一周活动预告: (10.10-10.16)
请查收。
目录：
The core-shell obstacle scattering in a multilayered medium （张磊）
腔体边界与内部点源的共同反演 （张德悦）
Simultaneous reconstruction of an acoustic obstacle and its excitation sources （郭玉坤）
Constrained Markov Decision Process and Some Extensions（Liangyu Zhang）
On the (Non)smoothness of neural networks training （张景昭）
Convergence of the Planewave Approximations for Quantum Incommensurate Systems（陈华杰）
Nonconforming Finite Element Stokes Complexes in Three Dimensions（黄学海）
How many PDE numerical eigenvalues can we trust, and how to get more out of it? （Zhimin Zhang）
Whyspectral methods are preferred in PDE eigenvalue computations in some cases? （Zhimin Zhang）
Empirical likelihood in single-index quantile regression with high dimensional and missing observations（梁汉营）
Euler-Maruyama scheme for the SDEdriven by stable (徐礼虎)
Multi-scale numerical modeling of radionuclides transport in sandy facies of the opalinus clay (Tao Yuan)
Some results on inverse problems to elliptic PDEs with solution data and their implications in operator learning (Prof. Kui Ren)
High-order bound preserving methods for compressible multi-species flow with chemical reactions （杜洁）
Geometric Deep Learning for Molecular Design（Wengong Jin）
Implicit Bias of Optimization Algorithms for Neural Networks and Their Effects on Generalization（Chao Ma）
A discrete data assimilation algorithm for the three dimensional planetary geostrophic equations of large-scale ocean circulation（尤波）
A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals （伍智璋）
随机观测数据线性反问题的正则化方法 (陈志明)
Overcoming the curse of dimensionality: from nonlinear Monte Carlo to the training of neural networks（Arnulf Jentzen）
From Calculus of Variations to Reinforcement Learning I (Diogo Gomes)
From Calculus of Variations to Reinforcement Learning II (Diogo Gomes)
Semi-Supervised Learning and the ∞-Laplacian I (José Miguel Urbano)
Semi-Supervised Learning and the ∞-Laplacian II (José Miguel Urbano)
Parallel efficient global optimization by using the minimun energy criterion （田玉斌）
“随机分析与量化金融”主题年活动通知 | 随机分析在典型领域中的应用专题研讨班 （需填写报名表，截止日期：10.18）
Long Feng Science Forum Seminar Series | Seminar #10（Simple implicit regularizations in deep learning）
天元系列活动 | Theory and application of computational methods in...
1. The core-shell obstacle scattering in a multilayered medium
报告人: 张磊 （浙江工业大学）
报告时间: 2022-10.10 9:00-10:00
报告链接: 腾讯会议ID: 767 452 246
信息来源:
https://math.seu.edu.cn/2022/1006/c16339a422428/page.htm
报告摘要:
Scattering from the coated targets embedded in a multilayered medium has attracted much interest during recent years for extensive applications, such as sea radar target detection and underwater radar surveillance. Over the past few decades, the scattering of electromagnetic waves by various composite targets has been investigated extensively by many researchers. We consider the scattering from 3D core-shell structures in a two-layered lossy medium with an interface by the boundary integral equation method in this talk. We present a novel boundary integral equation formulation with difference type integral kernels for the scattering problem that the singularity of the integral kernel is reduced. It leads to a well-posed integral operator system, as well as provides a convenience for numerical calculation. We prove the well-posedness of the scattering problem with the variational method, integral equation, and operator theory. Furthermore, we study the numerical solution for the integral equation.
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2. 腔体边界与内部点源的共同反演
报告人: 张德悦（吉林大学）
报告时间: 2022-10.10 10:00-11:00
报告链接: 腾讯会议ID: 767 452 246
信息来源:
https://math.seu.edu.cn/2022/1006/c16339a422429/page.htm
报告摘要:
针对利用测量的总场数据确定腔体边界与相应的入射点源位置的共同反演问题,提出利用单层位势方法解耦入射场与散射场,并通过优化方法与直接采样方法来分别重构腔体形状与确定点源的位置。所提出算法在传统数值方法单一测量曲线的基础上,增加了一条测量曲线,即本算法采用双测量曲线进行数据采集,确保了问题解的唯一性,且所提出算法是一种定性方法与定量方法的结合,也不需要求解正问题或迭代,因此计算量不大且易于实现。同时,我们从数学上分析了数据带有噪声时的逼近性质,从而为算法的鲁棒性奠定了理论基础。本报告还将展示若干数值算例来验证所提出算法的有效性。
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3. Simultaneous reconstruction of an acoustic obstacle and its excitation sources
报告人: 郭玉坤 （哈尔滨工业大学）
报告时间: 2022-10.10 11:00-12:00
报告链接: 腾讯会议ID: 767 452 246
信息来源:
https://math.seu.edu.cn/2022/1006/c16339a422430/page.htm
报告摘要:
This talk is concerned with the inverse acoustic scattering problem of simultaneously reconstructing an obstacle as well as the associated point sources from the total-field measurements. A novel sampling-optimization approach is proposed to recover the target obstacle and source. We first obtain an initial and qualitative imaging of the source locations and shape of the obstacle by the direct sampling method. Then the optimization method is utilized to quantitatively produce the final reconstruction. Theoretical analysis will be provided to justify the applicability of the proposed approach and several numerical examples will be presented to illustrate the effectiveness.
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4. Constrained Markov Decision Process and Some Extensions
报告人: Liangyu Zhang (PKU)
报告时间: 2022-10.10 16:00-17:00
报告地点:腾讯会议ID: 723 1564 5542
信息来源: https://www.math.pku.edu.cn/kxyj/xsbg/tlb/informationsciences/143784.htm
报告摘要:
Reinforcement learning (RL) has achieved great success in areas such as Game-playing, robotics, recommender systems, etc. However, due to safety concerns or physical limitations, in some real-world RL problems, we must consider additional constraints that may influence the optimal policy and the learning process. A standard framework to handle such cases is the constrained Markov Decision Process (CMDP). Within the CMDP framework, the agent has to maximize the expected cumulative reward while obeying a finite number of constraints, which are usually in the form of expected cumulative cost criteria.
In this talk, we will briefly review the formulation of CMDP model along with some RL algorithms for solving CMDP problems. Additionally, we will also present an extension of the CMDP model called semi-infinitely constrained Markov decision process, where we are allowed to consider RL under infinitely many constraints.
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5. On the (Non)smoothness of neural networks training
信息来源:
http://tmcc.whu.edu.cn/info/1102/2104.htm
6. Convergence of the Planewave Approximations for Quantum Incommensurate Systems
报告人: 陈华杰（北京师范大学）
报告时间: 2022-10.11 16:00-17:00
报告地点: 腾讯会议ID: 704967113 密码: 373822
信息来源:https://www.math.sjtu.edu.cn/Default/seminarshow/tags/MDAwMDAwMDAwMLGelNuHpKF2
报告摘要:
This work studies the spectrum distribution of incommensurate Schrödinger operators. We characterize the density of states for the incommensurate system and develop novel numerical methods to approximate them. In particular, we (i) justify the thermodynamic limit of the density of states in the real space formulation; and (ii) propose efficient numerical schemes to evaluate the density of states based on planewave approximations and reciprocal space sampling. We present both rigorous analysis and numerical simulations to support the reliability and efficiency of our numerical algorithms.
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7. Nonconforming Finite Element Stokes Complexes in Three Dimensions
Speaker: 黄学海（上海财经大学）
Time: 2022-10.12 09:00-10:30
Venue: Tencent Meeting ID: 889-101-639 Password: 221012
Info Source:
https://ins.sjtu.edu.cn/seminars/2171
Abstract:
Two fully nonconforming finite element Stokes complexes ended with the nonconforming P1-P0 element for the Stokes equation in three dimensions are constructed. The lower order H(gradcurl)-nonconforming finite element only has 14 degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The H(gradcurl)-nonconforming elements are applied to solve the quad-curl problem, and optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem is decoupled into two mixed methods of the Maxwell equation and the nonconforming P1-P0 element method for the Stokes equation, based on which a fast solver is developed.
Lower order H(gradcurl)-nonconforming but H(curl)-conforming finite elements are constructed, which are extended to nonconforming finite element Stokes complexes. Then H(gradcurl)-nonconforming finite elements are employed to discretize the quad-curl singular perturbation problem, which possess the sharp and uniform error estimates with respect to the perturbation parameter. The Nitsche’s technique is exploited to achieve the optimal convergence rate in the case of the boundary layers. In addition, the regularity of the quad-curl singular perturbation problem is established.
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8. How many PDE numerical eigenvalues can we trust, and how to get more out of it?
报告人: Zhimin Zhang （Wayne State University）
报告时间: 2022-10.12 10:00-11:00
报告链接: 腾讯会议ID: 359 771 916
信息来源:
http://www.amss.cas.cn/mzxsbg/202209/t20220922_6516142.html
报告摘要:
When approximating PDE eigenvalue problems by numerical methods such as finite difference and finite element, it is common knowledge that only a small portion of numerical eigenvalues are reliable. However, this knowledge is only qualitative rather than quantitative in the literature. In this talk, we will investigate the number of "trusted" eigenvalues by the finite element
(and the related finite difference method results obtained from mass lumping) approximation of 2mth order elliptic PDE eigenvalue problems. Our two model problems are the Laplace and bi-harmonic operators, for which a solid knowledge regarding magnitudes of eigenvalues are available in the literature. Combining this knowledge with a priori error estimates of the finite element method, we are able to figure out roughly how many "reliable" eigenvalues can be obtained from numerical approximation under a pre-determined convergence rate.
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9. Why spectral methods are preferred in PDE eigenvalue computations in some cases?
报告人: Zhimin Zhang （Wayne State University）
报告时间: 2022-10.12 11:00-12:00
报告链接: 腾讯会议ID: 359 771 916
信息来源:
http://www.amss.cas.cn/mzxsbg/202209/t20220922_6516144.html
报告摘要:
We know that only a small portion of numerical eigenvalues are reliable when approximating PDE eigenvalue problems by finite difference and finite element methods. As a comparison, spectral methods may perform extremely well in some situation, especially for 1-D problems. In addition, we demonstrate that
spectral methods can outperform traditional methods and the state-of-the-art method in 2-D problems even with singularities.
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10. Empirical likelihood in single-index quantile regression with high dimensional and missing observations
报告人: 梁汉营（同济大学）
报告时间: 2022-10.12 14:00-15:00
报告地点: 腾讯会议 ID: 109-934-668
信息来源: https://www.math.sjtu.edu.cn/Default/seminarshow/tags/MDAwMDAwMDAwMLGelNyGyqF2
报告摘要:
Based on empirical likelihood method, we investigate statistical inference in partially linear single-index quantile regression with high dimensional linear and single-index parameters when the observations are missing at random, which allows the response or covariates or response and covariates simultaneously missing. In particular, applying B-spline approximation to the unknown link function, we establish asymptotic normality of bias-corrected empirical likelihood ratio function and maximum empirical likelihood estimator of the parameters; variable selection are considered by using the SCAD penalty. Meanwhile, we propose a penalized empirical likelihood ratio statistic to test hypothesis, and prove its asymptotically chi-square distribution under the null hypothesis. Also, simulation study and a real data analysis are conducted to evaluate the performance of the proposed methods.
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11. Euler-Maruyama scheme for the SDE driven by stable
报告人: 徐礼虎 （澳门大学）
报告时间: 2022-10.12 15:00-16:00
报告链接: 腾讯会议ID: 171 487 367
信息来源:
http://www.amss.cas.cn/mzxsbg/202209/t20220927_6517888.html
报告摘要:
In this talk, we will develop an EM scheme to approximate the
ergodic measure of the SDE driven by stable process, and obtain
a bound for the Wasserstein-1 distance between the ergodic
measures of EM scheme and the SDE. We show that this bound
is optimal. The method is by a recently developed probability
approximation framework and Malliavin calculus. The work is
based on the joint work with Peng Chen, Changsong Deng and
Rene Schilling.
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12. Multi-scale numerical modeling of radionuclides transport in sandy facies of the opalinus clay
报告人: Tao Yuan（University of Tubingen, Germany）
报告时间: 2022-10.12 19:30-20:30
报告链接: 腾讯会议ID: 463 7874 0598
信息来源:
http://www.amss.cas.cn/mzxsbg/202210/t20221008_6520186.html
报告摘要:
Clay rock formations such as Opalinus Clay (OPA) are considered as host rocks for underground radioactive waste repositories. Molecular diffusion is an important transport mechanism for radionuclide migration in clay rock due to its very low hydraulic conductivity. Reliable predictions of diffusive transport heterogeneity are critical for assessing the sealing capacity of clay rocks. The predictive power of numerical approaches to flow field analysis and radionuclide migration depends on the quality of the underlying pore network geometry. Both sedimentary and diagenetic complexity in sandy facies are controlling factors.
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13. Some results on inverse problems to elliptic PDEs with solution data and their implications in operator learning
14. High-order bound preserving methods for compressible multi-species flow with chemical reactions
报告人: 杜洁 （清华大学）
报告时间: 2022-10.13 10:00
报告链接: 腾讯会议ID: 681 805 206
信息来源:
http://tianyuan.xmu.edu.cn/cn/letures/915.html
报告摘要:
In this talk, we consider bound preserving problems for multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction should be between 0 and 1. The mass fraction does not satisfy a maximum principle and hence it is not easy to preserve the upper bound 1. Also, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Some previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this work, we will construct third order conservative bound-preserving Rugne-Kutta and multistep methods to overcome all these difficulties. Moreover, we will discuss how to control numerical oscillations. Numerical experiments will be given to demonstrate the good performance of the bound-preserving technique and the stability of the scheme for problems with stiff source terms.
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15. Geometric Deep Learning for Molecular Design
Speaker: Wengong Jin（Broad Institute of MIT and Harvard）
Time: 2022-10.13 10:00-11:00
Venue:
Zoom Meeting ID: 824 5302 6226 Password: PSJAS1013
Tencent/ VooV Meeting ID: 904 526 516
Info Source:
https://ins.sjtu.edu.cn/topics/seminar
Abstract:
Molecules and proteins are geometric objects and their function relies on their structure (e.g. graph or 3D point cloud). The challenge of AI-driven molecular design includes prediction and generation. The prediction task (forward problem) aims to predict the property of a molecule/protein automatically based on its structure. The generation task (inverse problem) aims to generate molecules/proteins that have specific properties of interest. In this talk, I will present how to use geometric/graph deep learning to accelerate molecular design. The first half of the talk will focus on small molecule drug discovery, i.e. how to build graph convolutional networks for property prediction and fragment-based generative models for the de novo drug design. The second half of the talk will focus on antibody engineering, i.e., how to build equivariant geometric deep learning models to dock antibodies onto an antigen epitope and generate CDR sequences that bind to the epitope.
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16. Implicit Bias of Optimization Algorithms for Neural Networks and Their Effects on Generalization
Speaker: Chao Ma（Stanford University）
Time: 2022-10.13 10:00-11:00
Venue: Tencent Meeting ID: 192983794 Password: 184713
Info Source:
https://ins.sjtu.edu.cn/seminars/2173
Abstract:
Modern neural networks are usually over-parameterized—the number of parameters ex-ceeds the number of training data. In this case the loss functions tend to have many (or even infinite) global minima, which imposes an additional challenge of minima selection on optimization algorithms besides the convergence. Specifically, when training a neural network, the algorithm not only has to find a global minimum, but also needs to select minima with good generalization among many other bad ones. In this talk, I will share a series of works studying the mechanisms that facilitate global minima selection of optimi-zation algorithms. First, with a linear stability theory, we show that stochastic gradient descent (SGD) favors flat and uniform global minima. Then, we build a theoretical con-nection of flatness and generalization performance based on a special structure of neural networks. Next, we study the global minima selection dynamics—the process that an op-timizer leaves bad minima for good ones—in two settings. For a manifold of minima around which the loss function grows quadratically, we derive effective exploration dy-namics on the manifold for SGD and Adam, using a quasistatic approach. For a manifold of minima around which the loss function grows subquadratically, we study the behavior and effective dynamics for GD, which also explains the edge of stability phenomenon.
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17. A discrete data assimilation algorithm for the three dimensional planetary geostrophic equations of large-scale ocean circulation
报告人: 尤波 教授（西安交通大学）
报告时间: 2022-10.13 10:00-11:00
报告地点: 腾讯会议ID: 391-324-000
信息来源:
https://math.fudan.edu.cn/09/f0/c30472a461296/page.htm
报告摘要:
In this talk, we consider a discrete data assimilation algorithm for the three dimensional planetary geostrophic equations of large-scale ocean circulation in the case that the observable measurements, obtained discretely in time, may be contaminated by systematic errors, which works for a general class of observable measurements, such as low Fourier modes and local spatial averages over finite volume elements. We will provide some suitable conditions to establish asymptotic in time estimates of the difference between the approximating solution and the unknown exact (reference) solution in some appropriate norms for these two different kinds of interpolant operators, which also shows that the approximation solution of the proposed discrete data assimilation algorithm will converge to the unique unknown exact solution of the original system at an exponential rate, asymptotically in time if the observational measurements are free of error.
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18. A Filon-Clenshaw-Curtis-Smolyak rule for multi-dimensional oscillatory integrals
信息来源:
http://tmcc.whu.edu.cn/info/1102/2109.htm
19. 随机观测数据线性反问题的正则化方法
报告人: 陈志明（中科院）
报告时间: 2022-10.13 16:00-17:00
报告链接: 腾讯会议ID: 478 1365 3406
信息来源:
http://www.amss.cas.cn/mzxsbg/202210/t20221008_6520215.html
报告摘要:
Tikhonov正则化是求解线性反问题的一个得到广泛应用的数学模型。在实际应用中，如何选取正则化参数是一个关键。本报告将首先介绍Tikhonov正则化的发展历史和文献中Tikhonov正则化参数选取的几种方法。然后介绍处理带随机误差观测数据线性反问题的一个新的数学理论，基于该数学理论，我们提出了一个新的选取正则化参数的自洽迭代算法。我们将通过散乱数据插值的薄板样条模型和随机观测数据偏微分方程反源问题介绍该理论的相关结果。本报告基于与庹睿、张文龙和邹军的合作工作。
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20. Overcoming the curse of dimensionality: from nonlinear Monte Carlo to the training of neural networks
报告人: Arnulf Jentzen（香港中文大学，明斯特大学）
报告时间: 2022-10.14 15:00-16:00
报告地点:腾讯会议 ID 940-538-096
信息来源:
https://math.sustech.edu.cn/seminar_all/12493.html
报告摘要:
PDEs are among the most universal tools used in modelling problems in nature and man-made complex systems. Nearly all traditional approximation algorithms for PDEs in the literature suffer from the so-called “curse of dimensionality” in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms, it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In the first part of this talk, we present an efficient machine learning algorithm to approximate solutions of high-dimensional PDE and we also prove that suitable deep neural network approximations do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs. In the second part of the talk we present some recent mathematical results on the training of neural networks.
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21. From Calculus of Variations to Reinforcement Learning I
Speaker: Diogo Gomes, KAUST
Time: 2022-10-14, 9:30-10:30 Lisbon time
Registration Link and Info Source:
https://mpml.tecnico.ulisboa.pt
Abstract:
This course begins with a brief introduction to classical calculus of variations and its applications to classical problems such as geodesic trajectories and the brachistochrone problem. Then, we examine Hamilton-Jacobi equations, the role of convexity and the classical verification theorem. Next, we illustrate the lack of classical solutions and motivate the definition of viscosity solutions. The course ends with a brief description of the reinforcement learning problem and its connection with Hamilton-Jacobi equations.
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22. From Calculus of Variations to Reinforcement Learning II
Speaker: Diogo Gomes, KAUST
Time: 2022-10-14, 11:00-12:00 Lisbon time
Registration Link and Info Source:
https://mpml.tecnico.ulisboa.pt
Abstract:
This course begins with a brief introduction to classical calculus of variations and its applications to classical problems such as geodesic trajectories and the brachistochrone problem. Then, we examine Hamilton-Jacobi equations, the role of convexity and the classical verification theorem. Next, we illustrate the lack of classical solutions and motivate the definition of viscosity solutions. The course ends with a brief description of the reinforcement learning problem and its connection with Hamilton-Jacobi equations.
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23. Semi-Supervised Learning and the ∞-Laplacian I
Speaker: José Miguel Urbano, KAUST
Time: 2022-10-14, 14:30-15:30 Lisbon time
Registration Link and Info Source:
https://mpml.tecnico.ulisboa.pt
Abstract:
Motivated by a recent application in Semi-Supervised Learning (SSL), the minicourse is a brief introduction to the analysis of infinity-harmonic functions. We will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions and its several drawbacks, leading to the notion of AMLE (Absolutely Minimising Lipschitz Extension). We then explore the equivalence between being absolutely minimising Lipschitz, enjoying comparison with cones and solving the infinity-Laplace equation in the viscosity sense.
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24. Semi-Supervised Learning and the ∞-Laplacian II
Speaker: José Miguel Urbano, KAUST
Time: 2022-10-14, 16:00-17:00 Lisbon time
Registration Link and Info Source:
https://mpml.tecnico.ulisboa.pt
Abstract:
Motivated by a recent application in Semi-Supervised Learning (SSL), the minicourse is a brief introduction to the analysis of infinity-harmonic functions. We will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions and its several drawbacks, leading to the notion of AMLE (Absolutely Minimising Lipschitz Extension). We then explore the equivalence between being absolutely minimising Lipschitz, enjoying comparison with cones and solving the infinity-Laplace equation in the viscosity sense.
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25. Parallel efficient global optimization by using the minimun energy criterion
报告人: 田玉斌 （Beijing Institute of Technology）
报告时间: 2022-10.15 10:00-11:00
报告链接: 腾讯会议ID: 535 573 704
信息来源:
http://www.amss.cas.cn/mzxsbg/202210/t20221008_6520190.html
报告摘要:
In optimization problems, the expensive black-box function implies severely restricted budgets in terms of evaluation. Some Bayesian optimization methods have been proposed to solve this problem, such as expected improvement (EI)
and hierarchical expected improvement (HEI). However, most of these methods are nonparallel and use a one-point-at-a-time strategy. This study proposes a new parallel framework that uses the minimum energy criterion. This framework can reduce time cost by reducing the number of iterations and avoiding the local optimization trap by encouraging the exploration of the optimization space. We also propose a shrink-augment strategy to correct the local surrogate model for the black-box function by placing more points around the true optima, which could also benefit the optimization. Some numerical experiments are also presented to compare the new method with popular existing methods. The results show the superiority of our proposed method over other Bayesian methods due to delivering better results with fewer iterations.
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