Schedule for: 21w5035 - Women in Inverse Problems (Online)

All attendees will be giving a 1 min talk with 1 slide. Your slide should introduce yourself and what motivates you (as an applied mathematician working on Inverse Problems and more generally in life). Here is the dropbox invite where you can submit you 1 slide https://www.dropbox.com/request/1JbI9I2G7WgK66yxndpv This slide should be in pdf format with NO movies. We will be expecting the slides by Dec 1st.

Joint work with Huiyi Chen (UTD) and Bill Symes (Rice U.)

I will describe a construction of a reduced order model from wave scattering data collected by an array of sensors. The construction is based on interpreting the wave propagation as a dynamical system that is to be learned from the data. The states of the dynamical system are the snapshots of the wave at discrete time intervals. We only know these snapshots at the sensors in the array. The reduced order model is a Galerkin projection of the dynamical system that can be calculated from such knowledge. I will describe the construction and some properties of the reduced order model and I will show how it can be used for solving inverse wave scattering problems.

This talk introduces the problem of localizing electromagnetic sources from measurements of their radiated fields at two moving sensors. Two approaches are discussed, the first based on measuring quantities known as “time difference of arrivals“ and “frequency difference of arrivals”. This approach leads to some interesting geometrical problems. The second approach, a synthetic-aperture approach, is more promising but also involves some unsolved problems.

We develop a new empirical Bayesian inference algorithm for solving a linear inverse problem given multiple measurement vectors (MMV) of noisy observable data. Specifically, by exploiting the joint sparsity across the multiple measurements in the sparse domain of the underlying signal or image, we construct a new support informed prior. While a variety of applications can be modeled using this framework, our prototypical example comes from synthetic aperture radar (SAR) data, from which data are acquired from neighboring aperture windows. Hence a good test case is to consider the observations modeled as noisy Fourier samples. Our numerical experiments demonstrate that using the support informed prior not only improves accuracy of the recovery, but also reduces the uncertainty in the posterior when compared to standard sparsity producing priors. This is joint work with Theresa Scarnati formerly of the Air Force Research Lab Wright Patterson and now working at Qualis Corporation in Huntsville, AL, and Jack Zhang, recent bachelor degree recipient at Dartmouth College and now enrolled at University of Minnesota’s PhD program in mathematics.

Recently, a new paradigm has been introduced to the regularization of inverse problems, which derives regularization approaches for inverse problems in a data driven way. Here, regularization is not mathematically modelled in the classical sense, but modelled by highly over-parametrised models, typically deep neural networks, that are adapted to the inverse problems at hand by appropriately selected (and usually plenty of) training data. In this talk, I will review some machine learning based regularization techniques, present some work on unsupervised and deeply learned convex regularisers and their application to image reconstruction from tomographic measurements, and discuss some open mathematical problems.

Authors: Ege Ozsar, *Misha Kilmer, Eric Miller, Eric de Sturler, Arvind Saibaba *speaker Parametric level set (PaLS) methods for shape-based inverse problems provide flexibility in terms of their ability to represent shapes while avoiding many difficulties and numerical concerns associated with traditional level set methods. In this talk, we consider the restoration and reconstruction of piecewise constant objects in two and three dimensions using PaLEnTIR, a significantly enhanced PaLS model relative to the current state-of-the-art. Given only upper and lower limits of the contrast in the medium, our formulation requires only a single level set function to recover a scene with an unknown number of piecewise constant objects possessing multiple unknown contrasts. Relative to PaLS methods which employ radial basis functions (RBFs), our model employs non-isotropic basis functions, thereby expanding the class of shapes that a PaLS model of a given complexity can approximate. We demonstrate the performance of the new approach on linear and non-linear inverse problems, including 2D and 3D variants of X-ray computed tomography, diffuse optical tomography (DOT), and deconvolution problems.

Panel I: Future directions in inverse problems L. Borcea, D. Calvetti, C. Schönlieb, Y. Ou

Electrical impedance tomography (EIT) is a non-ionizing medical imaging technique in which electric fields are used to form real-time images of organ function and structure. To form these images, it is necessary to solve a severely ill-posed inverse problem with computational efficiency. However, the ill-posedness compromises image resolution, which is addressed by introducing regularization. Anatomical priors in the regularization term reduce the ill-posedness and give the opportunity to provide the reconstruction algorithm with known information about the conductivity distribution. In this talk, we propose a technique in which a statistical prior built from an anatomical atlas is used to post-process 2-D D-bar reconstructions of human chest data. The D-bar method, which has the attributes of being a direct (non-iterative) method with a proven nonlinear regularization strategy, is used to compute reconstructions of samples from the anatomical atlas and from the measured data. The resolution of the D-bar images are improved by maximizing the conditional probability density function of an image that is consistent with the a priori information and the statistical model. The effectiveness of the method is demonstrated on pulmonary EIT data from two patients with cystic fibrosis.

Machine learning tasks often utilize vast amounts of data in the form of multi-dimensional arrays. Standard techniques for processing this data have quickly become impractical due to the sheer size of the data sets at hand. In this talk, we will discuss stochastic iterative approaches for solving multilinear systems under the t-product and connect these approaches to their matrix counterparts. Such approaches are well known to be beneficial in the big data setting when entire data sets are too large to fit into working memory. We highlight regimes under which t-product measurements are more computationally efficient than alternative measurement schemes and create a foundation for designing and analyzing future iterative approaches for solving multilinear systems.

The estimation of the probability of a rare event is an important task in reliability and risk assessment of critical societal systems, for example, groundwater flow and transport, and engineering structures. Specifically, we consider rare events that are expressed in terms of a limit state function that depends on the solution of a partial differential equation (PDE). The main goal of the talk is to highlight how we might use algorithms and theoretical results in the inverse problems literature to design and analyse tools for rare event estimation. We discuss a selection of the following examples: (1) a multilevel Sequential Importance Sampling method which is based on a Sequential Monte Carlo method for Bayesian inversion, (2) the use of the Ensemble Kalman Filter for the estimation of failure probabilities, and (3) the analysis of the PDE approximation error in the failure probability estimate which uses results from PDE-constrained parameter identification problems. This is joint work with Fabian Wagner (TUM), Iason Papaioannou (TUM) and Jonas Latz (Heriot-Watt University).

Panel II: Women in Inverse Problems - community, challenges, open discussion M. Espanol, T. Vdovina, C. Tsogka, E. Resmerita