## PDE & Applied math seminar

Wednesday 10-10:50 am, Skye 268

Current Organizers : Weitao Chen / Heyrim Cho / Yat Tin Chow / Qixuan Wang

Past Organizers : Mark Alber / James Kelliher / Amir Moradifam

**Fall 2019 Schedule **

Oct 02 10:00 (Wed) Organizing meeting

Oct 09 10:00 (Wed) Dr. Jinsu Kim (University of California, Irvine) (Wang)

Oct 16 10:00 (Wed) Dr. Heyrim Cho (University of California, Riverside)

Oct 23 10:00 (Wed) Dr. Elaine Cozzi (Oregon State University) (Kelliher)

Oct 30 13:00 (Wed) Dr. Melvin Leok (University of California, San Diago) (Alber)

Nov 08 11:00 (Fri) Dr. Charles Doering (University of Michigan) (Kelliher)

Nov 13 10:00 (Wed) Dr. Franca Hoffmann (Caltech) (Cho)

Nov 20 10:00 (Wed) Dr. Zilong Song (University of California, Riverside)

Nov 27 10:00 (Wed) Thanksgiving?

Dec 04 10:00 (Wed) Dr. Helena Nussenzveig (Tentative) (Federal University of Rio de Janeiro) (Kelliher)

Dec 09 13:00 (Mon) Dr. Tien Khai Nguyen (North Carolina State University) (Chow)

**Title and Abstracts**

Nov 13 10:00 (Wed), 2019

Dr. Franca Hoffmann (Caltech)

**Title:** Kalman-Wasserstein Gradient Flows

**Abstract:** We study a class of interacting particle systems that may be used for optimization. By considering the mean-field limit one obtains a nonlinear Fokker-Planck equation. This equation exhibits a novel gradient structure in probability space, based on a modified Wasserstein distance which reflects particle correlations: the Kalman-Wasserstein metric. This setting gives rise to a methodology for calibrating and quantifying uncertainty for parameters appearing in complex computer models which are expensive to run, and cannot readily be differentiated. This is achieved by connecting the interacting particle system to ensemble Kalman methods for inverse problems. This is joint work with Alfredo Garbuno-Inigo (Caltech), Wuchen Li (UCLA) and Andrew Stuart (Caltech).

Nov 08 11:00 (Fri), 2019, Skye 284

Dr. Charles Doering (University of Michigan)

**Title:** Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

**Abstract:** For any quantity of interest in a system governed by nonlinear differential equations it is natural to seek the largest (or smallest) long-time average among solution trajectories. Upper bounds can be proved a priori using auxiliary functions, the best choice of which is a convex optimization. We show that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on maximal time averages. They also provide volumes in phase space where maximal trajectories must lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming which we illustrate using the Lorenz and Kuramoto-Sivashinsky equations. This is joint work with Ian Tobasco and David Goluskin, part of which appears in *Physics Letters A* **382**, 382-386 (2018).

Oct 30 13:00 (Wed), 2019, Skye 284

Dr. Melvin Leok (University of California, San Diago)

**Title:** Variational discretizations of gauge field theories using group-equivariant interpolation spaces.

**Abstract:** Variational integrators are geometric structure-preserving numerical methods that preserve the symplectic structure, satisfy a discrete Noether's theorem, and exhibit exhibit excellent long-time energy stability properties. An exact discrete Lagrangian arises from Jacobi's solution of the Hamilton-Jacobi equation, and it generates the exact flow of a Lagrangian system. By approximating the exact discrete Lagrangian using an appropriate choice of interpolation space and quadrature rule, we obtain a systematic approach for constructing variational integrators. The convergence rates of such variational integrators are related to the best approximation properties of the interpolation space. Many gauge field theories can be formulated variationally using a multisymplectic Lagrangian formulation, and we will present a characterization of the exact generating functionals that generate the multisymplectic relation. By discretizing these using group-equivariant spacetime finite element spaces, we obtain methods that exhibit a discrete multimomentum conservation law. We will then briefly describe an approach for constructing group-equivariant interpolation spaces that take values in the space of Lorentzian metrics that can be efficiently computed using a generalized polar decomposition. The goal is to eventually apply this to the construction of variational discretizations of general relativity, which is a second-order gauge field theory whose configuration manifold is the space of Lorentzian metrics.

Oct 23 10:00 (Wed), 2019

Dr. Elaine Cozzi (Oregon State University)

**Title:** Solutions to the 2D Euler equations with velocity growing at infinity

**Abstract:** In this talk, we outline a proof of existence of solutions to the 2D Euler equations with vorticity bounded and with velocity growing sufficiently slowly at spatial infinity. If time permits, we will also discuss uniqueness and continuous dependence on initial data for these solutions. This is joint work with James P. Kelliher.

Oct 16 10:00 (Wed), 2019

Dr. Heyrim Cho (University of California, Riverside)

**Title:** Models in mathematical oncology driven from high-dimensional genetic data to scarce patient data

**Abstract:** Mathematical oncology aims to study and fight against cancer by developing theoretical and computational tools in mathematics. Recent advances in biotechnology and genome sequencing, resulting surge of data, are bringing in new opportunities in mathematical modeling of biological systems. This motivates us to transfer the mathematical and computational models from a purely theoretical and qualitative setting to the clinic, to guide patient therapy via prediction. 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 patientâ€™s burden. In this talk, I will discuss modeling approaches on those two ends of the spectrum. First, I will develop models using high-dimensional single-cell gene sequencing data to model the normal hematopoiesis differentiation and abnormal processes of acute myeloid leukemia (AML) progression. The model can predict the emergence of cells in novel intermediate states of differentiation consistent with immunophenotypic characterizations of AML. Second, I will discuss parameter identifiability of cancer models under radiotherapy and immunotherapy, assuming scarce and noisy experimental data. We demonstrate that tools in mathematical analysis can help to improve the predictive power of mathematical models.

Oct 09 10:00 (Wed), 2019

Dr. Jinsu Kim (University of California, Irvine)

**Title:** Stochastically modeled reaction networks: stability and mixing times

**Abstract:** A reaction network is a graphical configuration that describes an interaction between species (molecules). If the abundances of the network system are small, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a jump by jump fashion continuous-time Markov chain. One of the challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. Thus, how to characterize network structures that induce characteristic behaviors of the system dynamics is one of the major open questions in this literature. In this talk, I will provide an analytic approach to find a class of reaction networks whose associated Markov process has a stationary distribution. Moreover, I will also talk about the convergence rate for the process to its stationary distribution with the mixing time.