Date/Time : 10 Sep 2025 16:30
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Wang Zhongjian, Nanyang Technological University

In this talk I will discuss the role of regularity estimate for viscous Hamilton Jacobi equation in showing the optimal iteration complexity (square root of dimension) of the Diffusion model. In our approach to the optimal bound, we only assume the Gaussian tail assumption, which does not require log-concaveness of the distribution. Our Gaussian tail assumption admits distribution of early stopping setting and scalable to infinite dimensions. More recently, we also provide analysis for Follmer process and show the deterministic heat flow is a Lipschitz change of variable between target and Gaussian under the Gaussian tail assumption.

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Date/Time : 04 Sep 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Khoat Than, Hanoi University of Science and Technology

Deep neural networks (DNNs) have driven many of the most significant breakthroughs in artificial intelligence in recent years, fundamentally transforming the way we live and work. Tasks that seemed intractable just a decade ago can now be solved with remarkable accuracy by large-scale DNNs. A central reason for their success lies in their ability to generalize: after learning from a finite training set, DNNs often perform surprisingly well on unseen data. Yet, despite this empirical success, the theoretical foundations of generalization in deep learning remain poorly understood, posing a long-standing challenge.

In this talk, I will provide an overview of recent theoretical advances, many of which, despite their elegance, turn out to yield vacuous or impractical guarantees for modern networks. I will then introduce our new approaches, which establish explicit connections between the local behaviors of a DNN across different regions of the data space, the complexity of the data distribution, and the model’s generalization performance. These results reveal novel structural insights and suggest that such perspectives could bring us closer to resolving one of the most fundamental open problems in deep learning.

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Date/Time : 03 Sep 2025 16:30
Venue : S17 #04-04 (SR3)
Speaker : Tuan Dam, Hanoi University of Science and Technology

This work presents novel Monte Carlo Tree Search (MCTS) algorithms for stochastic Markov Decision Processes (MDPs) that address fundamental limitations in existing methods like UCT and Fixed-Depth-MCTS. The proposed approaches incorporate power mean estimators as value backup operators to achieve more accurate value estimation between the biased extremes of weighted sum (underestimation) and maximum valuation (overestimation). Combined with polynomial exploration bonuses that properly balance exploration-exploitation in non-stationary settings, these algorithms achieve polynomial convergence rates of O(n^(-1/2)) for discrete spaces and O(n^(-ζ)), ζ ∈ (0,1/2) for continuous state-action spaces, where n represents visited trajectories. The methods extend existing theoretical guarantees from deterministic to stochastic environments, with empirical validation demonstrating effectiveness across various stochastic MDP tasks in both discrete and continuous domains.

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Date/Time : 28 Aug 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Zhang Zhen, Southern University of Science and Technology

We propose a novel phase-field model for N-phase problems by using a dichotomic representation. Derived from Onsager’s variational principle, the proposed model enjoys mechanic, energetic, algebraic and dynamic consistencies and is shown to asymptotically converge to its sharp interface limit. An operator-splitting based numerical method is developed to simulate the model with both second-order accuracy and energy stability. Various numerical results are presented to validate theoretical derivation and show interesting phenomena.

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Date/Time : 27 Aug 2025 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Sun Hongpeng, Renmin University of China

This paper introduces a preconditioned convex splitting algorithm enhanced with line search techniques for nonconvex optimization problems. The algorithm utilizes second-order backward differentiation formulas (BDF) for the implicit and linear components and the Adams-Bashforth scheme for the nonlinear and explicit parts of the gradient flow in variational functions. The proposed algorithm, resembling a generalized difference-of-convex function approach, involves a changing set of convex functions in each iteration. It integrates the Armijo line search strategy to improve performance. The study also discusses classical preconditioners such as symmetric Gauss-Seidel, Jacobi, and Richardson within this context. The global convergence of the algorithm is established through the Kurdyka-Łojasiewicz properties, ensuring convergence within a finite number of preconditioned iterations. Numerical experiments demonstrate its high efficiency. This is a joint work with Xinhua Shen and Zaijiu Shang.

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Date/Time : 13 Aug 2025 16:30
Venue : S17 #04-04 (SR3)
Speaker : Liu Wenjie, Harbin Institute of Technology

In this talk, we introduce a new theoretical framework built upon fractional spaces for optimal error estimates of Chebyshev/Legendre polynomial approximations to functions with limited regularity. It naturally arises from exact representations of orthogonal polynomial expansion coefficients. Here, the essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions); (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties, and (iii) asymptotic formulas or uniform upper bounds of GGF-Fs. Finally we introduce the hypothesis of Babuska and Hakula. The talk is based on joint work with Li-Lian Wang, Huiyuan Li, Boying Wu and Ruiyi Xie.

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Date/Time : 08 Aug 2025 10:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Timo Sprekeler, Texas A&M University

We begin by discussing the finite element approximation of stationary Fokker–Planck–Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings, based on the finite element scheme for stationary FPK problems developed in the first part. This talk includes joint work with Endre Süli (University of Oxford) and Zhiwen Zhang (The University of Hong Kong).

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Date/Time : 29 Apr 2025 16:00
Venue : S17 #04-05 (SR2)
Speaker : Liu Jinpeng, Tsinghua University

Fault-tolerant quantum computers are expected to excel in simulating unitary dynamics, such as the dynamics of a quantum state under a Hamiltonian. Most applications in scientific and engineering computations involve non-unitary dynamics. First, we propose a simple method for simulating a general class of non-unitary dynamics as a linear combination of Hamiltonian simulation (LCHS) problems. The LCHS method can achieve optimal cost in terms of state preparation. Second, we generalize the LCHS formula to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. We demonstrate the applicability of LCHS to a wide range of non-Hermitian dynamics, including linear parabolic PDEs and open quantum system dynamics. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.

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Date/Time : 23 Apr 2025 16:30
Venue : S17 #04-05 (SR2)
Speaker : Li Yifei, University of Tübingen

In this talk, I introduce a Sobolev-like space of order $3/2$, denoted as $\widehat{H}^{3/2}$, for Lagrangian finite elements, especially for $C^0$ elements. It is motivated by the limitations of current stability analysis of the evolving surface finite element method (ESFEM), which relies exclusively on energy estimate frameworks. When establishing a PDE-based analysis framework for ESFEM, we encounter a fundamental regularity mismatch: the ESFEM employs $C^0$ elements, while PDE regularity theory requires $H^{3/2}$ regularity for solutions. Our $\widehat{H}^{3/2}$ space resolves this challenge by demonstrating compatibility with elliptic PDE regularity theory, trace inequalities, and inverse inequalities. Notably, we extend the domain deformation estimates in ESFEM to the $\widehat{H}^{3/2}$ setting, thereby providing a rigorous foundation for PDE-based convergence analysis of ESFEM. The talk concludes with applications of the $\widehat{H}^{3/2}$ theory to numerical analysis for a coupled bulk-surface problem.

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