Distinguished Women in Mathematics Lecture Series

Upcoming Speakers

Liliana Borcea

Professor of Mathematics, University of Michigan

Colloquium

Date and Time: Monday, February 5th, 4:00 p.m.

Location: RLM 6.104

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

Abstract: 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.

Seminar

Date and Time: Tuesday, February 6th, 1:00 PM

Location: POB 6.304

Talk: Pulse reflection in a random waveguide with a turning point

Abstract: 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.

Lunch

All women faculty and graduate students are invited.

Date and Time: Monday, February 5th from 12:00-2:00 pm

Location: RLM 8.136

Pizza Seminar

Date and Time: Friday, February 2nd at noon

Location: RLM 12th floor lounge

Talk: Beyond the Born Approximation

Abstract: For a standard wave equation, the map from velocity c to the solution u is nonlinear although it is a linear PDE. In scattering theory, the linearization of this map to first order is called the Born approximation. It is accurate if the scattered field is small compared to the incident field on the scatterer, i.e., when the Born series converges. I will derive the Born approximation from an acoustic wave equation for illustration and show the convergence theorems. However, the Born approximation is not accurate in strongly scattering media. I will explain limitations encountered in practical applications which motivate the active study of going beyond it.