Date/Time : 22 Nov 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Anant Raj, University of Illinois Urbana-Champaign

In this talk, we delve into the intricate relationships between heavy-tailed distributions, generalization error, and algorithmic stability in the realm of noisy stochastic gradient descent. Recent research has illustrated the emergence of heavy tails in stochastic optimization and their intriguing links to generalization error. However, these studies often relied on challenging topological and statistical assumptions. Empirical evidence has further challenged existing theory, suggesting that the relationship between heavy tails and generalization is not always monotonic. In response, we introduce novel insights, exploring the relationship between tail behavior and generalization properties through the lens of algorithmic stability. Our analysis reveals that the stability of stochastic gradient descent (SGD) varies based on how we measure it, leading to interesting conclusions about its behavior.Expanding upon these findings, we extend the scope to a broader class of objective functions, including non-convex ones. Leveraging Wasserstein stability bounds for heavy-tailed stochastic processes, our research sheds light on the non-monotonic connection between generalization error and heavy tails, offering a more comprehensive perspective.Additionally, we introduce a unified approach for proving Wasserstein stability bounds in stochastic optimization, emphasizing time-uniform stability and its role in various scenarios, including convex and non-convex losses. Our approach is versatile and applicable to popular optimizers, highlighting the importance of ergodicity.

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Date/Time : 22 Nov 2023 16:30
Venue : S17 #04-05 (SR2)
Speaker : Wu Hanxin, National University of Singapore

In this talk I purpose a diffuse-interface model for dynamic soft wetting problem with Shuttleworth effect. The model consists of Cahn-Hilliard-Navier-Stokes equations for two-phase fluid on a moving domain and linear elastic equation for soft substrate, and they are coupled on the free fluid-solid interface. An efficient numerical scheme is purposed by combining a monolith fluid-structure-interaction solver and linear convex splitting for Cahn-Hilliard equation on the moving domain. Theorems regarding energy-preserving property of the scheme are proved. We compare numerical results for moving contact line on soft and rigid substrate.

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Date/Time : 20 Nov 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Julian Sester, National University of Singapore

In this talk I introduce a general framework for Markov decision problems (MDPs) under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one timestep robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure.

Based on the theoretical results, we are able to derive a novel Q-learning algorithm to solve numerically the associated robust MDPs where the corresponding ambiguity set of transition probabilities for the underlying  Markov decision process is a Wasserstein ball around a (possibly estimated) reference measure. Moreover, I will discuss two applications of the presented setting to problems in quantitative finance.

First, we apply this framework to portfolio optimization involving data of the S&P 500 where it turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.

Second, we study the optimal market-making problem in order-driven electronic markets while explicitly accounting for model uncertainty.

(The talk is based on joint works with Ying Chen, Ariel Neufeld, Mario Sikic, Hoang Hai Tran and Yijiong Zhang)

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Date/Time : 17 Nov 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Hiroshi Ishimoto, Osaka Metropolitan University

The original Ibukiyama isomorphism is an isomorphism between vector valued Siegel cusp forms of degree 2 of integral weight and those of half-integral weight. In this talk, I will present a similar isomorphism between vector valued Siegel cusp forms of degree 2 of integral weight and cuspidal automorphic forms on the metaplectic group of degree 2 generating a large discrete series representation at the real place.

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Date/Time : 17 Nov 2023 16:15
Venue : via Zoom
Speaker : Chen Ying, National University of Singapore

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https://nus-sg.zoom.us/j/83899499647?pwd=U2EvRVBPZXpxeXNmV3lsVi95Z2FTQT09

Meeting ID: 838 9949 9647
Passcode: math

In the digital era, complex data has emerged across various disciplines, characterized by high dimensions, nonlinear temporal and spatial dependencies, and irregular dynamic changes (non-stationarity). Understanding and effectively analyzing such complex temporal structures and achieving high-precision forecasting have become pressing challenges. Additionally, comprehending the stochastic behaviors of forecasting is paramount for executives and managers, aiding them in making resilient and robust decisions in real-life scenarios. In my presentation, I will introduce my research on advancing time series modeling and methodologies. These approaches leverage the power of data science, machine learning, and deep neural networks, all tailored to address the complexities of modern data. Some of the innovative techniques I will discuss include functional neural tangent kernel estimators, mixed-frequency spatio-temporal models, and deep switching state space models. Together, these methods offer insights and tools for tackling the challenges posed by high-dimensional, nonlinear, and nonstationary data in the contemporary landscape.

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Date/Time : 16 Nov 2023 10:00
Venue : via Zoom
Speaker : Yang Liu, University of California, Los Angeles

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https://nus-sg.zoom.us/j/83242625129?pwd=ZmRzME93QndWbTNZQUdPUVdIS080Zz09

Meeting ID: 832 4262 5129
Passcode: 157423

In this talk, I will first briefly introduce my work in Ph.D. years on physics-informed deep generative models. The general motivation is to exploit the flexibility and high-dimensional representability of deep generative models, and improve the generative models with physical and mathematical knowledge. I will present some numerical and theoretical results on PDE uncertainty quantification, optimal transport learning, particle dynamic inference, etc.

The main part of the talk will be on my recent work in UCLA. Realizing the limitations of task-specific models, I proposed a new paradigm for scientific machine learning, namely “In-Context Operator Networks” (ICON). A distinguishing feature of ICON is its ability to learn operators from data prompts during the inference phase, without weight adjustments. Much like how a single Large Language Model can solve a variety of natural language processing tasks specified by the language prompt, a single ICON model, trained as a generalist operator learner rather than being tuned to approximate a specific operator, can tackle a wide range of tasks involving different operators. I will show how a single ICON model manages 19 distinct problem types, encompassing forward and inverse ODE, PDE, and mean-field control (MFC) problems, each spanning a continuous spectrum of operators. I will also show the great accuracy of ICON on forward and backward predictions of conservation laws with discontinuities. The learning capability is generalized to operators beyond the training distribution without any fine-tuning.

Reference:
https://www.pnas.org/doi/10.1073/pnas.2310142120
https://arxiv.org/pdf/2308.05061.pdf

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Date/Time : 15 Nov 2023 16:30
Venue : S17 #05-11 (SR5)
Speaker : Zou Guijin, Institute of High Performance Computing

At the nano-bio interface, mechanics can play important roles in the development of health-related technologies. In this talk, we will highlight two examples of such applications. First, we will examine how the mechanical properties of flexible nanomaterials affect their encapsulation in lipid vesicles. Through molecular dynamics simulations complemented by theoretical analysis, we have mapped out the phase diagram for these systems, revealing how nanomaterial stiffness and size relative to the vesicle dictate encapsulation outcomes. These results enhance our understanding of cell-nanomaterial interactions such as nano-toxicity, help guide the design of artificial cells and biohybrid microrobots. In the second example, we will introduce the emerging field of membrane targeting nanomedicine (MTN), an area where our mechanistic understanding is still at a primitive stage. Collaborating with experimental groups, we demonstrate how atomistic modelling and simulation facilitate the identification of membrane targeting antimicrobials (MTAs) capable of selectively penetrating and aggregating in bacterial lipid membranes, causing membrane permeability/rupture. These studies suggest a general simulation-assisted platform to accelerate discovery and innovation in nanomedicine.

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Date/Time : 15 Nov 2023 10:00
Venue : via Zoom
Speaker : Alex Cohen, Massachusetts Institute of Technology

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https://nus-sg.zoom.us/j/81046462775?pwd=aDhaYnhQOTgybzdYS21yajlQdnBmZz09

Meeting ID: 810 4646 2775
Passcode: 23571113

Heilbronn’s triangle problem asks, how small is the smallest triangle formed by a set of points? Suppose n points are placed in the unit square and Delta is the smallest area triangle formed by three of these points. It is not too hard to see that Delta < C n^{-1}. Komlos, Pintz, and Szemeredi proved in 1981 that Delta << C n^{-8/7} by building on an ingenious method of Roth. We improve the bound to Delta < C n^{-8/7-1/2000}. The improvement comes from establishing new connections between Heilbronn’s problem and projection theory.

All joint with Dmitrii Zakharov and Cosmin Pohoata.

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Date/Time : 14 Nov 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Liu Shixiao,

Last semester I talked about the upper density Mathias forcing which we prove to be proper. This time we take a look at several similar forcings, each of which collapses the continuum to omega and thus fails to be proper.

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