Date/Time : 14 Dec 2023 16:20
Venue : S17 #04-06 (SR1)
Speaker : Xiao Manyu, Northwestern Polytechnical University

The common issue of surrogate models is to make good use of sampling data. In theory, the higher the fidelity of sampling data obtained, the more accurate the approximation model built. However, in practical engineering problems, high-fidelity data may be less available, and such data may also be computationally expensive. On
the contrary, we often obtain low-fidelity data under certain simplifications. Although low-fidelity data is less accurate, such data still contains much information about the real system. So, combining both high- and low-fidelity data in the construction of a surrogate model may lead to better representation of the physical phenomena.

In this work, the original principle of Multi-Fidelity Proper Orthogonal Decomposition will consist in projecting the high-dimensional system of equations into a low-dimensional space, with reduced basis vectors. These vectors are updated with incremental “on-the-fly” merges between costly high-fidelity simulation snapshots and more rapid lower-fidelity vectors. Successful achievements of this work will lead to potential benefits in the fields of large-scale mechanical engineering.

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Date/Time : 28 Nov 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Morris Ang, Columbia University

In the first half of the talk, I will introduce the central objects in random conformal geometry. Schramm-Loewner evolution (SLE) is a random planar curve describing the scaling limits of interfaces in statistical physics models (e.g. percolation, Ising model). Liouville quantum gravity (LQG) is a random 2D surface arising as the scaling limit of random planar maps. These fractal geometries have deep connections to bosonic string theory and conformal field theories. There are many powerful theorems which say that, roughly speaking, cutting LQG by independent SLE gives two independent LQG surfaces. In the second half, I present some new developments and their applications.

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Date/Time : 27 Nov 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Lai Lexiao, Columbia University

Lyapunov stability is an important notion for dynamical systems, which has been absent in the optimization literature. When studying optimization methods, the goal has always been proving convergence. However, convergence to critical points could sometimes be unrealistic or undesirable in practice. In this talk, I will introduce the notions of Lyapunov stability for optimization methods. I will then describe my works on characterizing the local and global stability of first-order methods. My results apply to problems arising in data science. In particular, I provide a guarantee of the SGD optimizer for training neural networks.

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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 11:00
Venue : S17 #04-05 (SR2)
Speaker : Eva Lütkebohmert, University of Freiburg

Sovereign loan portfolios of Multilateral Development Banks typically consist of only a small number of borrowers and hence are heavily exposed to single name concentration risk. Based on realistic MDB portfolios constructed from publicly available data, this paper quantifies the magnitude of the exposure to name concentration risk using exact Monte Carlo simulations. In comparing the exact adjustment for name concentration risk to its analytic approximation as currently applied by major rating agencies, we further investigate whether current capital adequacy frameworks for MDBs are overly conservative.

(This is joint work with Julian Sester and Hongyi Shen)

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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 : 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 : 12 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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Date/Time : 11 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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