# Ongoing Research Projects

### Bayesian Approaches for Strain Identification from Culture-Independent Samples

We develop new mathematical approaches, algorithms and systems for identifying pathogen strains using whole genome multilocus sequence typing (wgMLST) data in mixed samples. Strain identification is an underdetermined inverse problem when the samples are not cultured, i.e., contain multiple strains at unknown proportions. Therefore, we encode prior knowledge on both the strains and their proportion in a Bayesian framework. Our approaches use mixed-integer optimization to identify the most likely strains and Monte-Carlo sampling to quantify the uncertainty associated with those estimates. Motivating applications of our algorithms include the identification of *P. falciparum* (malaria) and *E.coli* (foodborne illness) strains to improve visibility and monitoring for public health officials.

PIs: Lars Ruthotto, Ymir Vigfusson, and Rebecca Mitchell

### Efficient Algorithms for PDE-Parameter Estimation

This project develops efficient numerical methods for solving big data parameter estimation problems that involve partial differential equations (PDEs). Parameter estimation problems of this kind impose key challenges in many scientific disciplines, e.g., in medical imaging, geophysical imaging, and deep learning. The inverse problem can be formulated as an optimization problem with constraints that are given by the PDEs. The unknowns are parameters of the PDEs (e.g., physical properties of the object to be measured). The objective is to minimize the misfit between PDE simulations and measured data plus some regularization term.

The project aims at deriving efficient algorithms and massively parallel solvers for solving big data PDE parameter estimation problems. Its main thrusts are

- Reduced-order modeling techniques
- Stochastic optimization methods
- Parallel and distributed optimization problems

PI: Lars Ruthotto

Funding: US National Science Foundation award DMS 1522599

### Algorithms for Inverse Problems that Exploit Kronecker Product and Tensor Structures

The aim of this project is to develop efficient singular value decomposition (SVD) approximation methods for large-scale matrices that arise in discrete ill-posed inverse problems. The approach will exploit inherent Kronecker product and tensor structures, and will be the basis for a computational platform for the efficient solution of large-scale ill-posed problems. Efficient approaches to solve Kronecker product and tensor structured SVD updating problems will be developed. Iterative methods that can incorporate regularization, sparse and low-rank constraints on the solution will also be considered. The SVD approximations and updating methods developed in this project can be used as tools to obtain approximate solutions of ill-posed inverse problems, as preconditioners to accelerate iterative solvers, or as tools to build solution methods for nonlinear problems. By developing a computational platform, based on SVD approximations, the work developed in this proposal will have a broad scientific impact for applications where it is necessary to compute solutions of large-scale ill-posed inverse problems, including astronomy, cosmology, geophysics, microscopy, and medical imaging.

PI: James Nagy

Funding: US National Science Foundation award DMS 1522760

### Numerical Methods for Multispectral Tomographic Image Reconstruction

Breast cancer is the most prevalent non-skin cancer in women in the US. Although mammography is currently the most common test for the early detection and diagnosis of breast cancer, its two-dimensional nature introduces various limitations in its capabilities for advanced imaging. Over the last decade, the introduction of digital imaging technology has resulted in the development of new tomographic and pseudo-tomographic methods for imaging the breast.

Digital tomographic image reconstruction uses multiple x-ray projections obtained along a range of different incident angles to reconstruct a 3D representation of an object. For example, computed tomography (CT) generally refers to the situation when a full set of angles are used (e.g., 360 degrees) while tomosynthesis refers to the case when only a limited (e.g., 30 degrees) angular range is used. In either case, most existing reconstruction algorithms assume that the x-ray source is monoenergetic. This results in a simplified linear forward model, which is easy to solve but can result in artifacts in the reconstructed images. It has been shown that these artifacts can be reduced by using a more accurate polyenergetic assumption for the x-ray source, but the polyenergetic model requires solving a large-scale nonlinear inverse problem. In addition to reducing artifacts, a full polyenergetic model can be used to extract additional information about the materials of the object; that is, to provide a mechanism for quantitative imaging. Similar mathematical models arise when using multispectral detectors.

The aim of this project is to develop algorithms and software to solve these very challenging inverse problems.

PI: James Nagy

This is joint work with Martin Andersen, Technical University of Denmark, and Ioannis Sechopoulos, Radboud University Medical Center, and is paritally funded by a grant from the US National Institute of Health.