Variable coefficients and numerical methods for electromagnetic waves

In the first part of the talk, we will discuss a numerical method for wave propagation in inhomogeneous media. The Trefftz method relies on basis functions that are solution of the homogeneous equation. In the case of variable coefficients, basis functions are designed to solve an approximation of the homogeneous equation. The design process yields high order interpolation properties for solutions of the homogeneous equation. This introduces a consistency error, requiring a specific analysis.

In the second part of the talk, we will discuss a numerical method for elliptic partial differential equations on manifolds. In this framework the geometry of the manifold introduces variable coefficients. Fast, high order, pseudo-spectral algorithms were developed for inverting the Laplace-Beltrami operator and computing the Hodge decomposition of a tangential vector field on closed surfaces of genus one in a three dimensional space. Robust, well-conditioned solvers for the Maxwell equations will rely on these algorithms.





Alexander Barnett

Dartmouth College

Integral equation based fast solvers for periodic flow and scattering problems

Boundary-value problems with periodic geometry arise in modeling diffraction gratings, meta-materials, and heat or fluid flow through composite media. I will explain a unified spectrally-accurate approach for solving such problems via 2nd-kind integral equation methods, that combines free-space Green's kernels with a small set of auxiliary particular solutions, whose coefficients are solved in the least squares sense. The scheme is compatible with fast algorithms, avoids non-robustness and other issues with the periodic Green's function approach, and directly applies physical boundary conditions.  I will illustrate this with solvers for doubly-periodic Stokes flow in 2D (with up to thousands of inclusions per unit cell), and for various Helmholtz and Maxwell wave diffraction problems in 2D and 3D.