The Ohio State University Computational Math Seminar

Thursday at 1:50-2:45 PM (unless otherwise noted)

For questions, contact Dr. Maria Han Veiga, Dr. Yulong Xing or Dr. Dongbin Xiu, Email: hanveiga dot 1@osu.edu, xing dot 205@osu.edu or xiu dot 16@osu.edu



Year 2023-2024 Schedule

DATE and TIME  Location  SPEAKER  TITLE 
August 31 
Thursday, 1:50pm 
In person 
Math Tower 154
François Ged  
(EPFL) 
Matryoshka policy gradient for max-entropy reinforcement learning 
November 16  
Thursday, 1:50pm 
In person 
Math Tower 154
Johnathan Weare 
(NYU) 
Iterative random sparsification and convergence of a fast randomized Jacobi iteration 
January 11 
Thursday, 1:50pm 
In person 
University Hall 0090
Steven Wise  
(Univ of Tennessee) 
A non-isothermal phase field crystal model with lattice expansion 
February 29  
Thursday, 1:50pm 
In person 
Math Tower 100A
Eirik Endeve 
(ORNL) 
Application of Euler equations to model astrophysical flows 
March 22 
Friday, 4:00pm 
In person 
Math Tower 154
Qingguo Hong  
(Missouri Univ. of Science and Technology) 
On the Activation Function Dependence of the Spectral Bias of Neural Networks 
April 4  
Thursday, 10:20am 
In person 
TBD
Mrinal Kumar 
(OSU) 
TBD 
April 12  
Friday, 4:10pm 
In person 
TBD
Deep Ray 
(Univ of Maryland) 
Learning WENO for entropy stable schemes to solve conservation laws  


Abstracts:

August 31
François Ged
Title: Matryoshka policy gradient for max-entropy reinforcement learning

Reinforcement Learning (RL) is the area of Machine Learning addressing tasks where an agent interacts with its environment through a sequence of actions, chosen according to its policy. The agent’s goal is to maximize the rewards collected along the way, and in this talk, entropy bonuses are added to the rewards. This regularization technique has become more common, with benefits such as: enhancement of the exploration of the environment, uniqueness and stochasticity of the optimal policy, and more robustness of the agent to adversarial modifications of the rewards. Policy gradient algorithms are well suited to deal with large (possibly infinite) state and action spaces but theoretical guarantees have been lacking or obtained in rather specific settings. The case of infinite (continuous) state and action spaces remains mostly unsolved. In this talk, I will present a novel algorithm called Matryoshka Policy Gradient (MPG) that is both very intuitive and mathematically tractable. It uses so-called softmax policies and relies on the following idea: by fixing in advance a maximal horizon N, the agent with MPG learns to optimize policies for all smaller horizons simultaneously, that is from 1 to N, in a nested way (recalling the image of Matryoshka dolls). Theoretically, under mild assumptions, our most important results can be summarized as follows: 1. training converges to the unique optimum when the optimum belongs to the parametric space; 2. training converges to an explicit orthogonal projection of the unique optimum when it does not belong to the parametric space, this projection being optimal within that space; 3. for policies parametrized by a neural network, we provide a simple sufficient criterion at convergence for the global optimality of the limit, in terms of the neural tangent kernel of the neural network. Most notably, these convergence guarantees hold for infinite continuous state and action spaces. Numerically, we confirm the potential of our algorithm by successfully training an agent on two basic standard benchmarks from Open AI Gym, namely, frozen lake and cart pole. No background in RL is needed to understand the talk. Based on joint work with Prof. Maria Han Veiga.

November 16
Jonathan Q Weare
Title: Iterative random sparsification and convergence of a fast randomized Jacobi iteration

The traditional methods of numerical linear algebra are prohibitively expensive for high-dimensional problems for which even a single matrix multiplication by a dense vector may be too costly. In this talk I will discuss a general framework for reducing the cost of classical iterative numerical linear algebra schemes by randomly sparsifying the approximate solution at each iteration. In the specific case of Jacobi iteration I will provide a thorough characterization of the randomized scheme's error properties. The talk is based on joint work with Tim Berkelbach, Sam Greene, Lek-Heng Lim, James Smith, and Rob Webber.

January 11
Steven Wise
Title: A Non-Isothermal Phase Field Crystal Model with Lattice Expansion

The phase field crystal modeling framework describes materials at atomic space scales on diffusive time scales. It has been used to study grain growth, fracture, crystallization, and other phenomena. In this talk I will describe some recent work with collaborators developing a thermo- dynamically consistent phase field crystal model that includes heat transport and lattice expansion and contraction. We use the theory of non-equilibrium thermodynamics, a formalism developed by Alt and Pawlow, and Onsager’s principle to give consistent laws of entropy production, and mass and energy conservation. I will show some preliminary numerical simulation results involving heat transport during solidification, and I will discuss some ideas on developing entropy and energy stable numerical methods.

February 29
Eirik Endeve
Title: Application of Euler Equations to Model Astrophysical Flows

The Euler equations are a key component of multi-physics models of many astrophysical systems, including core-collapse supernovae and binary neutron star mergers. While the Euler equations alone do not provide a realistic description of these systems, they can sometimes be used to study some of their aspects that are intractable with full-physics models. The study of the so-called standing accretion shock instability (SASI), which operates in a stalled supernova shock wave, and was discovered using idealized models based on the Euler equations, is a prime example. In this talk, I will first briefly discuss the Euler equations and a discontinuous Galerkin method to solve them numerically. Then I will discuss the application of the Euler equations to model the SASI. I will conclude by showing results from a recent study (Dunham et al., arXiv:2307.10904) comparing simulations using relativistic and non-relativistic implementations of the Euler equations.

March 22
Qingguo Hong
Title: On the Activation Function Dependence of the Spectral Bias of Neural Networks

Neural networks are universal function approximators which are known to generalize well despite being dramatically overparameterized. We study this phenomenon from the point of view of the spectral bias of neural networks. We provide a theoretical explanation for the spectral bias of ReLU neural networks by leveraging connections with the theory of finite element methods. Based upon this theory we predict that switching the activation function to a piecewise linear B-spline, namely the Hat function, will remove this spectral bias, which we verify empirically in a variety of settings. Our empirical studies also show that neural networks with the Hat activation function are trained significantly faster using stochastic gradient descent and ADAM. Combined with previous work showing that the Hat activation function also improves generalization accuracy on image classification tasks, this indicates that using the Hat activation provides significant advantages over the ReLU on certain problems.

March 28
Mrinal Kumar
Title: TBD

April 12
Deep Ray
Title: Learning WENO for entropy stable schemes to solve conservation laws

Entropy stable solvers for hyperbolic conservation laws ensure the selection of a physically relevant (weak) solution of the underlying PDE. Among such methods, the TeCNO schemes [Fjordholm et al, 2012] form a class of high-order finite difference-based solvers that utilize reconstruction algorithms satisfying a critical “sign-property” at the cell-interfaces. However, only a handful of existing reconstructions are known to satisfy this property. In [Fjordholm-Ray, 2016], the first weighted essentially non-oscillatory (WENO) reconstruction satisfying the sign- property was developed. However, despite leading to provably entropy stable schemes, the numerical solutions using this reconstruction suffered from large under/overshoots near discontinuities.

In this talk, we propose an alternate approach to constructing WENO schemes possessing the sign-property. In particular, we train a neural network to determine the polynomial weights of the WENO scheme, while strongly constraining the network to satisfy the sign-property. The training data comprises smooth and discontinuous data that represent the local solution features of conservation laws. Additional constraints are built into the network to guarantee the expected order of convergence (for smooth solutions) with mesh refinement. We present several numerical results to demonstrate a significant improvement over the existing variants of WENO with the sign-property.


Year 2022-2023 Seminar

Year 2021-2022 Seminar

Year 2019-2020 Seminar


Does this page seem familiar?