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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Date/Time : 26 Mar 2025 16:30
Venue : S17 #06-11 (SR6)
Speaker : Wang Chushan, National University of Singapore

The nonlinear Schrödinger equation (NLSE) arises from various applications in quantum physics and chemistry, nonlinear optics, plasma physics, Bose-Einstein Condensates, etc. In these applications, it is necessary to incorporate low-regularity or singular potential and nonlinearity into the NLSE. Typical examples of such potential and nonlinearity include the discontinuous square-well potential, the singular Coulomb potential, the non-integer power nonlinearity, and the logarithmic nonlinearity. Such low regularity and singularity pose significant challenges in the analysis of standard numerical methods and the development of accurate, efficient, and structure-preserving schemes.

In this talk, I will introduce several new analysis techniques to establish optimal error bounds for some widely used numerical methods under optimally weak regularity assumptions. Based on the analysis, we also propose novel temporal and spatial discretizations to handle the low regularity and singularity more effectively.

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Date/Time : 12 Mar 2025 16:30
Venue : S17 #05-12 (SR4)
Speaker : Zhang Junyu, National University of Singapore

Mirror descent (MD) and its stochastic variants are well-understood first order methods in convex optimization under appropriate smoothness conditions and have been extended to general relative smooth and nonconvex problems in recent years. Despite being widely studied, there are still many issues in the existing research.

In this talk, we will first reveal the inappropriateness of the currently widely adopted optimality measures, which casts doubts about the validity of current complexity results, for both deterministic and stochastic MD. Second, due to the absence of Lipschitz continuity, most existing results on nonconvex stochastic mirror descent (SMD) do not allow popular acceleration techniques such as random shuffling and variance reduction, etc. To fix these issues, we propose a novel regularity condition that is widely satisfied by most of the popular kernels. Combined with the dual space perspective of mirror descent, a new optimality measure has been proposed and a Lipschitz-like gradient bound has been established. We elaborate how these concepts can help the analysis SMD by establishing new and improved complexity results for SMD with variance reduction.

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Date/Time : 05 Mar 2025 16:30
Venue : S17 #05-12 (SR4)
Speaker : Anthony Chen, University of Michigan

Fast Summation refers to a family of techniques for the fast approximation of N-body sums. While traditionally fast summation has been applied to problems coming from astrophysics or electrodynamics, many problems in geophysical fluid dynamics can be rewritten as the computation of a spherical convolution, and when these integrals are discretized, the resulting problem is an N-body problem. In this talk, I discuss a novel spherical tree code/fast multipole method based on barycentric Lagrange interpolation, as well as applications to problems coming from geophysical fluid dynamics, including tidal modeling and vortex dynamics.

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