Date/Time : 17 Jun 2022 16:30
Venue : S17 #04-06 (SR1)
Speaker : John Harlim , Penn State University

Abstract:  I will discuss recent efforts in using the Diffusion Maps (DM) algorithm to solve elliptic PDEs on unknown manifolds using point clouds data. The key idea rests on the fact that away from the boundary, the second-order elliptic differential operators can be approximated by integral operators defined with appropriate Gaussian kernels. On manifolds with boundary, however, such an approximation is only valid for functions that satisfy the Neumann boundary condition. Motivated by the classical ghost-point correction in the finite-difference method for solving Neumann problems, we extend the diffusion maps algorithm with ghost points such that it is a consistent estimator in the pointwise sense even near the boundary. Applying the proposed algorithm, which we called the Ghost Points Diffusion Maps (GPDM), to solve the well-posed elliptic PDEs with Dirichlet, Neumann, or Robin boundary conditions, we establish the convergence of the approximate solution under appropriate smoothness assumptions. I will also discuss a neural-network regression solution on an algebraic equation induced by the DM/GPDM discretization. In addition to theoretical convergence for a wide enough neural network model, we numerically found that the proposed solver avoids the classical large matrix inversion problem. I will also discuss the application of DM/GPDM on Bayesian elliptic inverse problems. If time permits, I will also present symmetric Graph-Laplacian matrices that spectrally converge to the Laplace-Beltrami operator defined on compact manifolds with homogeneous Neumann and Dirichlet boundary conditions.

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Date/Time : 13 May 2022 16:00
Venue : via Zoom
Speaker : Jan S Hesthaven, EPFL

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https://nus-sg.zoom.us/j/82587724360?pwd=dGJ4RTZVb25rK1dGRFI5Y0FyTUJiUT09

Meeting ID: 825 8772 4360
Passcode: 015672

Abstract: The development of reduced order models for complex applications promises rapid and accurate evaluation of the output of complex models under parameterized variation with applications to problems which require many evaluations, such as in optimization, control, uncertainty quantification and applications where near real-time response is needed. However, many challenges remain to secure the flexibility, robustness, and efficiency needed for general large scale applications, in particular for nonlinear and/or time-dependent problems.

After a brief introduction to reduced order models, we discuss the development of methods which seek to conserve chosen invariants for nonlinear time-dependent problems. We develop structure-preserving reduced basis methods for a broad class of Hamiltonian dynamical systems, including canonical problems and port-Hamiltonian problems, before considering the more complex situation of Hamiltonian problems endowed with a general Poisson manifold structure which encodes the physical properties, symmetries and conservation laws of the dynamics.

Time permitting we also discuss the extension of structure preserving models within a framework for nonlinear reduced order models in which a local basis allows to maintain a small basis even for problems with a slowly decaying Kolmogorov n-width such a transport dominated problems. We shall demonstrate the efficiency of such techniques for nonlinear transport dominated problems, including problems in kinetic plasma physics,

This work is done in collaboration with B, Maboudi (DTU), C. Pagliantini (TU/e), N. Ripamonti (EPFL).

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Date/Time : 08 Apr 2022 09:00
Venue : via Zoom
Speaker : Konstantin Tikhomirov, Georgia Institute of Technology

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https://nus-sg.zoom.us/j/6124909748?pwd=TDFJYitzZ3ZSeUZUcTBEWnhTZWtSUT09

Meeting ID: 612 490 9748
Passcode: 007151

Abstract: The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Although the worst-case loss of precision in GEPP due to roundoff errors is known to be very significant, empirical evidence suggests that for a typical coefficient matrix, GEPP is numerically stable. Until recently, this phenomenon has received little theoretical justification. We make a progress on this problem by showing that, given the random $n\times n$ standard Gaussian coefficient matrix, the {\it growth factor} of the Gaussian Elimination with Partial Pivoting is at most polynomially large in $n$ with probability close to one. This implies that with high probability the GEPP results only in logarithmic (in $n$) loss of bits of precision, which improves an earlier estimate of Sankar of order $O(\log^2 n)$. This is joint work with Han Huang.

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Date/Time : 11 Mar 2022 09:00
Venue : via Zoom
Speaker : Laura DeMarco, Harvard University

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https://nus-sg.zoom.us/j/86723583031?pwd=WDBVNVpqbU8xWWI3amhKMTdPdk45Zz09

Meeting ID: 867 2358 3031
Passcode: 232023

Abstract: The periodic orbits and their structure are a fundamental feature of a dynamical system. In an algebraic setting, where the system is defined by polynomials, we can use tools from algebraic or arithmetic geometry to study these orbits. As the past year has seen major breakthroughs in complex and arithmetic geometry — for example in the uniform versions of the Mordell or Manin-Mumford Conjectures proved recently by Dimitrov-Gao-Habegger, Kuhne, Yuan and others, studying families of abelian varieties — I will take this opportunity to describe some parallel questions and progress in complex and arithmetic dynamical systems.

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Date/Time : 11 Feb 2022 09:00
Venue : via Zoom
Speaker : Geordie Williamson, University of Sydney

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https://nus-sg.zoom.us/j/86817881669?pwd=T1dVRWRQVjJRTjZNbDE0QUpsa3E0QT09

Meeting ID: 868 1788 1669
Passcode: 432904

Abstract: The combinatorial invariance conjecture is a fascinating conjecture in Representation Theory. Basically it says that certain important polynomials (“Kazhdan-Lusztig polynomials”) are determined in a very non-trivial way by a directed graph. About 2 years ago, I began working on this problem with DeepMind, an AI lab based in London. (DeepMind is famous for their AlphaGo program which was the first program to defeat the best human Go players.) Our goal was to try to discover whether modern Machine Learning techniques are helpful in approaching problems in Pure Mathematics. I will outline how the Machine Learning models work, how successful they were and (by far the trickiest part) how we went about extracting new mathematics from these models. The result is a formula which sheds light on the combinatorial invariance conjecture. This is joint work with the DeepMind team: Charles Blundell, Lars Buesing, Alex Davies and Petar Veličković.

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Date/Time : 14 Jan 2022 09:00
Venue : via Zoom
Speaker : Alexander Kiselev, Duke University

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https://nus-sg.zoom.us/j/81290641127?pwd=eUowVGZYN3REWndWVHRKK2piQ1pUUT09

Meeting ID: 812 9064 1127
Passcode: 612753

Abstract: The Euler equation describing motion of ideal fluid goes back to 1755. The analysis of the equation is challenging since it is nonlinear and nonlocal. Its solutions are often unstable and spontaneously generate small scales. The fundamental question of global regularity vs finite time singularity formation remains open for the Euler equation in three spatial dimensions. In this lecture, I will review the history of this question and its potential connection with the arguably greatest unsolved problem of classical physics, turbulence. Results on small scale and singularity formation in two dimensions and for a number of related models will also be presented.

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Date/Time : 19 Nov 2021 10:00
Venue : via Zoom
Speaker : Po-Shen Loh, Carnegie Mellon University

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https://nus-sg.zoom.us/j/84707855085?pwd=dVVUa0puWjlMT3NvN3hRWUV4N016QT09

Meeting ID: 847 0785 5085
Passcode: 687520

Abstract: We introduce a categorically new approach for pandemic control, which transforms selfish human behavior from a liability to an asset. Amidst a high-lethality pandemic, it could leverage individual survival instincts across the entire population, naturally incentivizing system-wide behavior that mitigates pandemic spread.

One of the central tools in pandemic control is contact tracing, which informs people after they have already been exposed to an infected person, and asks all of those people to remove themselves from society, to protect everyone else from themselves.

Informed by Game Theory, we reverse the incentives with a new infrastructure based on mobile devices which helps individuals directly reduce their own chance of infection (protecting themselves from everyone else). The core concept is to inform every participating user of their network-theoretic distance to each newly reported case, against the network of frequent close physical contacts.

This is fundamentally different from contact tracing, because it does not seek to control individuals after they are exposed, but rather empowers not-yet-exposed individuals to take selfish action to avoid exposure in the first place. It is analogous to arming every individual with their personal pandemic network radar. This opportunity was only recently unlocked by the mass proliferation of connected mobile devices, and opens a new category of intervention at the intersection of Game Theory, Technology, and Health.

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Date/Time : 12 Nov 2021 16:00
Venue : via Zoom
Speaker : Noga Alon, Princeton and Tel Aviv

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https://nus-sg.zoom.us/j/84886693609?pwd=ODArd3JMUDBlVjB2RTBTZlBjVnViUT09

Meeting ID: 848 8669 3609
Passcode: 821993

Abstract:  I will show how basic properties of polynomials can be used in the study of problems in Combinatorics, Additive Number Theory, Combinatorial Geometry and Graph Theory, describing recent and less recent results, problems and algorithmic challenges.

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Date/Time : 08 Oct 2021 16:00
Venue : via Zoom
Speaker : Siddhartha Mishra, ETH Zürich

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https://nus-sg.zoom.us/j/88995092115?pwd=UmxidFJqTzZyQ2JRLyt5SGNnaHhnUT09

Meeting ID: 889 9509 2115
Passcode: 309203

Abstract:  We will describe PINNs and illustrate several examples for using PINNs for solving PDEs. Our aim would be to elucidate mechanisms that underpin the success of PINNs in approximating classical solutions to PDEs by deriving rigorous bounds on the resulting error. Examples of a variety of linear and nonlinear PDEs will be provided including PDEs in high dimensions and inverse problems for PDEs.

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