Fall 2021
Speaker: Prof. Pedram Rooshenas, UNC Charlotte
Date and Time: Aug 27, 2021, 1:00 p.m. -2:00 p.m (Fretwell 315)
Title: Energy-Based Models: Promises and Challenges
Abstract: Recent advances in deep neural networks (DNNs) have revolutionized fields such as natural language processing, computer vision, and robotics, while also recently impacting other fields such as biology, computational mechanics, and health care.
Many problems in these domains require a joint model to describe the correlation among a set of variables, such as the correlation of words in a translated sentence. Classically, these correlations were modeled using graphical models (such as Markov random fields) defined over the variables. This talk will show how we can leverage DNNs to describe these correlations using energy-based models (EBMs). I will also show how we can use EBMs for different problem settings and discuss their promises and challenges.
Speaker: Son Tu, University of Wisconsin – Madison
Date and Time: Sep 24th, 2021 , 10 a.m. – 11 a.m. (Fretwell 315)
Title: Rate of convergence on the vanishing viscosity process of state-constraint Hamilton–Jacobi equations.
Abstract: We investigate the convergence rate in the vanishing viscosity process of the solutions to the subquadratic state-constraint Hamilton-Jacobi equations. We give two different proofs of the fact that, for nonnegative Lipschitz data that vanish on the boundary, the rate of convergence is O(√ε) in the interior. Moreover, the one-sided rate can be improved to O(ε) for nonnegative compactly supported data and O(ε^(1/p)) (where 1 < p < 2 is the exponent of the gradient term) for nonnegative data f ∈ C^2(Ω) such that f = 0 and Df = 0 on the boundary. Our approach relies on deep understanding of the blow-up behavior near the boundary and semiconcavity of the solutions. This is a joint-work with Yuxi Han.
Video Recording: https://drive.google.com/file/d/1e_iy7h66fHELEKqcZkaddHkC9CbS5xCf/view?usp=sharing
Speaker: Dr. Deep Ray, University of Southern California
Date and Time: Oct 8th, 2021, 10:00 a.m. – 11:00 a.m. (Fretwell 315)
Title: Solving physics-based inverse problems using generative adversarial networks.
Abstract: Inverse problems are notoriously hard to solve and are often ill-posed. Bayesian inference provides a principled approach to resolve this by posing the problem in a statistical framework. However, this approach can be challenging to implement when the field being inferred is high-dimensional, or when the prior information/data is too complex to represent using simple distributions. In this talk, we will discuss two strategies using generative adversarial networks (GANs) to overcome these issues. First, we demonstrate that by training a GAN to learn complex priors, and reformulating the inference problem in the low-dimensional latent space of the GAN, we can efficiently compute the solution of large-scale Bayesian inverse problems. Second, we show how a conditional GAN can be trained to directly learn and sample from the posterior distribution. We present several physics-based numerical experiments to demonstrate the efficacy of these deep learning approaches in the Bayesian framework.
Speaker: Dr. Pratima Hebbar, Duke University
Date and Time: Oct 15th, 2021, 1 p.m. – 2 p.m.
Title: Branching diffusion processes.
Abstract: We investigate the asymptotic behavior of solutions to parabolic partial differential equations in R^d with space-periodic diffusion matrix, drift, and potential. Using this asymptotics, we describe the behavior of branching diffusion processes in periodic media. For a super-critical branching process, we distinguish two types of behavior for the normalized number of particles in a bounded domain, depending on the distance of the domain from the region where the bulk of the particles is located. At distances that grow linearly in time, we observe intermittency (i.e., the k−th moment dominates the k−th power of the first moment for some k), while, at distances that grow sub-linearly in time, we show that all the moments converge.
Speaker: Prof. Qixuan Wang, UC Riverside
Date and Time: Oct 22nd, 2021, 1 p.m. – 2:00 p.m.
Title: Roles of cellular anisotropy and heterogeneity in life.
Abstract: Cells can be structurally anisotropic, and they can be heterogeneous due to either genetic or environment clues. Cellular anisotropy and heterogeneity might lead to interesting behaviors of both an individual cell and a collection of cells. In this talk we will discuss the roles of cellular anisotropy and heterogeneity in two systems. In the first part, we will discuss how anisotropic flagella bending rigidity affects the flagellar beating dynamics. Flagellar beating is controlled by molecular motors that exert forces along the length of the flagellum and are regulated by a feedback mechanism coupled to the flagellar motion. We build on previous work on sliding-controlled motor feedback to develop a fully three-dimensional description of flagellar beating, accounting for both bending and twist. We show that with isotropic bending, three-dimensional spiral modes are spontaneously generated beyond a critical molecular activity. On the other hand, when a bias in the bending directions presents, the three-dimensional spiral modes give way to planar beating. In the second part, we will discuss how hair follicle heterogeneous responses to signals regulate the follicle temporal growth dynamics. Hair follicles are mini skin organs rich of stem cells, and they undergo cyclic growth. The growing phase – anagen of a hair follicle is tightly controlled by a group of epithelial transient amplifying (TA) cells. Using an interdisciplinary approach combined of multi-scale modeling and lineage tracing experiments, we show that cellular heterogeneity based on cell division generations generates the clonal drift phenomenon that prolongs the anagen.
Speaker: Ihor Borachok, Ivan Franko National University of Lviv
Date and Time: Oct 29th, 2021, 10:30 a.m. -11:00 a.m.
Title: A method of the fundamental solutions for non-stationary Cauchy problems.
Abstract: We derive a method of fundamental solutions (MFS) for the numerical solution of an ill-posed lateral Cauchy problem for the hyperbolic wave and parabolic heat equations in bounded planar annular domains. The Laguerre transform is applied to reduce the time-dependent lateral Cauchy problem to a sequence of elliptic Cauchy problems with a known set of fundamental solutions termed a fundamental sequence. The solution of the elliptic problems is approximated by linear combinations of the elements in the fundamental sequence. Source points are placed outside of the solution domain, and by collocating on the boundary of the solution domain itself a sequence of linear equations is obtained for finding the coefficients in the MFS approximation. Tikhonov regularization is applied to get a stable solution to the obtained systems of linear equations.
Speaker: Dr. Heyrim Cho, UC Riverside
Date and Time: Nov 5th, 2021, 1:00 p.m. -2:00 p.m.
Title: Overcome the challenge of limited temporal data in clinical applications of mathematical modeling .
Abstract: Recent advances in biotechnology and genome sequencing, resulting in a surge of data, are bringing in new opportunities in mathematical modeling of biological systems. However, the amount of data that can be practically collected in everyday patients during the therapy is very limited due to the cost and the patient’s burden. Especially the amount of data that can be collected in the time domain is very limited. This motivates us to transfer the mathematical and computational models to meet the challenges in clinical data, to guide patient therapy via prediction. In this talk, I will discuss modeling approaches on the two ends of the spectrum of data. In the first part, I will discuss a Bayesian information-theoretic approach to determine effective scanning protocols of cancer patients. We propose a modified mutual information function with a temporal penalty term to account for the loss of temporal data. The effectiveness of our framework is demonstrated in determining image scanning scheduling for radiotherapy patients. In the second part, I will discuss modeling work using high-dimensional single-cell gene sequencing data. Due to the high cost of obtaining gene sequencing data, temporal data also lacks. We show that our cell state dynamics model can be used to incorporate genetic alteration with low cost, where we show an example of modeling hematopoiesis system and simulating abnormal differentiation that corresponds to acute myeloid leukemia.
Speaker: Dr. Elizabeth Newman, Emory University
Date and Time: Nov 12th, 2021, 1:00 p.m. -2:00 p.m.
Title: How to Train Better: Exploiting the Separability of Deep Neural Networks
Abstract: You would be hard-pressed to find anyone who hasn’t heard the hype about deep neural networks (DNNs). These high-dimensional function approximators, composed of simple layers parameterized by weights, have shown their success in countless applications. What the hype-sters won’t tell you is this: DNNs are challenging to train. Typically, the training problem is posed as a stochastic optimization problem with respect to the DNN weights. With millions of weights, a non-convex and non-smooth objective function, and many hyperparameters to tune, solving the training problem well is no easy task.
In this talk, our goal is simple: we want to make DNN training easier. To this end, we will exploit the separability of commonly-used DNN architectures; that is, the weights of the final layer of the DNN are applied linearly. We will leverage this linearity using two different approaches. First, we will approximate the stochastic optimization problem via a sample average approximation (SAA). In this setting, we can eliminate the linear weights through partial optimization, a method affectionately known as Variable Projection (VarPro). Second, in the stochastic approximation (SA) setting, we will consider a powerful iterative sampling approach to update the linear weights, which notably incorporates automatic regularization parameter selection methods. Throughout the talk, we will demonstrate the efficacy of these two approaches to exploit separability using numerical examples.
Speaker: Dr. Weiqi Chu, UCLA
Date and Time: Nov 19th, 2021, 1:00 p.m. – 2:00 p.m.
Title: A reduced-order method for electron transport with long-range interactions
Abstract: In the study of electron transport, one typical situation is a molecular junction, where single molecules are bound to two semi-infinite leads that are regarded as quantum baths. This necessarily introduces a large number of electronic degrees of freedom to the system. Another challenge is the long-range interactions from Coulomb potential. In the density matrix representation, Coulomb potential introduces a highly non-linear Hamiltonian to the Liouville–von Neumann equation. In this talk, we will introduce a model reduction approach using Petrov–Galerkin projection. In order to recover the global electron density profile as a vehicle to compute the Coulomb potential, we propose a domain decomposition approach, where the computational domain also includes segments of the bath that are selected using logarithmic grids. This approach leads to a multi-component self-energy that enters the effective Hamiltonain.
Speaker: Madhu Gupta, UT Arlington
Date and Time: Dec 3rd, 2021, 1p.m. -2 p.m.
Title: Sparsity-based nonlinear reconstruction of optical parameters in two-photon
photoacoustic computed tomography
Abstract:
Photoacoustic computes tomography (PACT) is a hybrid imaging modality, and we will
see a new nonlinear optimization approach for the sparse reconstruction of single-photon
absorption and two-photon absorption coefficients in PACT. This framework comprises
of minimizing an objective functional involving least squares fit of the interior pressure
field data corresponding to two boundary source functions, where the absorption
coefficients and the photon density are related through a semi-linear elliptic partial
differential equation (PDE) arising in photoacoustic tomography. Further, the objective
functional consists of an 𝐿1 regularization term that promotes sparsity patterns in
absorption coefficients. We will discuss the theoretical results which includes the
existence and uniqueness of the solution to the semi-linear PDE, and computational
techniques involve proximal method and Picard solver. Finally, to demonstrate the
effectiveness of our proposed framework we will see the numerical experiments