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Links : Interdisciplinary Center for Quantitative Modeling in Biology (ICQMB) seminar

             UCR Math Department seminar

In Fall 2023, the PDE & Applied Math seminar will be held through Zoom or in person (Skye 268). Specific information about the format of each talk will be provided in the email announcement and posted below. If you're interested in attending the seminar, please contact Dr. Yat Tin Chow (yattinc@ucr.edu) and Dr. Yiwei Wang (yiweiw@ucr.edu).

Fall 2023 Schedule 

Oct 4 10:00 AM (Wed) Organizational meeting

Oct 11 10:00 (Wed) Rene Cabrera (University of Texas at Austin)  In-person/Hybrid

Oct 18 10:00 AM (Wed) Li Wang (University of Minnesota, Twin Cities) Online only

Oct 25 10: 00 AM (Wed) Shu Liu (University of California, Los Angeles) Online only

Nov 1 10:00 AM (Wed) Arkadz Kirshtein (Tufts University) Online only

Nov 8 10:00 AM (Wed) Artur Stephan (Weierstrass Institute) Online only

Nov 15 10:00 AM (Wed) Siming He (University of South Carolina) In-person/Hybrid

Nov 29 10:00 AM (Wed) Maziar Raissi (University of California, Riverside) In-person/Hybrid

Dec 6 10:00 AM (Wed) Roberto Santoprete (L'Oréal Research & Innovation Labs)  Joint with ICQMB seminar

Upcoming Talk:

Nov 29, 2023, 10:00 AM - 10:50 AM PT 

Dr. Maziar Raissi (University of California, Riverside)

Title: Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations

Abstract: We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where: (1) all we observe are noisy data on black-box initial conditions, and (2) we are interested in quantifying the uncertainty associated with such noisy data in our solutions to time-dependent partial differential equations. Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors. This is an attempt to construct structured and data-efficient learning machines, which are explicitly informed by the underlying physics that possibly generated the observed data. The effectiveness of the proposed approach is demonstrated through several benchmark problems involving linear and nonlinear time-dependent operators. In all examples, we are able to recover accurate approximations of the latent solutions, and consistently propagate uncertainty, even in cases involving very long time integration.

Title and Abstracts:

Oct 11, 2023, 10:00 AM - 10:50 AM PT 

Dr.Rene Cabrera, University of Texas at Austin

Title: Smoothing estimates for a nonlocal diffusion equation.

Abstract: The Landau-Coulomb equation is regarded as one of the most studied equations in PDEs. It models the evolution of a particle distribution in the theory of collisional plasma.  At the present moment, it remains uncertain whether the Landau-Coulomb equation possesses a unique smooth solution for large times. With its diffusion term, this equation contains a reaction term that could rapidly transform ``nice" configurations into singularities. In this talk, we'll give a bit of history regarding well posedness of this equation, then present a ``new" result regarding the diffusion operator in the Landau-Coulomb equation that provides much stronger regularization effects than its linear counterpart, the Laplace operator. This is work in collaboration with Maria Pia Gualdani.

Oct 18, 2023, 10:00 AM - 10:50 AM PT 

Prof.Li Wang, University of Texas at Austin

Title: New perspectives on inverse problems: stochasticity and Monte Carlo method

Abstract: In this talk, I will introduce two new aspects of inverse problems formulated as PDE-constrained optimization. Firstly, while current approaches assume deterministic parameters, many real-world problems exhibit stochastic behavior. We present a novel approach that treats the PDE solver as a push-forward map to recover the full distribution of unknown random parameters. We introduce a gradient-flow equation to estimate the ground-truth parameter probability distribution. Secondly, as problem dimensions increase, Monte Carlo methods regain relevance. However, directly applying them to gradient-based PDE-constrained optimization poses challenges due to the product of forward and adjoint solutions involving Dirac deltas. We propose strategies to rescue Monte Carlo methods and make them compatible with gradient-based optimization.

Oct 25, 2023, 10:00 AM - 10:50 AM PT 

Dr.Shu Liu, University of California, Los Angeles

Title: Wasserstein gradient flows and Hamiltonian flows on the push-forward generative model

Abstract: In this talk, we introduce a scalable, sampling-friendly algorithm for computing Wasserstein gradient flows (e.g. Fokker-Planck equations, Aggregation equations, etc.) and Wasserstein Hamiltonian flows (e.g. classical Hamiltonian systems with random initial conditions) on the probability manifold by leveraging the push-forward generative models. By projecting the corresponding probability flows onto the parameter space of the generative model, we obtain finite-dimensional ordinary differential equations (ODEs), which can be directly computed by classical numerical integrations. The computed generative models can then efficiently generate samples from the probability flows via the pushforward maps. We discuss numerical analysis results that guarantee the accuracy of our algorithm. Some numerical examples will also be demonstrated to verify the effectiveness of the proposed method.

Nov 1, 2023, 10:00 AM - 10:50 AM PT 

Dr. Arkadz Kirshtein, Tufts University

Title:  Variational modeling of fluid in poroelastic medium

Abstract: In this talk I will discuss modeling fluid flow through a deformable porous medium. I will start from introducing a variational approach for fluids and elasticity in Lagrangian coordinates. Next I will discuss an existing approach based on Biot's consolidation model. Ultimately I will introduce a system derived using energetic variational approach and discuss numerical methods and simulations based on it.

Nov 8, 2023, 10:00 AM - 10:50 AM PT 

Dr. Artur Stephan, Weierstrass Institute

Title: Gradient systems and time-splitting methods

Abstract: Gradient systems provide an important and versatile modelling framework. Gradient flows describe an evolution in the direction of the steepest descent of a driving functional (often the energy or entropy), and enjoy many applications in continuum mechanics, semiconductor physics, and also chemical processes. Mathematically, gradient systems consist of a state space, a driving functional, and a geometric or dissipation mechanism. In my talk, we are interested in the situation where the dissipation is given by two parts. This splitting provides also a decomposition of the right-hand side of the gradient-flow evolution equation and thus enable to construct solutions via a split-step method. The main result is the convergence analysis of the time-splitting method. It relies on methods from the calculus of variations, and the usage of the energy-dissipation principle for gradient flows. The talk is based on joint work with Alexander Mielke (Berlin) and Riccarda Rossi (Brescia).

Nov 15, 2023, 10:00 AM - 10:50 AM PT 

Dr.Siming He, University of South Carolina

Title: On the chemotaxis-fluid system

Abstract: In this talk, we will present a coupled Patlak-Keller-Segel-Navier-Stokes (PKS-NS) system that models chemotaxis phenomena in the fluid. The system exhibits critical threshold phenomena. For example, if the total population of the cell density is less than $8\pi$, then the solutions exist globally in time. Moreover, finite-time blowup solutions exist if this population constraint is violated. We further show that globally regular solutions with arbitrary large cell populations exist. The primary blowup suppression mechanism is the shear flow mixing-induced enhanced dissipation phenomena. 

Nov 29, 2023, 10:00 AM - 10:50 AM PT 

Dr. Maziar Raissi (University of California, Riverside)

Title: Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations

Abstract: We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where: (1) all we observe are noisy data on black-box initial conditions, and (2) we are interested in quantifying the uncertainty associated with such noisy data in our solutions to time-dependent partial differential equations. Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors. This is an attempt to construct structured and data-efficient learning machines, which are explicitly informed by the underlying physics that possibly generated the observed data. The effectiveness of the proposed approach is demonstrated through several benchmark problems involving linear and nonlinear time-dependent operators. In all examples, we are able to recover accurate approximations of the latent solutions, and consistently propagate uncertainty, even in cases involving very long time integration.