## Current Series

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Mon Nov 04

## Applied and Computational Mathematics Seminar

3:35pm - Vincent Hall 6
Applied differential geometry and harmonic analysis in deep learning regularization
Wei Zhu, Duke University

Deep neural networks (DNNs) have revolutionized machine learning by gradually replacing the traditional model-based algorithms with data-driven methods. While DNNs have proved very successful when large training sets are available, they typically have two shortcomings: First, when the training data are scarce, DNNs tend to suffer from overfitting. Second, the generalization ability of overparameterized DNNs still remains a mystery.

In this talk, I will discuss two recent works to inject the modeling flavor back into deep learning to improve the generalization performance and interpretability of the DNN model. This is accomplished by DNN regularization through applied differential geometry and harmonic analysis. In the first part of the talk, I will explain how to improve the regularity of the DNN representation by enforcing a low-dimensionality constraint on the data-feature concatenation manifold. In the second part, I will discuss how to impose scale-equivariance in network representation by conducting joint convolutions across the space and the scaling group. The stability of the equivariant representation to nuisance input deformation is also proved under mild assumptions on the Fourier-Bessel norm of filter expansion coefficients.

Mon Nov 18

## Applied and Computational Mathematics Seminar

3:35pm - Vincent Hall 6
Scalable Algorithms for Data-driven Inverse and Learning Problems
Tan Bui-Thanh, UT-Austin

Inverse problems and uncertainty quantification (UQ) are pervasive in scientific
discovery and decision-making for complex, natural, engineered, and societal systems.
They are perhaps the most popular mathematical approaches for enabling predictive scientific simulations that integrate observational/experimental data, simulations and/or
models. Unfortunately, inverse/UQ problems for practical complex systems possess these the simultaneous challenges: the large-scale forward problem challenge, the high dimensional parameter space challenge, and the big data challenge.

To address the first challenge, we have developed parallel high-order (hybridized) discontinuous Galerkin methods to discretize complex forward PDEs. To address the second challenge, we have developed various approaches from model reduction to advanced Markov chain Monte Carlo methods to effectively explore high dimensional parameter spaces to compute posterior statistics. To address the last challenge, we have developed a randomized misfit approach that uncovers the interplay between the Johnson-Lindenstrauss and the Morozov's discrepancy principle to significantly reduce the dimension of the data without compromising the quality of the inverse solutions.

In this talk we selectively present scalable and rigorous approaches to tackle these challenges for PDE-governed Bayesian inverse problems. Various numerical results for simple to complex PDEs will be presented to verify our algorithms and theoretical findings. If time permits, we will present our recent work on scientific machine learning for inverse and learning problems.

Mon Jan 27

## Applied and Computational Mathematics Seminar

3:35pm - Vincent Hall 311
An inverse problem on Light Sheet Fluorescence Microscopy
Benjamin Palacios, University of Chicago

In Light Sheet Fluorescence Microscopy a density of fluorescent material (fluorophores) needs to be reconstructed through a process that consists in the application of a thin sheet of light that stimulates fluorophores, inducing the emission of fluorescent light that is recorderded and which constitute our measurements. In this talk I will present a mathematical model for this two-step process as well as the inverse problem arising from it. Uniqueness and stability of the inverse problem will be discussed.

Mon Feb 17

## Applied and Computational Mathematics Seminar

3:35pm - Vincent Hall 311
Direct Sampling Algoritmis in Inverse Scattering
Isaac Harris, Purdue University

In this talk, we will discuss a recent qualitative imaging method referred to as the Direct Sampling Method for inverse scattering. This method allows one to recover a scattering object by evaluating an imaging functional that is the inner-product of the far-field data and a known function. It can be shown that the imaging functional is strictly positive in the scatterer and decays as the sampling point moves away from the scatterer. The analysis uses the factorization of the far-field operator and the Funke-Hecke formula. This method can also be shown to be stable with respect to perturbations in the scattering data. We will discuss the inverse scattering problem for both acoustic and electromagnetic waves.

Mon Mar 02

## Applied and Computational Mathematics Seminar

3:35pm - Vincent Hall 311
Equilibration of aggregation-diffusion equations with weak interaction forces
Ruiwen Shu, University of Maryland

I will talk about the large time behavior of aggregation-diffusion equations. For one spatial dimension with certain assumptions on the interaction potential, the diffusion index $m$, and the initial data, we prove the convergence to the unique steady state as time goes to infinity (equilibration), with an explicit algebraic rate. The proof is based on a uniform-in-time bound on the first moment of the density distribution, combined with an energy dissipation rate estimate. This is the first result on the equilibration of aggregation-diffusion equations for a general class of weakly confining potentials $W(r)$: those satisfying $\lim_{r\rightarrow\infty}W(r)<\infty$.