# Inverse problems

Friday, September 8, 2017 - 10:40am - 11:15am

Luis Tenorio (Colorado School of Mines)

Since in Bayesian inversion data are often informative only on a low-dimensional subspace of the parameter space,

significant computational savings can be achieved using such subspace to characterize and approximate the posterior distribution of the parameters.

We study approximations of the posterior covariance matrix defined as low-rank updates of the prior covariance matrix and

prove their optimality for a broad class of loss functions which includes the Forstner

significant computational savings can be achieved using such subspace to characterize and approximate the posterior distribution of the parameters.

We study approximations of the posterior covariance matrix defined as low-rank updates of the prior covariance matrix and

prove their optimality for a broad class of loss functions which includes the Forstner

Wednesday, September 6, 2017 - 2:55pm - 3:30pm

Youssef Marzouk (Massachusetts Institute of Technology)

Many inverse problems may involve a large number of observations. Yet these observations are seldom equally informative; moreover, practical constraints on storage, communication, and computational costs may limit the number of observations that one wishes to employ. We introduce strategies for selecting subsets of the data that yield accurate approximations of the inverse solution. This goal can also be understood in terms of optimal experimental design.

Thursday, February 16, 2017 - 10:15am - 11:05am

Faouzi Triki (Université Grenoble-Alpes)

In the talk I will present recent results on multifrequency electrical impedance tomography. The inverse problem consists in identifying a conductivity inclusion inside a homogeneous background medium by injecting one current. I will use an original spectral decomposition of the solution of the forward conductivity problem to retrieve the Cauchy data corresponding to the extreme case of perfect conductor.

Thursday, February 16, 2017 - 11:30am - 12:20pm

Francis Chung (University of Kentucky)

The standard problem of optical tomography is to obtain information about the optical properties of an object by making measurements on the boundary. Acousto-optic tomography is a variation of this problem where the object is perturbed by an acoustic field, and optical boundary measurements are taken as the parameters of the acoustic field vary. In this talk I'll give a short introduction to the idea of acousto-optic tomography, and discuss some inverse problems that arise from this imaging technique.

Tuesday, February 14, 2017 - 11:30am - 12:20pm

P. Scott Carney (University of Illinois at Urbana-Champaign)

Optical coherence tomography (OCT) provides an alternative to physical sectioning that allows for imaging of living samples and even in vivo examination of cell structure and dynamics. There is, in the OCT community, a widely held belief that there exists a trade-off between transverse resolution and the thickness of the volume that may be imaged with a fixed focal plane. Efforts to overcome this trade-off have focused on the design optical elements and imaging hardware.

Friday, June 10, 2016 - 9:00am - 10:00am

Benham Jafarpour (University of Southern California)

In this talk, I will present an overview of sparse representations and their applications in solving inverse modeling problems involving PDEs that describe multi-phase flow in heterogeneous porous media. The related PDE-constrained inverse problems are often formulated to infer spatially distributed material properties from dynamic response measurements at scattered source/sink locations.

Thursday, June 9, 2016 - 9:00am - 10:00am

Paul Barbone (Boston University)

Inverse problems are often formulated as constrained optimization problems. The constraint is in the form of a forward problem. The forward model is repeatedly solved and iteratively updated in order to bring its predictions in conformity with a set of data, or observations. We identify two distinct classes of deficiencies present in standard forward solvers (i.e. variational formulations) when used in this way. These deficiencies has motivated the development of two new variational formulations to address them.

Tuesday, June 7, 2016 - 2:50pm - 4:05pm

Wilkins Aquino (Duke University)

1. Introduction and Motivation

-What are Inverse problems?

-Examples: detection of contaminant sources, image and voice recognition, medical imaging, subsurface imaging, materials identification

2. Theoretical aspects of (discrete) inverse problems

-Why are inverse problems (oftentimes) difficult to solve?

-Well-posed and ill-posed problems: existence, uniqueness, and stability of solutions

-Linear vs nonlinear inverse problems

-Singular Value Decomposition: a path to understanding inverse problems

-Regularization:

-What are Inverse problems?

-Examples: detection of contaminant sources, image and voice recognition, medical imaging, subsurface imaging, materials identification

2. Theoretical aspects of (discrete) inverse problems

-Why are inverse problems (oftentimes) difficult to solve?

-Well-posed and ill-posed problems: existence, uniqueness, and stability of solutions

-Linear vs nonlinear inverse problems

-Singular Value Decomposition: a path to understanding inverse problems

-Regularization:

Tuesday, March 15, 2016 - 11:00am - 11:30am

Barbara Kaltenbacher (Universität Klagenfurt)

Parameter identification problems typically consist of a model equation, e.g. a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated via the parameter-to-state map. Alternatively, one might consider both sets of equations (model and observations) as one large system, to which some regularization method is applied.

Tuesday, March 15, 2016 - 9:00am - 10:00am

Allen Tannenbaum (State University of New York, Stony Brook (SUNY))

In this talk, we will review some key inverse problems in medical imaging and vision, in particular, segmentation and registration. The approach will be via energy minimization through the calculus of variations as well Bayesian. We will also show how these techniques made be made interactive through the use of feedback. This will allow us to make contact with ideas from distributed parameter systems. All of the techniques will be illustrated through some real medical imagery including MRI and CT data.