Computational and Applied Mathematics Seminar
Mondays at 2:15-3:05p.m (ZOOM/WebEx or in-person (Room Carver 401) talks)
The CAM Seminar is organized in the ISU Mathematics Department. It brings speakers from inside and outside of ISU, raising issues and exchanging ideas on topics of current interest in the are of computational and applied mathematics.
- Apr. 10 (in-person, Carver 401)
Title: HPC at ISU: Using High Performance Computing Resources on Campus
Marina Kraeva, High Performance Computing, Iowa State University
Abstract: In this talk Dr. Marina Kraeva from the High Performance Computing (HPC) group will give an overview of HPC resources available at ISU and discuss best practices of using those.
- Apr. 17
Title: VarMiON: A variationally mimetic operator network
Deep Ray, University of Maryland, College Park
Abstract: Operator networks have emerged as promising deep learning tools for building fast surrogates of PDE solvers. These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE, i.e., they learn the solution operator of the PDE. In this talk, we consider a new type of operator network called VarMiON, that mimics the variational or weak formulation of PDEs. A precise error analysis of the VarMiON solution reveals that the approximation error contains contributions from the error in the training data, the training error, quadrature error in sampling input and output functions, and a “covering error” that measures how well the training dataset covers the space of input functions. Numerical experiments are presented for a canonical elliptic PDE to demonstrate the efficacy and robustness of the VarMiON as compared to a standard Deep Operator Network (DeepONet).
SHORT BIO: Deep Ray is an Assistant Professor of Mathematics at the University of Maryland, College Park. He obtained his PhD in Mathematics from the Tata Institute of Fundamental Research (Bangalore, India), followed by postdoctoral positions at EPFL (Switzerland), Rice University and University of Southern California. His research interests lie at the interface of conventional numerical analysis and machine learning. He has worked on the judicious integration of deep learning tools to overcome computational bottlenecks in areas such as fluid flow simulations, reduced order modeling, PDE constrained optimization, and Bayesian inference.
- Apr. 24
- Jan. 16
- Jan. 23
Title: Organizational Meeting
- Jan. 30
- Feb. 06 (2:30-3:10pm, Carver 401)
Title: Group Discussion
Abstract: Discussion on qualifying exams and graduate courses (Applied Math and Numerical Analysis).
- Feb. 13
Title: Spacetime Discontinuous Galerkin Methods for Hyperbolic Systems
Yifan Hu, Iowa State University
Abstract: This talk describes a family of high-order spacetime discontinuous Galerkin (DG) methods for hyperbolic systems. We formulate them in a prediction-correction fashion, where the prediction step is a regionally-implicit discontinuous Galerkin (RIDG) method producing reconstructed solutions in spacetime, and the correction step is an explicit update based on those reconstructed solutions. This method is equivalent to the Lax-Wendroff DG method (Gassner et al. 2011) when we limit the region to a single element, and it becomes the original RIDG method (Guthrey and Rossmanith 2019) when the region is a three-element stencil. We demonstrate the improved stability region for RIDG via linear stability analysis and show that the stable timestep size does not decay exponentially as dimensionality increases.
Further, we develop a novel hierarchical formulation of RIDG to reduce computation cost while the stability traits are mostly preserved. We show, by numerical simulations for advection equations and wave equations, that this new formulation is high-order accurate and presents better performance for systems with high dimensionality.
- Feb. 20
Title: A Machine Learning Study of the Wind-Driven Water Runback Characteristics Pertinent to Aircraft Icing Phenomena
Hui Hu, Iowa State University (Department of Aerospace Engineering)
Abstract: Aircraft icing is widely recognized as one of the most serious weather hazards to flight safety in cold weather. The behavior of unfrozen water on an accreting ice surface can directly and indirectly influence the shape of the resulting glaze ice accretion. The transport behavior of unfrozen water prior to freezing has a direct impact on the ice shape due to its effect of redistributing the impinging water mass. The local heat transfer is the controlling factor in wet surface glaze ice accretion, where there is sufficient impinging water such that ice accretion rate is limited by the ability to remove latent heat of fusion from the surface. The behavior of unfrozen surface water can also influence the ice accretion process through its impact on surface roughness, which modifies local convective heat transfer as well as the transport of surface water. The characteristics of the transient wind-driven runback water flow over airframe surfaces are highly dependent on the complicated multiphase interactions between incoming airflow, surface water film flow, and solid ice accreting airframe surface. For complicated multiphase flow predictions and inverse problems, machine learning-based methods provide an attractive alternative to standard theoretical and numerical methods. In the present study, advanced machine learning (ML) architectures are designed to reveal the dynamic interactions between complex multiphase systems (i.e., air, water, and aircraft surface) and to estimate the front contact line and full-field film thickness of wind-driven water runback flows over a test surface. The effectiveness of the proposed machine learning algorithms is shown by their high precision and cheap computation time. Then, a physics-informed neural network is developed to identify the spatial-temporal shear stress distribution at the water/air interface from the restricted knowledge of interface locations of experimental wind-driven runback water flows. The research findings highlight the ability of machine learning to extract additional vital physics information from experimental flow data in order to improve knowledge of the underlying physics behind the multiphase wind-driven runback water transport process.
- Feb. 27
Title: Controlling Regularized Conservation Laws via Entropy-Entropy Flux Pairs
Wuchen Li, University of South Carolina
Abstract: In this talk, we study a class of variational problems for regularized conservation laws with Lax's entropy-entropy flux pairs. We first introduce a modified optimal transport space based on conservation laws with diffusion. Using this space, we demonstrate that conservation laws with diffusion are "flux-gradient flows". We next construct variational problems for these flows, for which we derive dual PDE systems for regularized conservation laws. Several examples, including traffic flow, Burgers' equation, and barotropic compressible Navier–Stokes equations, are presented. Incorporating both primal-dual hybrid gradient algorithms and monotone schemes, we successfully compute the control of conservation laws. This is based on a joint work with Siting Liu and Stanley Osher.
- Mar. 06
Title: Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs
Chi-Wang Shu, Brown University
Abstract: In scientific and engineering computing, we encounter time-dependent
partial differential equations (PDEs) frequently. When designing high
order schemes for solving these time-dependent PDEs, we often first
develop semi-discrete schemes paying attention only to spatial
discretizations and leaving time $t$ continuous. It is then important
to have a high order time discretization to main the stability
properties of the semi-discrete schemes. In this talk we discuss several
classes of high order time discretization, including the strong stability
preserving (SSP) time discretization, which preserves strong stability from
a stable spatial discretization with Euler forward, the implicit-explicit
(IMEX) Runge-Kutta or multi-step time marching, which treats the more
stiff term (e.g. diffusion term in a convection-diffusion equation)
implicitly and the less stiff term (e.g. the convection term in such an
equation) explicitly, for which strong stability can be proved under the
condition that the time step is upper-bounded by a constant under
suitable conditions, the explicit-implicit-null (EIN) time marching,
which adds a linear highest derivative term to both sides of the PDE
and then uses IMEX time marching, and is particularly suitable for
high order PDEs with leading nonlinear terms, and the explicit
Runge-Kutta methods, for which strong stability can be proved in many cases
for semi-negative linear semi-discrete schemes. Numerical examples will be
given to demonstrate the performance of these schemes.
- Mar. 20
Title: Anderson acceleration of gradient methods with energy for optimization problems
Xuping Tian, Iowa State University
Abstract: Anderson acceleration (AA) as an efficient technique for speeding up the
convergence of fixed-point iterations may be designed for accelerating an optimization
method. We propose a novel optimization algorithm by adapting Anderson acceleration
to the energy adaptive gradient method (AEGD) [arXiv:2010.05109]. The feasibility of
our algorithm is examined in light of convergence results for AEGD, though it is not
a fixed-point iteration. We also quantify the accelerated convergence rate of AA for
gradient descent by a factor of the gain at each implementation of the Anderson mixing.
Our experimental results show that the proposed algorithm requires little tuning of
hyperparameters and exhibits superior fast convergence.
- Mar. 27
Title: Target Identification through Wave-Form Optimization
Mark Lyon, University of New Hampshire
Abstract: A computational study is presented in which optimized incident wave-forms are used in a machine learning based target identification algorithm that relies solely on the backscattered returns after the model is trained. Training data is quickly obtained using a recently developed fast wave equation solver, which will be discussed, to efficiently generate time-domain far-field backscattered returns from a set of target geometries at various incident directions. The optimization of the wave-form is integrated into the stochastic decent algorithm so that the neural network parameters and the incident wave-form parameters are optimized at the same time. One goal of the study was to determine if the optimal waveform for machine learning identification would differ from current wave-forms which were designed for human observation. We show that even the optimal frequency of range of the wave-form for machine learning identification changes with the quantity and quality of the training data.
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- Apr. 03 (CAM Seminar Carver 401)
Title: The mean-field limit of non-exchangeable integrate and fire systems
Pierre Jabin, Penn State University
Abstract: We investigate the mean-field limit of large networks of interacting biological neurons. The neurons are represented by the so-called integrate and fire models that follow the membrane potential of each neurons and captures individual spikes. However we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. We are able to extend the concept of extended graphons, introduced in Jabin-Poyato-Soler, by introducing a novel notion of discrete observables in the system. This is a joint work with D. Zhou.
- Apr. 04 (Mathematics-CAM Colloquium Carver 268)
Title: A new approach to the mean-field limit of Vlasov-Fokker-Planck equations
Pierre Jabin, Penn State University
Abstract: We introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension 2 together with a partial result in dimension 3. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted L p norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in L^2. This is a joint work with D. Bresch and J. Soler.