SIAM Central States Section(CSS) Computational and Applied Mathematics Forum

[Ying Wang]

Dear colleagues,

We will start a new online (via Zoom) research seminar called SIAM Central States Section(CSS) Computational and Applied Mathematics Forum. These talks are part of the SIAM CSS activities, and will be addressed to a general audience.

The forum time is Wednesday 3:30-4:30pm CST, with an alternative time Friday 3:30-4:30pm CST. 

The schedule of the forum can be found at https://math.ou.edu/~wang/SIAM_CSS_CAF.html. 

The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

You are invited to attend the forum. Please help to spread the word among interested colleagues, postdocs and students.

Dr. Chi-Wang Shu will deliver the inauguration lecture on Friday 2/24 3:30-4:30 CST. The title and abstract of his talk is given at the end of this message.

Best regards,
Ying Wang (wa...@ou.edu)
Profess of Mathematics
University of Oklahoma




              Stability of time discretizations for
       semi-discrete high order schemes for time-dependent PDEs

                      Chi-Wang Shu
             Division of Applied Mathematics
                    Brown University
               Providence, RI 02912, USA

    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.  

[Ying Wang]

Dear colleagues,

This week's SIAM CSS Computational and Applied Mathematics Forum speaker is Dr. Eitan Tadmor. The title and abstract of his talk is given at the end of this message. 

Time: Friday 3/3 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html. 

The Zoom link for the forum is 
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463) 

Best regards,
Ying Wang (wa...@ou.edu)
Profess of Mathematics
University of Oklahoma

Swarm-Based Gradient Descent Method for Non-Convex Optimization
Eitan Tadmor
University of Maryland
 
We introduce a new swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents, identified with positions x and masses m.  There are three key aspects to the SBGD dynamics: (i) transition of mass from high to lower ground; (ii) a random choice of marching direction aligned with the orientation of the steepest gradient descent; and (iii) a time stepping protocol, h(x,m), which decreases with m.
The interplay between positions and masses leads to dynamic distinction between `leaders’ and `explorers’. Heavier agents lead the swarm near local minima with small time steps. Lighter agents explore the landscape in random directions with large time steps,  and lead to improve position, i.e., reduce the ‘loss’ for the swarm. Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer. 

[Ying Wang]

Dear SIAM CSS colleagues,

This week's SIAM CSS Computational and Applied Mathematics Forum speaker is Dr. Zhihong Jeff Xia. The title and abstract of his talk is given at the end of this message. 

Time: Friday 3/10 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html. 

The Zoom link for the forum is 
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463) 

Best regards,
Ying Wang (wa...@ou.edu)
Profess of Mathematics
University of Oklahoma

A reformulated KAM theorem and its applications
Zhihong Jeff Xia
Northwestern University

The celebrated KAM (Kolmogorov-Arnold-Moser) Theorem shows stability for majority of trajectories for near-integrable Hamiltonian systems. We present a simple reformulated KAM theorem on existence of invariant tori. This new formulation uses parametrization to overcome difficulties arising from degeneracies. We will show some applications of this new formulation, particularly to systems with degeneracies, and we also show simple proofs to some classical results.

This talk is aimed at a general audience, no technical background on KAM is required.

[Ying Wang]

Dear SIAM CSS colleagues,

This week's speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. Hong Qian from University of Washington. The title and abstract of his talk is given at the end of this message.

Time: Wednesday 3/22 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Ying Wang (wa...@ou.edu)
Profess of Mathematics
University of Oklahoma

Toward a New Mathematical Foundation of Statistical Thermodynamics
Hong Qian
University of Washington

Abstract:  No background in physics is needed for this talk.  For recurrent dynamical systems with an invariant measure and repeated measurements {\it ad infinitum}, an entropy function and it’s maximization are emergent phenomena. In the theory of probability, this is known as Sanov’s large deviation theory and its contraction. Convex optimization leads to a thermodynamic like manifold and the concept of “ensembles”. Legendre transform follows Lagrange duality; Fenchel-Young inequality provides a brand new notion of “equilibrium”. We argue the concept of energy is the “information” hidden in counting frequency data; and a statistical analysis naturally gives rise to a macro-micro dichotomy.

[Ying Wang]

Dear SIAM CSS colleagues,

This week's speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. Leo Rebholz from Clemson University. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 3/29 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Ying Wang (wa...@ou.edu)
Profess of Mathematics
University of Oklahoma

On behalf of the organizing committee
Alexander Grigo (University of Oklahoma)
Ying Wang (University of Oklahoma)



Anderson acceleration for nonlinear solvers
Leo Rebholz
Clemson University

Abstract:
Anderson acceleration (AA) is an extrapolation technique developed in 1965 that recombines the most recent iterates and update steps in a fixed point iteration to improve the convergence properties of the sequence. Despite being successfully used for many years to improve nonlinear solver behavior on a wide variety of problems, a theory that explains the often-observed accelerated convergence was lacking. In this talk, we give an introduction to AA, then present a proof of AA convergence which shows how it improves the linear convergence rate based on a gain factor of an underlying optimization problem, but also introduces higher order terms in the residual error bound. We then discuss improvements to AA based on our convergence theory, show numerical results for the algorithms applied to several application problems including Navier-Stokes, Boussinesq, and nonlinear Helmholtz systems, and also the effect of AA on superlinear and sublinear converging iterations.

[Ying Wang]

Dear SIAM CSS colleagues,

Next week's speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. Xiaobing Feng from the University of Tennessee. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 4/5 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,

Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma


A New Theory of Fractional Calculus and Fractional Sobolev Spaces and Applications
Xiaobing H. Feng
The University of Tennessee


Abstract:
In this talk, I shall first discuss a newly developed theory of weak fractional (differential) calculus and fractional Sobolev spaces. The focus is the introduction of a weak fractional derivative concept which is a natural generalization of integer order weak derivatives and helps to unify multiple existing fractional derivative concepts. Based on the weak fractional derivative concept, new fractional-order Sobolev spaces can be naturally defined and many essential properties of those Sobolev spaces can also be established. I shall then introduce a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations.   It leads to new fractional differential equations, including one-side fractional Laplace operators and future value problems. Finally, if time permits, I shall also briefly introduce some new finite element (and DG) methods for approximating the weak fractional derivatives and the solutions of fractional calculus of variations problems and their associated fractional differential equations.

[Ying Wang]

Dear SIAM CSS colleagues,

Today, the speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. Chun Liu from Illinois Institute of Technology. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 4/26 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Active Fluids and Applications: An Energetic Variational Approach
Chun Liu
Illinois Institute of Technology

Abstract:
I will present a general theory for active fluids which convert chemical energy into various type of mechanical energy. This is the extension of the classical energetic variational approaches for mechanical systems. The methods will cover a range of both chemical reaction kenetics and mechanical processes. This is a joint project with many collaborators, in particular, Bob Eisenberg, Yiwei Wang and Tengfei Zhang.

[Ying Wang]

Dear SIAM CSS colleagues,

Next week's speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. Chiu-Yen Kao from Claremont Mckenna College. The title and abstract of her talk are given at the end of this message.

Time: Wednesday 5/17 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Geometric Optimization Involving Partial Differential Equations
Chiu-Yen Kao
Claremont Mckenna College


Abstract:  
Optimal geometric design for energy functionals, which depends on solutions to partial differential equations, provides a vast number of interesting and challenging mathematical problems. Numerical approaches for these kinds of problems require both forward solvers and optimization solvers. The forward solvers are numerical approaches, such as finite element methods, boundary integral methods, method of fundamental solutions, and spectral methods, to solve problems on a given domain efficiently and accurately. The optimization solvers aim to find the optimal geometric design which maximizes the energy functional. With a given initial choice of the coefficients, the domain, or the metrics, an iterative scheme which combines both forward and optimization solvers is then used to generate a sequence of the coefficients, domains, or metrics, to approach the optimizer of the energy functional. In this talk, we will discuss the recent progress in optimal geometric design of Laplace Beltrami eigenvalue problems and Steklov problems. 

[Ying Wang]

Dear SIAM CSS colleagues,

For the week of May 24th, the speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. James Nagy from Emory University. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 5/24 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Numerical Methods for Large-Scale Ill-Conditioned Linear Systems
James Nagy
Emory University


Abstract: Inverse problems arise in a variety of applications: image processing, finance, mathematical biology, machine learning, and more. Mathematical models for these applications may involve integral equations, partial differential equations, and dynamical systems, and solution schemes are formulated by applying algorithms that incorporate regularization techniques and/or statistical approaches. In most cases these solution schemes involve the need to solve a large-scale ill-conditioned linear system that is corrupted by noise and other errors. This talk will begin with a brief introduction to inverse problems and describe a few applications. We will then discuss considerations that are needed to compute an approximate solution, and describe some details about new efficient iterative solvers that can exploit efficiency of modern computer architectures. 

[Ying Wang]

Dear SIAM CSS colleagues,

This week, the speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. Albert Chern from UCSD. The title and abstract of his talk are given at the end of this message.

This is a joint event with Claremont McKenna College
Time: Wednesday 6/9 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Geometric Measure Theory for Convexifying and Compactifying Computational Problems
Albert Chern
University of California San Diego


Abstract: This talk consists of two geometric techniques for computational problems, both motivated by Geometric Measure Theory. They are applied to geometric optimizations and solving partial differential equations (PDEs) on unbounded domains, which are fundamental in shape synthesis and physical simulations. Shape optimization problems often involve non-convex geometric functionals such as surface area. Directly minimizing these energies can lead to stuck at local minima, and initializing a correct surface topology is challenging. Geometric measure theory provides new ways of approaching geometric optimization problems. Classical curves and surfaces are generalized to differential forms representing superpositions of infinitely many curves and surfaces. Under such a representation, the minimal surface problem becomes convex, and standard convex optimization techniques apply.

The second part of the talk focuses on simulations aided by Kelvin transformation. Many physical simulation problems take place in an unbounded space, requiring solving PDEs on a non-compact domain. Standard numerical approaches rely on coordinate mapping or domain truncation, yielding coordinate singularity or artifacts on the truncation boundary. We describe a general Kelvin transformation technique, which maps the infinite domain to a bounded one without creating singularities. The method is made possible by factoring out an asymptotic of the singularity induced by the coordinate stretching. The resulting transformation of functions can be understood as the natural transformation for fractional densities in geometric measure theory. In the viewpoint of Klein's Erlangen Program, the analysis reveals a “Kelvin Geometry,” where objects are functions subject to Kelvin transforms, leaving the PDE of interest invariant. The key to solving the infinite domain problem is to recognize that the boundedness quality of the domain is not a geometrically invariant notion under Kelvin Geometry. Therefore, we can transform the infinite domain problem into a compact one without sacrificing numerical accuracy.

[Ying Wang]

Dear SIAM CSS colleagues,

Correction: this week's talk is on Friday, not Wednesday.

Time: Friday 6/9 3:30-4:30pm CST

Sorry about the inconvenience.

Alexander Grigo

Ying Wang (wa...@ou.edu)

University of Oklahoma

[Ying Wang]

Dear SIAM CSS colleagues,

For the week of June 21st, the speaker of SIAM Central States Section(CSS) Computational and Applied Mathematics Forum is Dr. Ronen Basri from Weizmann Institute of Science. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 6/21 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

On the Connection between Deep Neural Networks and Kernel Methods
Ronen Basri
Weizmann Institute of Science

Abstract: Recent theoretical work has shown that under certain conditions, massively overparameterized neural networks are equivalent to kernel regressors with a family of kernels called Neural Tangent Kernels (NTKs). My work in this subject aims to better understand the properties of NTK for various network architectures and relate them to the inductive bias of real neural networks. In particular, I will argue that for input data distributed uniformly on the sphere NTK favors low-frequency predictions over high-frequency ones, potentially explaining why overparameterized networks can generalize even when they perfectly fit their training data. I will further discuss the behavior of NTK when data is distributed nonuniformly and show that NTK (with ReLU activation) is tightly related to the classical Laplace kernel, which has a simple closed-form. Finally, I will discuss our analysis of NTK for convolutional networks, which indicates that these networks are biased toward learning low frequency target functions with any higher frequencies concentrated in local regions. Overall, our results suggest that much insight about neural networks can be obtained from the analysis of NTK.

[Ying Wang]

Dear SIAM CSS colleagues,

We will resume the SIAM Central States Section(CSS) Computational and Applied Mathematics Forum by a sequence of lectures next week (9/25-9/29) featuring the following speakers and more:

All the times are US Central Time
9/25/2023 9-10am Hongkun Zhang, U Mass Amherst
          Machine learning of Hamiltonian dynamical systems
9/26/2023 2:30-3:30pm Xingjie Li, UNC Charlotte
          Coarse – Graining of stochastic system
9/27/2023 12:30-1:30pm Haiyan Wang, Arizona State
          Combining network theory and partial differential equation to improve prediction of epidemic spreading
9/27/2023 3-4pm Yulong Xing, Ohio State University
9/28/2023 9-10am Ilse Ipsen, North Carolina State
          An Introduction to Randomized Matrix Computations
9/28/2023 10-11am Yongtao Zhang, Nortre Dame
          High order WENO fast sweeping methods for sparse grids and for factored Eikonal equations
9/28/2023 3:30-4:30pm Dongbin Xiu, Ohio State University
          Data Driven Modeling of Unknown Systems with Deep Neural Networks

Please check the forum website for updates
https://math.ou.edu/~wang/SIAM_CSS_CAF.html

The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Hope to see you all!

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

[Ying Wang]

Dear SIAM CSS colleagues,

Just a quick update. Two talks have been added to Wednesday's schedule. Please see the updated program below

All the times are US Central Time

9/25/2023 9-10am Hongkun Zhang, Amherst

          Machine learning of Hamiltonian dynamical systems
9/26/2023 2:30-3:30pm Xingjie Li, UNC Charlotte
          Coarse – Graining of stochastic system

9/27/2023 9-10am Yulong Lu, University of Minnesota
          Two-scale gradient descent ascent dynamics finds
          mixed Nash equilibria of continuous games:
          A mean-field perspective
9/27/2023 10:30-11:30am Haitao Fan, Georgetown University
          Hysteretic conservation laws, with application to
          traffic jam analysis

9/27/2023 12:30-1:30pm Haiyan Wang, Arizona State
          Combining network theory and partial differential
          equation to improve prediction of epidemic spreading
9/27/2023 3-4pm Yulong Xing, Ohio State University

          High Order Structure Preserving Numerical Methods
          for Euler Equations with Gravitation
9/28/2023 9-10am Else Ipsen, North Carolina State

          An Introduction to Randomized Matrix Computations
9/28/2023 10-11am Yongtao Zhang, Nortre Dame
          High order WENO fast sweeping methods for sparse
          grids and for factored Eikonal equations
9/28/2023 3:30-4:30pm Dongbin Xiu, Ohio State University
          Data Driven Modeling of Unknown Systems with
          Deep Neural Networks

All the talks are given via zoom

https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Forum website
https://math.ou.edu/~wang/SIAM_CSS_CAF.html

Everyone is welcome!


Best regard,

Ying Wang (wa...@ou.edu)
University of Oklahoma

[Ying Wang]

Dear SIAM CSS colleagues,

This week, we have Dr. Elisabeth Ullmann from Technische Universität München to present to us at the SIAM Central States Section(CSS) Computational and Applied Mathematics Forum. The title and abstract of her talk are given at the end of this message.

Time: Wednesday 10/25/2023 10-11am CST (please note the unusual time)

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Title: Particle dynamics for rare event estimation with PDE-based models
Elisabeth Ullmann, Technische Universität München (TUM)

Abstract: The estimation of the probability of rare events is an important task in reliability and risk assessment of critical societal systems, for example, groundwater flow and transport, and engineering structures. In this talk we consider rare events that are expressed in terms of a limit state function which depends on the solution of a partial differential equation (PDE). We present two novel estimators for the rare event probability based on (1) the Ensemble Kalman filter for inverse problems, and (2) a consensus-building mechanism. Both approaches use particles which follow a suitable stochastic dynamics to reach the failure states. The particle methods have historically been used for Bayesian inverse problems. We connect them to rare event estimation.

This is joint work with Konstantin Althaus, Fabian Wagner and Iason Papaioannou (TUM).

[Ying Wang]

Dear SIAM CSS colleagues,

This week, we have Dr. Michael Herty from RWTH Aachen University to present to us at the SIAM Central States Section(CSS) Computational and Applied Mathematics Forum. The title and abstract of his talk are given at the end of this message.

Time: Thursday 11/2/2023 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Title: Recent Results on Ensemble Methods For Optimization

Michael Herty (RWTH Aachen University)

Abstract:
We are interested in the construction of numerical methods for constrained high-dimensional constrained nonlinear optimization problems by gradient free techniques. Gradients are replaced by particle approximations and recently different methods have been proposed, e.g. consensus-based, swarm-based or ensemble Kalman based methods. We discuss recent extensions to the constrained and the parametric case as well as their corresponding mean field descriptions in the many particle  limit. Those allow to show convergence as well as the analysis of properties of the new algorithm. Several numerical examples, also in high dimensions, illustrate the theoretical findings as well as the performance of those methods.

[Ying Wang]

Dear SIAM CSS colleagues,

This week, we have Dr. Maryam Yashtini from Georgetown University to present to us at the SIAM Central States Section(CSS) Computational and Applied Mathematics Forum. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 11/8/2023 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Counting Objects by Diffused Index: geometry-free and training-free approach

Maryam Yashtini, Georgetown University

Abstract. Counting objects is a fundamental but challenging problem. In this talk, I propose diffusion-based, geometry-free, and learning-free methodologies to count the number of objects in images. The main idea is to represent each object by a unique index value regardless of its intensity or size, and to simply count the number of index values. First, we place different vectors, referred to as seed vectors, uniformly throughout the mask image. The mask image has boundary information of the objects to be counted. Secondly, the seeds are diffused using an edge-weighted harmonic variational optimization model within each object. We propose an efficient algorithm based on an operator splitting approach and alternating direction minimization method, and theoretical analysis of this algorithm is given. An optimal solution of the model is obtained when the distributed seeds are completely diffused such that there is a unique intensity within each object, which we refer to as an index. For computational efficiency, we stop the diffusion process before a full convergence, and propose to cluster these diffused index values. We refer to this approach as Counting Objects by Diffused Index (CODI). We explore scalar and multi-dimensional seed vectors. For Scalar seeds, we use Gaussian fitting in histogram to count, while for vector seeds, we exploit a high-dimensional clustering method for the final step of counting via clustering. The proposed method is flexible even if the boundary of the object is not clear nor fully enclosed. We present counting results in various applications such as biological cells, agriculture, concert crowd, and transportation. Some comparisons with existing methods are presented.

[Ying Wang]

Dear SIAM CSS colleagues,

Next week, we have Dr. Daniel Reynolds from Southern Methodist University to present to us at the SIAM Central States Section(CSS) Computational and Applied Mathematics Forum. The title and abstract of his talk are given at the end of this message.

Time: Monday 11/13/2023 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Flexible Time Integration Methods for Multiphysics PDE Systems

Daniel Reynolds, Southern Methodist University

Abstract:
In recent years computational simulations have rapidly evolved in complexity (high order discretizations, spatial adaptivity, wide arrays of physical processes), placing ever larger strains on the time integration methods on which they rely.  High spatial order necessitates comparably high order time integration.  Spatial adaptivity and multiphysics processes give rise to subsets of the solution that evolve at differing time scales, or to simulations that combine nonstiff/nonlinear processes with others that may be highly stiff but that are frequently linear.  In this talk, I will discuss recent work on time integration methods that allow the flexibility to apply different techniques to distinct physical processes.  While techniques for flexible time integration have existed for some time, including additive Runge--Kutta ImEx, multirate (a.k.a. multiple time stepping), and operator-splitting methods, there have been comparably few that combine these types of flexibility into a single family, while also supporting high orders of accuracy and temporal adaptivity.  In this talk, I focus on the newly developed IMEX-MRI-GARK (Chinomona and R., 2022) and IMEX-MRI-SR (Fish, R. and Roberts, 2023) families of methods, along with novel techniques for time adaptivity in multirate infinitesimal time integration methods (Fish and R., 2023).  While some of these methods are already available in the ARKODE time integration library within SUNDIALS, I will point out our release plans for the remainder.

[Ying Wang]

Dear SIAM CSS colleagues,

This week, we have Dr. Zhaojun Bai from UC Davis to present to us at the SIAM Central States Section(CSS) Computational and Applied Mathematics Forum. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 11/29/2023 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

Interplay between optimization on matrix manifolds
and matrix eigenvalue problems

Zhaojun Bai
University of California, Davis

Optimization on matrix manifolds (OptMM) are ubiquitous in scientific computing, such as energy minimization in electronic structure calculations, finding projection matrices for dimensionality reduction in high dimensional data analysis, and robustification for the effects of data variability. In this talk, I will provide a perspective on intriguing interplay between OptMM and matrix eigenvalue problems (EVP).  We will show how to exploit underlying properties of OptMM and EVP to reveal variational characterization of EVP, and design efficient numerical algorithms for solving large scale OptMM with EVP. The success of the perspective will be shown in solving challenging problems in real-life applications, such as low-rank approximation of tensor networks, robust common spatial pattern analysis in brain-computer interfaces.

[Ying Wang]

Dear SIAM CSS colleagues,

This week, we have Dr. Melvin Leok from UCSD to present to us at the SIAM Central States Section(CSS) Computational and Applied Mathematics Forum. The title and abstract of his talk are given at the end of this message.

Time: Wednesday 12/6/2023 3:30-4:30pm CST

Forum website https://math.ou.edu/~wang/SIAM_CSS_CAF.html.
The Zoom link for the forum is
https://oklahoma.zoom.us/j/96270425796?pwd=K3djTHg5U25kdWFRcGtISHhFcDFqUT09
(Meeting ID: 962 7042 5796   Passcode: 03939463)

Best regards,
Alexander Grigo
Ying Wang (wa...@ou.edu)
University of Oklahoma

The Connections Between Discrete Geometric Mechanics, Information Geometry, Accelerated Optimization and Machine Learning

Melvin Leok, Mathematics, University of California, San Diego

Abstract: Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.