Time/Location: April 25th, 2:30-3:30 pm, LeConte 440

Speaker: Ying Liang, Duke University

Website: https://yingliang94.github.io/

Title: Theoretical and Computational Studies on Inverse Random Source Problems

Abstract: In the field of inverse problems, the estimation of an unknown source term from indirect observations is a fundamental challenge. Inverse source problems play a significant role in various fields, including antenna synthesis, seismology, and medical imaging. Stochastic inverse problems emerge when uncertainties within the model need to be considered, and the introduction of random sources further amplifies the complexity due to their inherent ill-posedness and uncertainties.

This talk covers both theoretical and computational investigations into the inverse random source problems. In our theoretical findings, we establish a unified framework for stability estimates of inverse source problems governed by the stochastic acoustic, biharmonic, and elastic wave equations driven by white noise. Our computational exploration focuses on a data-assisted approach that utilizes boundary measurement data to reconstruct the statistical properties of the random source. This method requires fewer realizations compared to traditional methods. We will also highlight potential directions for future research involving stochastic inverse scattering problems.

When: 2:30 pm - 3:30 pm, April 18th, 2025, LeConte 440

Speaker: Hanye Zhu, Duke University

Abstract: When two conducting or insulating inclusions are closely located in a composite, the gradient of the solution to the conductivity problem may become arbitrarily large in the narrow region in between them as the distance between the inclusions tends to zero. This phenomenon is known as field concentration, a central topic in the theory of composite material. We study the field concentration between two closely spaced perfect conductors with imperfect bonding interfaces of low conductivity type, where the potential is allowed to be discontinuous across the interfacial boundaries. The imperfect bonding interface condition holds significant practical relevance, as it approximates the membrane structure in biological systems. We discover a new dichotomy for the field concentration depending on the bonding parameter of the interfaces. The results are surprisingly different from the perfect interface setting. Based on joint work with Hongjie Dong and Zhuolun Yang.

When: 2:30-3:30 pm, April 4th,  LeConte 440

Speaker: Boyang Su, University of South California

Abstract: The existence of global solutions for the nonlinear Schrödinger equation (NLS) has been extensively studied. The problem becomes more complicated as we consider a general power-type nonlinearity of degree p. So far, global well-posedness for small data is known for p strictly greater than the Strauss exponent, as the dispersive effect becomes weaker for smaller p. In dimension 3, this Strauss exponent is 2, making 3D NLS with quadratic nonlinearity particularly interesting.

In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized initial data. We will review the space-time resonance method developed in previous works, which have all required 0-dimensional space-time resonance set. In applying the space-time resonance method to a general system of quadratic NLS, there are two cases when the space-time resonance sets are 3-dimensional, which cannot be handled by existing method. I will explain how we fully resolved one case and tackled the other case, which arises from the $u\Bar{u}$-type nonlinearity, by introducing an $\epsilon$ regularization to the lower frequency part of the nonlinear term.

When:  2:30 pm -- 3:30 pm, March 28th,  LeConte College 440,

Speaker: Eric C. Cyr, Sandia National Lab

Title: Exploiting time-domain parallelism to accelerate neural network training and PDE constrained optimization

Abstract: This talk will explore methods for accelerating numerical optimization constrained by transient problems using parallelism. Two types of transient problems will be considered. In the first case training algorithms for Neural ODEs will be discussed. Neural ODEs are a class of neural network architecture where the depth of the neural network (the layers) is modeled as a continuous time domain. For the second case, transient PDE-constrained optimization problems will be described. In either case, simulation-based optimization requires repeated executions of the simulator’s forward and backward (adjoint) time integration schemes. Consequently, the arrow of time creates a major sequential bottleneck in the optimization process. Second, for performance these methods rely strongly on the available parallelization for the forward and adjoint solves. Thus, when forward and adjoint solvers are already operating at the limit of strong scaling and hardware utilization, the arrow-of-time bottleneck cannot be overcome by additional parallelization across the spatial grid or network layers.

In the first half of this talk we consider deep neural network models. Deep neural networks are a powerful machine learning tool with the capacity to‚ learn complex nonlinear relationships described by large data sets. Despite their success training these models remains a challenging and computationally intensive undertaking. We will present a layer-parallel training algorithm that exploits a multigrid scheme to accelerate both forward and backward propagation. Introducing a parallel decomposition between layers requires inexact propagation of the neural network. The multigrid method used in this approach stitches these subdomains together with sufficient accuracy to ensure rapid convergence. We demonstrate an order of magnitude wall-clock time speedup over the serial approach, opening a new avenue for parallelism that is complementary to existing approaches. We also discuss applying the layer-parallel methodology to recurrent neural networks. We study the generalized recurrent unit (GRU) architecture. We demonstrate its relation to a simple ODE formulation that facilitates application of the layer-parallel approach. Results are demonstrating performance improvements on a human activity recognition (HAR) data set are presented.

The second half of this talk focuses on PDE-constrained optimization formulations.

Solving optimization problems with transient PDE-constraints is computationally costly due to the number of nonlinear iterations and the cost of solving large-scale KKT matrices. These matrices scale with the size of the spatial discretization times the number of time steps. We propose a new 2-level domain decomposition preconditioner to solve these linear systems when constrained by the heat equation. Our approach leverages the observation that the Schur-complement is elliptic in time, and thus amenable to classical domain decomposition methods. Further, the application of the preconditioner uses existing time integration routines to facilitate implementation and maximize software reuse. The performance of the preconditioner is examined in an empirical study demonstrating the approach is scalable with respect to the number of time steps and subdomains.

When: 2:30pm, March 7th, LeConte College 440

Speaker: Khai T. Nguyen, Department of Mathematics, North Carolina State University

Title: Scalar balance laws with nonlocal singular sources

Abstract: In this presentation, I will establish the global existence of entropy weak solutions for scalar balance laws with nonlocal singular sources, along with a partial uniqueness result. A detailed description of the solution is provided for a general class of initial data in a neighborhood where two shocks interact. Additionally, some open questions will be discussed.

When: Feb. 28th, 2:30 pm, LeConte 440,

Speaker: Dehua Wang, University of Pittsburgh

Title: Euler equations and transonic flows

Abstract: In this talk, we will consider the Euler equations of gasdynamics and applications in transonic flows. First, the basic theoryof Euler equations will be reviewed. Then we will present the resultson the transonic flows past obstacles, transonic flows in the fluiddynamic formulation of isometric embeddings, and the transonic flowsin nozzles. We will discuss global solutions and stability obtainedthrough various techniques and approaches.

When:  November 22, 2024, from 2:30 – 3:30 p.m.

Where: LeConte 440

Speaker: Xuelong Gu

Abstract: The sine-Gordon and Allen-Cahn equations are representative models in the fields of conservative Hamiltonian systems and dissipative gradient flows, respectively. As the demand for numerical methods that respect intrinsic energy conservation/dissipation laws turns into a fundamental principle, the subsequent computational efficiency is getting more and more desired. Linearly implicit methods, which only require solutions of linear algebraic systems at each time step, have become a popular way to design efficient schemes. However, the overall computational cost of these methods depends heavily on the speed at which the linear system can be solved. The Saul’yev method is a highly efficient numerical approach for pointwise solutions. We have demonstrated that it possesses the discrete gradient property and have used it to develop structure-preserving algorithms, exemplified by the sine-Gordon and Allen-Cahn equations. Our approach is fully implicit at first glance, but actually it is completely decoupled point by point, allowing for the implementation of a scalar nonlinear equation successively with a lower complexity that is comparable only to the degrees of freedom. Moreover, we establish connections between this method and the classic Itoh–Abe discrete gradient and partitioned averaged vector field methods. Based on these insights, we propose a generalized Itoh–Abe discrete gradient method, which provides significant flexibility in determining the order of updates for each unknown. One notable advantage is the ability to design update orderings that enable natural parallelization of the resulting scheme, thereby enhancing computational efficiency. Various numerical experiments are presented to illustrate the performance of the proposed schemes.

When: November 15, 2024, from 2:30 – 3:30 p.m.

Where: LeConte 440

Speaker: Li Xiantao (Penn State University)

Abstract: Quantum computing has recently emerged as a potential tool for large-scale scientific computing. In sharp contrast to their classical counterparts, quantum computers use qubits that can exist in superposition, potentially offering exponential speedup for many computational problems.  Current quantum devices are noisy and error-prone, and in near term, a hybrid approach is more appropriate. I will discuss this hybrid framework using three examples: quantum machine learning, quantum algorithms for density-functional theory and quantum optimal control. In particular, this talk will outline how quantum algorithms can be interfaced with a classical method, the convergence properties and the overall complexity.

When: November 8, 2024, from 2:30 – 3:30 p.m.

Where: LeConte 440

Speaker: Yuming Zhang (Auburn University)

Abstract: In tumor growth models, two primary approaches are commonly used. The first, described by Porous Medium type equations, models the tumor cells as a distribution evolving over space. The second, based on Hele-Shaw type flows, focuses on the evolution of the domain occupied by the cells. These two models are connected through the incompressible limit. In this talk, I will discuss the uniform strict propagation of the free boundaries and their convergence in the incompressible limit. As an outcome, we provide an upper bound on the Hausdorff dimension of free boundaries and show that the limiting free boundary has finite \((d-1)\)-dimensional Hausdorff measure. This is joint work with Jiajun Tong.

When: November 1, 2024, from 2:30 – 3:30 p.m.

Where: LeConte 440

Speaker: Yupei Huang (Duke University)

Abstract: Classification of the steady states for the 2D Euler equation is a classical topic in fluid mechanics. In this talk, we consider the rigidity of the analytic steady states in bounded, simply connected domains. By studying an over-determined elliptic problem in Serrin type, we show the stream functions for the steady state are either radial functions or solutions to semi-linear elliptic equations.  This is the joint work with Tarek Elgindi, Ayman Said, and Chunjing Xie.

When: October 25, 2024, from 2:30 – 3:30 p.m.

Where: LeConte 440

Speaker: Mitchel J. Colebank (University of South Carolina)

Abstract: Mathematical modeling of the cardiovascular system is now recognized as a tool for disease management and is prominent in the development of physiological digital twins. A common model is the one-dimensional (1D) pulse wave propagation model, which is a hyperbolic system of PDEs derived from Naiver-Stokes in cylindrical coordinates with only one spatial coordinate. These models then need to be personalized, requiring the solution of a limited-data, noisy inverse problem. In this talk, we will introduce the 1D system and its application in computational blood flow modeling. We will discuss how emulation techniques (e.g., polynomial chaos expansions and Gaussian processes) can help speed up computation time for both forward and inverse problems. We will then see the application of this speed up in determining parameter identifiability using a robust method called the profile-likelihood, and show how emulation is necessary for near real time predictions as part of the future of digital twin technologies.

When: October 11, 2024, from 2:30 – 3:30 p.m.

Where: LeConte 440

Speaker: Xiaoqian Xu (Duke Kunshan University)

Abstract: In the study of incompressible fluid, one fundamental phenomenon that arises in a wide variety of applications is dissipation enhancement by so-called mixing flow. In this talk, I will give a brief introduction to the idea of mixing flow and the role it plays in the field of advection-diffusion-reaction equation. I will also discuss the examples of such flows in this talk.

When: August 30, 2024, from 3:40 – 4:30 p.m.

Where: LeConte 440

Speaker: Viktor Stein (TU Berlin)

Abstract: We give a comprehensive description of Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals \(F_\nu := \text{MMD}_K^2(\cdot, \nu)\) towards given target measures \(\nu\) on the real line, where we focus on the negative distance kernel \(K(x,y) := -|x-y|\). In one dimension, the Wasserstein-2 space can be isometrically embedded into the cone \(C(0,1) \subset L_2(0,1)\) of quantile functions leading to a characterization of Wasserstein gradient flows via the solution of an associated Cauchy problem on \(L_2(0,1)\). Based on the construction of an appropriate counterpart of \(F_\nu\) on \(L_2(0,1)\) and its subdifferential, we provide a solution of the Cauchy problem. For discrete target measures \(\nu\), this results in a piecewise linear solution formula. We prove invariance and smoothing properties of the flow on subsets of \(C(0,1)\). For certain \(F_\nu\)-flows this implies that initial point measures instantly become absolutely continuous, and stay so over time. Finally, we illustrate the behavior of the flow by various numerical examples using an implicit Euler scheme and demonstrate differences to the explicit Euler scheme, which is easier to compute, but comes with limited convergence guarantees. This is joint work with Richard Duong (TU Berlin), Robert Beinert (TU Berlin), Johannes Hertrich (UCL) and Gabriele Steidl (TU Berlin).

Joint Seminar with the RTG Seminars on Data Science

Organized by: Changhui Tan (tan@math.sc.edu) & Siming He (siming@mailbox.sc.edu)

Unless otherwise noted, the seminar will be held on Fridays from 2:30pm to 3:30pm in LeConte 440.

This page will be update as new seminar information becomes available. Check back often for the most up to date information!