Math/ICES Center of Numerical Analysis Seminar (Spring 2018)

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: Fall 2017 Spring 2017 Fall 2016 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


Li Wang

SUNY Buffalo

Front capturing schemes for nonlinear PDEs with a free boundary limit

Evolution in physical or biological systems often involves interplay between nonlinear interaction among the constituent “particles”, and convective or diffusive transport, which is driven by density dependent pressure. When pressure-density relationship becomes highly nonlinear,  the evolution equation converges to a free boundary problem as a stiff limit. In terms of numerics, the nonlinearity and degeneracy bring great challenges, and there is lack of standard mechanism to capture the propagation of the front in the limit. In this talk, I will introduce a numerical scheme for tumor growth models based on a prediction-correction reformulation, which naturally connects to the free boundary problem in the discrete sense. As an alternative, I will present a variational method for a class of continuity equations (such as Keller-Segel model) using the gradient flow structure, which has built-in stability, positivity preservation and energy decreasing property, and serves as a good candidate in capturing the stiff pressure limit.




RLM 5.104

(Mathematics Colloqium)

Liliana Borcea

University of Michigan

Untangling the nonlinearity in inverse scattering using data-driven reduced order models

We discuss an inverse problem for the wave equation, where an array of sensors probes an unknown, heterogeneous medium with pulses and measures the scattered waves. The goal in inversion is to determine from these measurements scattering structures in the medium, modeled mathematically by a reflectivity function. Most imaging methods assume a linear mapping between the unknown reflectivity and the array data. The linearization, known as the Born (single scattering) approximation is not accurate in strongly scattering media, so the reconstruction of the reflectivity may be poor. We show that  it is possible to remove the multiple scattering (nonlinear) effects from the data using a reduced order model (ROM). The ROM is defined by an orthogonal projection of the wave propagator operator on the subspace spanned by the time snapshots of the solution of the wave equation. The snapshots are known only at the sensor locations, which is enough information to construct the ROM. The main result discussed in the talk is a novel, linear-algebraic algorithm that uses the ROM to map the data to its Born approximation. 






RLM 10.176

Liliana Borcea

University of Michigan

Pulse Reflection in a Random Waveguide with a Turning Point

Guided waves arise in a variety of applications like underwater acoustics, optics, the design of musical instruments, and so on. We present an analysis of wave propagation and reflection in an acoustic waveguide with random sound soft boundary and a turning point. The waveguide has slowly bending axis and variable cross section. The variation consists of a slow and monotone change of the width of the waveguide and small and rapid fluctuations of the boundary, on the scale of the wavelength. These fluctuations are modeled as random. The turning point is many wavelengths away from the source, which emits a pulse that propagates toward the turning point, where it is reflected.  We consider a regime where scattering at the random boundary has a significant effect on the reflected pulse. We determine from first principles when this effects amounts to a deterministic pulse deformation. This is known as a pulse stabilization result. The reflected pulse shape is not the same as the emitted one. It is damped, due to scattering at the boundary, and is deformed by dispersion in the waveguide. An example of an application of this result is in inverse problems, where the travel time of reflected pulses at the turning points can be used  to determine the geometry of the waveguide.


Alexander Mamonov

University of Houston










Alexei Novikov

Penn State University 











Braxton Osting

University of Utah