UCSB Applied Math/PDE/Data Science Seminar

Abstract: Motivated by non-equilibrium thermodynamics, the framework of the energetic variational approach (EnVarA) provides a paradigm for building thermodynamically consistent variational models for many complicated systems in soft matter physics, material science, biology, and machine learning. In this talk, we'll present a numerical framework for developing structure-preserving variational discretizations for these variational models based on their energetic variational forms. The numerical approach starts with the energy-dissipation law, which describes all the physics and the assumptions in each system and can combine distinct types of spatial discretizations, including Eulerian, Lagrangian, particle, and neural-network-based discretizations. The resulting semi-discrete equation inherits the variational structures from the continuous energy-dissipation law. The numerical procedure guarantees the developed scheme is energy stable and preserves the intrinsic physical constraints, such as the conservation of mass and the maximum principle. We'll discuss several applications of this numerical approach, including variational Lagrangian schemes for phase-field models and generalized diffusions, and particle-based energetic variational inference for machine learning. The talk is mainly based on several joint works with Prof. Chun Liu (IIT) and Prof. Lulu Kang (IIT). 

Host person: Paul Atzberger 

Abstract: Inference by means of mathematical modeling from a collection of observations remains a crucial tool for scientific discovery and is ubiquitous in application areas such as signal compression, imaging restoration, and supervised machine learning. With ever-increasing model complexities and growing data size, new specially designed methods are urgently needed to recover meaningful quantities of interest. This work describes a new approach that uses deep neural networks (DNN) to obtain regularization parameters for solving inverse problems. We consider a supervised learning approach, where a network is trained to approximate the mapping from observation data to regularization parameters. Once the network is trained, regularization parameters for newly obtained data can be computed by efficient forward propagation of the DNN. We show that various regularization functionals, forward, and noise models may be considered. Further, we provide a new perspective on sparsity-induced regularized inverse problems by exploring the generalized lasso problem through variable projection methods. 

Host person: Paul Atzberger 

Abstract: We first formulate the transition path problem for Markov jump processes as a stochastic optimal control problem in an infinite time horizon. Using the Girsanov transformation for pure jump process on general standard Borel space, we choose the certain relative-entropy type running cost and a terminal cost for the stochastic optimal control problem with a stopping time. We prove the discrete committor function gives an optimal control which guides the transition between local minimums efficiently. Moreover, disintegration formula puts both finite time and infinite time optimal control into one framework, which are convex optimization problem for measures.  

Host person: Sui Tang

Abstract: In decision-making applications where multiple forward simulations are needed, such as parameter study, design optimization, optimal control, uncertainty quantification, and inverse problems, we need to iteratively solve forward problems. However, subject to the model complexity and the fineness of the discretization, the computational cost of forward simulations can be high. It may take a long time to complete a single forward simulation with the available computing resource. In this talk, we will introduce various reduced order modeling techniques, which aim to lower the computational complexity and maintain a good accuracy, including projection-based intrusive nonlinear model reduction and non-intrusive model reduction approaches. We will demonstrate the implementation of these reduced order modeling techniques in libROM (www.librom.net) and its application to numerical solvers for solving various physics problems.

Host person: Paul Atzberger 

Abstract: In this work, we consider a novel inverse problem in mean-field games (MFG). We aim to recover the MFG model parameters that govern the underlying interactions among the population based on a limited set of noisy partial observations of the population dynamics under the limited aperture. Due to its severe ill-posedness, obtaining a good quality reconstruction is very difficult. Nonetheless, it is vital to recover the model parameters stably and efficiently in order to uncover the underlying causes for population dynamics for practical needs. Our work focuses on the simultaneous recovery of running cost and interaction energy in the MFG equations from a finite number of boundary measurements of population profile and boundary movement. To achieve this goal, we formalize the inverse problem as a constrained optimization problem of a least squares residual functional under suitable norms with L1 regularization. We then develop a fast and robust operator splitting algorithm to solve the optimization using techniques including harmonic extensions, three-operator splitting scheme, and primal-dual hybrid gradient method. Numerical experiments illustrate the effectiveness and robustness of the algorithm. A future direction will be to develop a technique for algorithmic speedup for inverse problems in higher dimensions with the help of bilevel optimization, machine learning techniques and neural network architecture. This is a joint work with Samy W. Fung (Colorado School of Mines), Siting Liu (UCLA), Levon Nurbekyan (Emory University), and Stanley J. Osher (UCLA).

Host person: Paul Atzberger 

Abstract: We consider stochastic three-dimensional Navier-Stokes equations + Waves. Regularity results are established by bootstrapping from global regularity of the averaged stochastic resonant equations and convergence theorems. The averaged covariance operator couples stochastic and wave effects. The regularization time horizon is long. Infinite time regularity is proven for the deterministic case. Regularization is the consequence of precise mechanisms of relevant three-dimensional nonlinear interactions. We establish multi-scale stochastic averaging, convergence and regularity theorems in a general framework. We also present theoretical and computational results for three-dimensional nonlinear dynamics. 

Host person: Bjorn Birnir 

Abstract:  Cell polarization, in which a uniform distribution of substances becomes asymmetric due to internal or external stimuli, is a fundamental process underlying cell mobility and cell division. Budding yeast provides a good system to study how biochemical signals and mechanical properties coordinate with each other to achieve stable cell polarization and give rise to certain morphological change in a single cell. Recent experimental data suggests yeast budding develops into two trajectories with different bud shapes as mother cells become old. We first developed a 2D model to simulate biochemical signals on a shape-changing cell and investigated strategies for robust yeast mating. Then we extended and coupled this biochemical signaling model with a 3D subcellular element model to take into account cell mechanics, which was applied to investigate how the interaction between biochemical signals and mechanical properties affects the cell polarization and budding initiation. This 3D mechanochemical model was also applied to predict mechanisms underlying different bud shape formation due to cellular aging. 

Host person: Paul Atzberger  

Abstract:  This talk concerns an equation of Fisher-KPP type with a Keller-Segel chemotaxis term. The goal is to determine how propagation properties of solutions are affected by strong repelling chemotaxis. To this end, we study traveling wave solutions. We provide an almost complete picture of the asymptotic dependence of the traveling wave speed on parameters representing the strength and length-scale of chemotaxis. Our study is based on the convergence, in certain asymptotic regimes, to traveling waves of the porous medium Fisher-KPP equation and to those of a hyperbolic Fisher-KPP-Keller-Segel equation. The talk is based on joint work with C. Henderson and Q. Griette. 

Host person: Katy Craig 

Abstract: We consider an electrodiffusion model describing the nonlinear and nonlocal evolution of several ionic concentrations in a two-dimensional incompressible fluid perturbed by an additive stochastic noise. We address the global well-posedness, long-time behavior, and ergodicity of the Markov transition kernels associated with the model. This is based on joint work with Ruimeng Hu and Quyuan Lin. 

Host person: Ruimeng Hu

Abstract: Optimal feedback synthesis for nonlinear dynamics -a fundamental problem in optimal control- is enabled by solving fully nonlinear Hamilton-Jacobi-Bellman type PDEs arising in dynamic programming. While our theoretical understanding of dynamic programming and HJB PDEs has seen a remarkable development over the last decades, the numerical approximation of HJB-based feedback laws has remained largely an open problem due to the curse of dimensionality. More precisely, the associated HJB PDE must be solved over the state space of the dynamics, which is extremely high-dimensional in applications such as distributed parameter systems or agent-based models. In this talk we will review recent approaches regarding the effective numerical approximation of very high-dimensional HJB PDEs. We will explore modern scientific computing methods based on tensor decompositions of the value function of the control problem, and the construction of data-driven schemes in supervised, and semi-supervised learning environments. We will highlight some novel research directions at the intersection of control theory, scientific computing, and statistical machine learning. 

Host person: Sui-Tang