# Minisymposium-4: Computational Inverse Problems: Theory and Application

Inverse problems seek to infer causal factors from observations. They arise naturally in a wide range of scientific fields. Computational methods for inverse problems are crucial and of unique flavor due to their ill-posedness. Recently, the enormous increase in computing power and the development of cutting-edge numerical methods have made it possible to simulate forward problems of growing complexity, which in turn leads to demand for novel computational methods to solve the resulting inverse problems. This mini-symposium will focus on new challenges that have appeared recently in inverse problems, including new applications in imaging sciences, life sciences, physical sciences and industry. It will bring together applied mathematicians and engineers to discuss the recent progress of computational methods in inverse problems. The mini-symposium is expected to promote the development of novel ideas and new research collaborations through knowledge dissemination and interdisciplinary discussion.

## Organizers:

Wei Li, DePaul University
Yang Yang, Michigan State University
Yimin Zhong, Duke University

## Saturday, October 2, 2021 at 10:20 – 11:40 am (CST) (Session 1)

Fei Lu
Johns Hopkins University
10:20 am

Nonparametric Learning of Interaction Kernels in Mean-field Equations of Particle Systems

Systems of interacting particles/agents arise in multiple disciplines, such as particle systems in physics, flocking birds and swarming cells in biology, and opinion dynamics in social science. We consider the learning of the distance-based interaction kernels between the particles/agents from data. A challenging case is when the system is large with millions of particles, and we can only observe the population density. We present an efficient regression algorithm to estimate the interaction kernel, along with a systematic learning theory addressing identifiability and convergence of the estimators. We demonstrate our algorithm on three typical examples: the opinion dynamics with a piecewise linear kernel, the granular media model with a quadratic kernel, and the aggregation-diffusion with a repulsive-attractive kernel.

Arvind Saibaba
North Carolina State University
10:40 am

Bayesian Level Set Approach for Inverse Problems with Piecewise Constant Reconstructions

There are several challenges associated with inverse problems in which the unknown parameters can be modeled as piecewise constant functions. We model the unknown parameter using multiple level sets to represent the piecewise constant function. Adopting a Bayesian approach, we impose prior distributions on both the level set functions that determine the piecewise constant regions as well as the parameters that determine their magnitudes. We develop a Gauss-Newton approach with a backtracking line search to efficiently compute the maximum a priori (MAP) estimate as a solution to the inverse problem. We use the Gauss-Newton Laplace approximation to construct a Gaussian approximation of the posterior distribution and use preconditioned Krylov subspace methods to sample from the resulting approximation. To visualize the uncertainty associated with the parameter reconstructions we compute the approximate posterior variance using a matrix-free Monte Carlo diagonal estimator. We will demonstrate the benefits of our approach and solvers on synthetic test problems (photoacoustic and hydraulic tomography, respectively a linear and nonlinear inverse problem) as well as an application to X-ray imaging with real data. Joint work with William Reese (NC State), and Jonghyun Lee (University of Hawaii Manoa).

Junshan Lin
Auburn University
11:00 am

A Super-resolution Imaging Approach by Using Subwavelength Hole Resonances

Based on our recent studies on subwavelength hole resonances, we present a new imaging modality with illumination patterns generated from a collection of coupled resonant holes. When the incident frequencies are close to the resonant frequencies, the corresponding patterned illuminations encompass both low frequency and highly oscillatory waves, which allow for probing both the low and high spatial frequencies components of the imaging sample to achieve super-resolution. Under the weak scattering scenario, the linear imaging problem essentially boils down to a deconvolution problem that can be solved efficiently. The imaging setup, the underlying mathematical framework and the computational results will be exemplified in two dimensions.

Miao-Jung Yvonne Ou
University of Delaware
11:20 am

Non-negative Least Squares Using Multi-regularization and a Gaussian Basis, with Application to Magnetic Resonance Relaxometry

Non-negative least squares is an effective method for recovery of the probability distribution function (PDF) of a distributed parameter from a discretized Fredholm integral equation of the first kind. Regularization is often employed to stabilize this procedure with respect to noise, leading to the significant challenge of selection of an optimal regularization parameter $\lambda$. To avoid this treacherous process, we propose a new method for incorporating regularization into the solution of such inverse problems; rather than seeking to \textit{replace} the native problem with a suitable mathematically close, regularized, version, we instead \textit{augment} the native formulation with regularization, and introduce multiple simultaneous regularization parameters. We term this new approach multiple-regularization (Multi-Reg). This permits incorporation of several degrees of regularization into the solution. We illustrate the method with extensive simulation results as well as application to experimental magnetic resonance data. Joint work with: Chuan Bi, Wenshu Qian, Kenneth W. Fishbein, Mustapha Bouhrara and Richard G. Spencer at National Institute on Aging, Baltimore, MD 21224, U.S.A

## Saturday, October 2, 2021 at 4:40 – 6:00 pm (CST) (Session 2)

Kshitij Tayal
University of Minnesota
4:40 pm

Inverse Problems, Deep Learning, and Symmetry Breaking

In many physical systems, inputs related by intrinsic system symmetries generate the same output. Hence, when inverting such systems, an input is mapped to multiple symmetry-related outputs. This causes fundamental difficulties for tackling these inverse problems by the emerging end-to-end deep learning approach. Using phase retrieval as an illustrative example, we show that careful symmetry breaking on the training data can help eliminate the difficulties and significantly improve learning performance in real data experiments. We also extract and highlight the underlying mathematical principle of the proposed solution, which is directly applicable to other inverse problems.

Tian-Ming Fu
Howard Hughes Medical Institute
5:00 pm

Advanced Microscope Technologies for Biological Discovery and Related Computational Challenges: From Single-Molecule Dynamics to Whole-Organism Development

“Seeing is believing”. Advances in fluorescence microscopy over the past decades set off a revolution in visualizing and quantifying biological organizations and dynamics with unprecedented spatiotemporal resolution. The quantity and complexity of resulting images also pose new computational challenges and opportunities. As an experimentalist, I will first introduce our group’s recent efforts in the development of a multimodal imaging platform, MOSAIC, that can observe molecular and subcellular dynamics inside multicellular organisms. Specifically, adaptive optics, a concept borrowed from telescope, is implemented for each imaging modality of the MOSAIC to recover resolution and contrast deep inside optical heterogenous specimens. Then I will describe some of the computational challenges we faced in high resolution in vivo imaging with a focus on spatially dependent deconvolution. Last, I will briefly discuss the opportunities and challenges in seamless integrating hardware and software to achieve “intellectual microscope”—a microscope that can be automatically reconfigured and optimized for the specific biological samples and extract complementary information across spatiotemporal scales.

Wei Li
DePaul University
5:20 pm

Acousto-Electric Inverse Source Problem

We propose a method to reconstruct the electrical current density inside a conducting medium from acoustically-modulated boundary measurements of the electric potential. We show that the current can be uniquely reconstructed with Lipschitz stability. We also perform numerical simulations to illustrate the analytical results, and explore the partial data setting when measurements are taken only on part of the boundary.