Date/Time : 26 Aug 2020 16:00
Venue : via Zoom
Speaker : Tong Xin, NUS

Join Zoom Meeting https://nus-sg.zoom.us/j/92076578514?pwd=aWk4Nm1Cc3htS1hGWDVpTlZ2cUgrdz09 Meeting ID: 920 7657 8514 Password: 459566 Gradient descent (GD) is known to converge quickly for convex objective functions, but it can be trapped at local minima. On the other hand, Langevin dynamics (LD) can explore the state space and find global minima, but in order to give accurate estimates, LD needs to run with a small discretization step size and weak stochastic force, which in general slows down its convergence. This paper shows that these two algorithms can “collaborate” through a simple exchange mechanism, in which they swap their current positions if LD yields a lower objective function. This idea can be seen as the singular limit of the replica-exchange technique from the sampling literature. We show that this new algorithm converges to the global minimum linearly with high probability, assuming the objective function is strongly convex in a neighborhood of the unique global minimum. By replacing gradients with stochastic gradients, and adding a proper threshold to the exchange mechanism, our algorithm can also be used in online settings. We further verify our theoretical results through some numerical experiments, and observe superior performance of the proposed algorithm over running GD or LD alone. We will further explain how does replica exchange method improve LD’s spectral gap using Poincare type of inequality.

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Date/Time : 19 Aug 2020 16:00
Venue : via Zoom
Speaker : Pang Tongyao , National University of Singapore

Join Zoom Meeting https://nus-sg.zoom.us/j/97080214054?pwd=cnJtSXF0YTZWZ1U2QU5YQWpwaU1tUT09 Meeting ID: 970 8021 4054 Password: 632494 Abstract: In this talk, I will start with the classical sparse estimation, whih has been extensively studied and applied to a wide range of applications, e.g., image restoration, in the past few decades. While most of the existing methods targeted at solving a maximum a posterior (MAP) variational model with the sparsity prior, we proposed to find the minimum mean square error (MMSE) solution which provides a good balance between sparsity and unbiasedness. We develop a coordinate minimization algorithm and connect it to a variational model to establish a comprehensive convergence analysis. In recent years, deep learning has shown its advantages over the traditional sparsity prior based methods for image restoration problems. However, most deep learning methods are supervised which require a high-quality training dataset. Bringing the idea of Bayesian inference from sparse estimation to deep learning further, we are able to develop a self-supervised deep learning method that calls no other training samples but still achieves SOTA performance. The key to success of our method lies in the approximation to the MMSE estimate, which helps to reduce model bias and enables ensemble prediction scheme with lower variance.

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Date/Time : 12 Aug 2020 16:00
Venue : via Zoom
Speaker : Dong Bin, Peking University

Join Zoom Meeting https://nus-sg.zoom.us/j/93373644793?pwd=Y2VrdnJRazZnRXllN2N5K1BneWE1UT09 Meeting ID: 933 7364 4793 Password: 711028 In this talk, I will start with a brief review of the dynamics and optimal control perspective on deep learning (including supervised learning, reinforcement learning, and meta-learning), especially the so-called unrolled dynamics approach and its applications in medical imaging. Then, I will present some of our recent studies on how this new approach may help us to advance CT imaging and image-based diagnosis further. Specifically, I will focus on our thoughts on how to combine the wisdom from mathematical modeling with ideas from deep learning. Such combination leads to new data-driven image reconstruction models and new data-driven scanning strategies for CT imaging, and with a potential to be generalized to other imaging modalities.

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Date/Time : 22 Jan 2020 16:00
Venue : S17 #05-11 SR5
Speaker : Lucas Lestandi, Nanyang Technological University

Date/Time:22 Jan 2020 16:00 Venue: S17 #05-11 SR5 Speaker: Lucas Lestandi, Nanyang Technological University Breaking the curse of dimensionality : from large data to reduced order models. It is known to every scientist who works with numerical simulation – that increasing complexity in the computing task leads to exploding computing times. This can then be materialized in many ways e.g. very fine meshes, extremely small time steps, multiscale or multiphysics problems and most importantly, large number of dimensions. The last issue (large of dimensions) is often referred as the curse of dimensionality. It is very common in fields as diverse as finance, chemistry or “simple” mechanical engineering optimization problem, parametric studies, etc. In this talk, we will cover some of the essential aspects in building reduced order models that can overcome some of the above difficulties. First, we introduce low rank approximation of high dimensional data. These tools enable both data reduction (by several orders) and yield reduced bases that are used to build the actual reduced order model (ROM). Then we will see how common ROMs are built and the issues that remain to be solve. Finally, we will have a look at the future of ROM, in which deep learning is set to play a large part. Add to calendar: 

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Date/Time : 03 Jan 2020 16:00
Venue : S17 #04-06 SR1
Speaker : HAN Jiequn, Princeton University

Date/Time:03 Jan 2020 16:00 Venue: S17 #04-06 SR1 Speaker: HAN Jiequn, Princeton University Deep BSDE: A Unified Framework for Solving High-Dimensional PDEs, FBSDEs, and Control Problems Developing algorithms for solving high-dimensional partial differential equations (PDEs), forward-backward stochastic differential equations (FBSDEs), and control problems has been an exceedingly difficult task for a long time, due to the notorious difficulty known as the curse of dimensionality. In this talk we introduce the “deep BSDE method” as a unified framework to solve general high-dimensional PDEs and FBSDEs. Starting from the BSDE formulation, we approximate the unknown component by neural networks and design a proper loss function to optimize parameters. Numerical results of a variety of examples, including applications in game theory and eigenvalue problem, demonstrate that the proposed algorithm is quite effective in high- dimensions, in terms of both accuracy and speed. We furthermore provide a theoretical error analysis to illustrate the validity and property of the objective function. Add to calendar: 

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Date/Time : 26 Dec 2019 16:00
Venue : S17 #05-11 SR5
Speaker : Hu Jingwei, Purdue University

Date/Time:26 Dec 2019 16:00 Venue: S17 #05-11 SR5 Speaker: Hu Jingwei, Purdue University A second-order asymptotic-preserving and positivity-preserving exponential Runge-Kutta method for a class of stiff kinetic equations We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) — can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving — can preserve the non- negativity of the solution which is a probability density function for arbitrary Knudsen numbers. The method is based on a new formulation of the exponential Runge-Kutta method and can be applied to a large class of stiff kinetic equations including the BGK/ES-BGK equation (relaxation type), the Fokker-Planck equation (diffusion type), and even the full Boltzmann equation (nonlinear integral type). Furthermore, we show that when coupled with suitable spatial discretizations the fully discrete scheme satisfies an entropy-decay property. Various numerical results are provided to demonstrate the theoretical properties of the method. This is joint work with Ruiwen Shu (University of Maryland). Add to calendar: 

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Date/Time : 13 Nov 2019 16:00
Venue : S17 #05-11 SR5
Speaker : Li Ji, National University of Singapore

Date/Time:13 Nov 2019 16:00 Venue: S17 #05-11 SR5 Speaker: Li Ji, National University of Singapore Numerical Algorithms of Phase Retrieval Phase retrieval is of paramount importance in a broad range of applications in engineering and imaging sciences. Mathematically, it amounts to solving a system of nonlinear quadratic equations, which is generally NP-hard. In this talk, we will give a numerical review of the problem and emphasize the different interests between the applied mathematicians and physicians. In other words, much research achievement of applied mathematicians has not intrigue the interests of physicians. One such reason is the simpler landscape of the resulting formulation. For example, the ambiguity solutions for practical phase retrieval is more complicated than phase retrieval from random sensing measurements. Thus the sound good solvers for random phase retrieval fails in locating a satisfying solution for practical phase retrieval. After we review the numerical algorithms for phase retrieval in literature, we introduce our recent work on phase retrieval, which tries its best to close the gap of random and practical phase retrieval. Our work gives some implications for designing the numerical algorithm for practical phase retrieval. Add to calendar: 

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Date/Time : 30 Oct 2019 16:00
Venue : S17 #05-11 SR5
Speaker : Luminita Danaila, University of Rouen Normandy

Date/Time:06 Nov 2019 16:00 Venue: S17 #05-11 SR5 Speaker: Ionut Danaila, University of Rouen Normandy Numerical methods and finite-element toolboxes for the simulation of Bose-Einstein condensates We present several numerical tools using classical finite elements with mesh adaptivity for solving different models used for the study of Bose-Einstein condensates. The programs are written as toolboxes for FreeFem++ (www.freefem.org), a free finite-element software, allowing to easily implement various numerical algorithms [1]. For solving the stationary (imaginary-time) Gross-Pitaevskii equation, we use two robust and optimised numerical methods: a steepest descent method based on Sobolev gradients and a minimization algorithm based on the state-of-the-art optimization library IPOPT. A very recent conjugate-gradient method using concepts of Riemannian optimization is also presented [2]. For the Bogoliubov-de Gennes system, representing a linearisation of the Gross-Pitaevskii equation, a Newton method and a fast algorithm based on ARPACK for the calculation of eigenvalues are available. For the real-time Gross-Pitaevskii equation, classical splitting and relaxation methods were implemented and intensively tested. Validations and illustrations are presented for computing difficult configurations with vortices observed in physical experiments: single-line vortex, Abrikosov lattice, giant vortex, dark/anti-dark solitons in one or two-component Bose-Einstein condensates [3]. [1] G. Vergez, I. Danaila, S. Auliac, F.Hecht, A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation, Computer Physics Communications, 209, p. 144-162, 2016. [2] I. Danaila, B. Protas, Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization, SIAM J. Sci. Comput. Vol. 39, No. 6, pp. B1102-B1129, 2017. [3] I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels and P. G. Kevrekidis, Vector dark-antidark solitary waves in multicomponent Bose-Einstein condensates, Phys. Rev. A 94, 053617, 2016. Add to calendar: 

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Date/Time : 09 Oct 2019 16:00
Venue : S17 #05-11 SR5
Speaker : CAI Yongyong, Beijing Computational Science Research Center

Date/Time:09 Oct 2019 16:00 Venue: S17 #05-11 SR5 Speaker: CAI Yongyong, Beijing Computational Science Research Center Numerical methods for Bogoliubov excitations of Bose-Einstein condensates We study the analytical properties and the numerical methods for the Bogoliubov-de Gennes equations (BdGEs) describing the elementary excitation of Bose-Einstein condensates around the mean field ground state, which is governed by the Gross-Pitaevskii equation (GPE). Three numerical methods are proposed to solve the BdGEs, including sine-spectral method, central finite difference method and compact finite difference method. Extensive numerical tests are provided to validate the algorithms and confirm that the sine-spectral method has spectral accuracy in spatial discretization, while the central finite difference method and the compact finite difference method are second-order and fourth-order accurate, respectively. Finally, sine spectral method is extended to study the elementary excitations under optical lattice potential and solve the BdGEs around the first excited states of the GPE. Add to calendar: 

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