Applied Math
PhD Seminar

2026 Spring

The seminar of this semester is organized by Jiamao Wu and Zhuangao He, and co-organized by the graduate student union in the School of Mathematical Sciences at Fudan. This section is partially sponsored by Shanghai Key Laboratory for Contemporary Applied Mathematics.

2026-05-28 15:10:00 - 16:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We report a $k$-point extension of the second-order co-iterative augmented Hessian (CIAH) algorithm, termed $k$-CIAH, for Pipek--Mezey (PM) localization of Wannier functions (WFs). By exploiting an efficient evaluation of the Hessian--vector product, $k$-CIAH achieves $O(N_k^2 n^3)$ scaling in both CPU time and memory, matching that of previously reported first-order $k$-space approaches while improving upon the $O(N_k^3 n^3)$ scaling of $\Gamma$-point CIAH, where $N_k$ denotes the number of $k$-points sampling the first Brillouin zone and $n$ characterizes the unit-cell size. Benchmark calculations on a diverse set of solids---including insulators, semiconductors, metals, and surfaces---demonstrate the fast and robust convergence of $k$-CIAH-based PMWF optimization, which yields an overall computational efficiency approximately $2$--$3$-fold higher than first-order $k$-space methods and orders of magnitude higher than $\Gamma$-point CIAH for localizing $1000$--$5000$ orbitals. The quality of the resulting PMWFs is further validated by accurate electronic band structures obtained via PMWF-based Wannier interpolation.

2026-05-28 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Non-Hermitian many-body systems can be spectrally unstable, so small perturbations may induce large eigenvalue shifts. The pseudospectrum quantifies this instability and provides a perturbation-robust diagnostic. For inverse-polynomially small $\epsilon$, we show that deciding whether a point $z\in\mathbb{C}$ is $\epsilon$-close to the spectrum is PSPACE-hard for $5$-local operators, whereas deciding whether $z$ lies in the $\epsilon$-pseudospectrum is QMA-complete for $4$-local operators. This identifies pseudospectrum membership as a natural computational target. We then present a concrete end-to-end quantum framework for deciding pseudospectrum membership, which combines a singular-value estimation step with a dissipative state preparation algorithm. Our Quantum Singular-value Gaussian-filtered Search (QSIGS) combines quantum singular value transformation (QSVT) with classical post-processing to achieve Heisenberg-limited query scaling for singular-value estimation. To prepare suitable input states, we introduce an algorithmic Lindbladian protocol for approximate ground right singular vectors and prove its effectiveness for the Hatano--Nelson model. Finally, we demonstrate the full pipeline on a trapped-ion quantum computer and distinguish points inside and outside the target pseudospectrum near the exceptional point of a minimal non-Hermitian qubit model.

2026-06-04 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The high-resolution ODE framework provides a useful perspective for analyzing accelerated gradient methods by capturing higher-order effects such as Hessian-driven damping. While such damping can reduce oscillations and improve convergence, gradient-correction terms alone may sometimes slow down function-value convergence. We study a generalized perturbed ODE and show that suitable combinations of gradient and gradient-correction perturbations can preserve accelerated behavior. We also investigate Gauss-Seidel splitting with a predictor-corrector scheme (GS-PC). The corrector step naturally introduces Hessian-driven damping into both the discrete algorithms and their associated high-resolution ODEs. This viewpoint extends to composite optimization and operator zero-finding problems, offering a unified interpretation of several accelerated methods, including Nesterov-type schemes and fast Krasnosel'skii-Mann iterations.

Past Presentations

2026-03-05 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Currently, several finite difference methods are used to solve the Dirac equation in the semiclassical regime. However, finite difference methods have certain limitations in terms of accuracy and stability. The CNFD method, often employed to enhance stability, usually requires solving large linear systems, leading to significant computational costs. In this work, we conduct a rigorous analysis of the numerical error for the time-splitting methods applied to the Dirac equation in the semiclassical region. The dimensionless parameter $\epsilon \in (0,1]$, representing the reduced Planck constant, causes the solution field to exhibit rapid oscillations with characteristic wavelengths of order $O(\epsilon)$. We demonstrate that both $S_1$ and $S_2$ schemes preserve total mass conservation in a discrete sense. Furthermore, we establish error estimates for the time-splitting methods, clearly describing how the approximation error varies with time resolution $\tau$, spatial discretization step $h$, and the parameter $\epsilon$. We also derive error bounds for physical observables, and through numerical experiments, we validate the reliability of the error estimates.

2026-03-12 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we consider the numerical approximation of the Abels-Garcke-Grün model for a binary mixture of two viscous incompressible fluids with unmatched densities and viscosities. The system consists of a Navier-Stokes equation for the volume-averaged fluid velocity and a convective Cahn-Hilliard equation with Flory-Huggins potential for the phase-field variable. Based on the pressure stabilization method and an additional first-order stabilization term to the advective velocity in the Cahn-Hilliard equation, the scheme is fully decoupled and preserves the physical properties, i.e. the positivity-preserving property, which means that the discrete solution of the phase-field always stays in the physical interval (-1, 1) at a point-wise level, and the unconditional energy stability. Additionally, we perform a detailed optimal rate convergence analysis and derive error estimates. Numerical results are presented to validate the convergence rate and energy stability.

2026-03-19 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We study optimal learning rate schedules (LRSs) under the functional scaling law (FSL) framework, where the loss is written explicitly as a functional of the schedule. This formulation reveals a sharp phase transition governed by two exponents: a source exponent $s>0$ controlling signal learning and a capacity exponent $\beta>1$ controlling noise forgetting. In the easy-task regime, the optimal schedule follows a power decay to zero, with exponent determined by $\beta$. In the hard-task regime, the optimal schedule takes a warmup-stable-decay (WSD) form: it maintains the largest admissible learning rate for most of training and decays only near the end, with a vanishing decay fraction. We further analyze shape-fixed schedules, showing how the tail exponent governs both their optimality and their limitations through capacity saturation. This yields a principled evaluation of commonly used schedules such as cosine and linear decay. Finally, we apply the power-decay LRS to one-pass SGD for kernel regression and show that the last iterate attains the exact minimax-optimal rate.

2026-03-26 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand For a given matrix \( A \) in the real Schur form, the eigenvalue reordering algorithm updates \( A \) by an orthogonal matrix \( Q \) such that \( Q^{\top} AQ \) maintains the real Schur form while the eigenvalues of two adjacent diagonal blocks are swapped. By a series of updates, the diagonal blocks will eventually be arranged in a desired order. In LAPACK, this operation in double precision is performed by the routine DLAEXC, which is based on a direct swapping algorithm proposed by Bai and Demmel in 1993. In this work, we present a thorough error analysis of a modified direct swapping algorithm. By a carefully designed pivoting Givens QR factorization strategy, this algorithm can achieve an \( O(1) \cdot \mathbf{u} \Vert A \Vert \) error bound under quite mild assumptions, where \( \mathbf{u} \) is the machine precision. We present empirical evidence to illustrate the superior stability of our algorithm over existing approaches.

2026-04-02 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This report introduces a machine learning method that combines Tensor Neural Networks (TNN) with homogenization theory for solving elliptic multiscale equations. The core advantage of TNN lies in its unique tensor structure, which allows the computation of high-dimensional neural network function integrals to be reduced to one-dimensional integrals. This enables the design of highly accurate high-dimensional integration methods, whose computational complexity scales only polynomially with the number of dimensions. Leveraging this feature, we design a high-precision solver for multiscale problems. Specifically, the original problem is first transformed via homogenization into a series of cell problems and a homogenized equation. These are then solved separately using TNN-based methods. Unlike conventional machine learning methods that rely on Monte Carlo sampling, our approach employs deterministic numerical integration, achieving high computational accuracy. In particular, for cases where the multiscale coefficients depend on both fast and slow variables, the corresponding cell problems are defined on high-dimensional domains; the TNN-based approach enables efficient and accurate computation for such cases compared to traditional methods, thereby extending the applicability of homogenization techniques. We also generalize this approach to elliptic multiscale eigenvalue problems.

2026-04-09 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This work analyzes the inverse optimal transport (IOT) problem under Bregman regularization. We establish well-posedness results, including existence, uniqueness (up to equivalence classes of solutions), and stability, under several structural assumptions on the cost matrix. On the computational side, we investigate the existence of solutions to the optimization problem with general constraints on the cost matrix and provide a sufficient condition guaranteeing existence. In addition, we propose an inexact block coordinate descent (IBCD) method for the problem with a strongly convex penalty term. In particular, when the penalty is quadratic, the subproblems admit a diagonal Hessian structure, which enables highly efficient element-wise Newton updates. We establish a linear convergence rate for the algorithm and demonstrate its practical performance through numerical experiments, including the validation of stability bounds, the investigation of regularization effects, and the application to a marriage matching dataset.

2026-04-16 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Precise prediction of spatiotemporal dynamics over predictive horizons is constrained by the computational cost of high-fidelity solvers and the sparsity, noise, and irregularity of data. We introduce MERLIN, a Koopman-based framework that lifts dynamics to the evolution of learned observation functionals with near-linear progression, enabling full-field reconstruction at arbitrary resolutions. Theoretically, we develop a functional Koopman theory for PDEs and compensate for the loss of finite-dimensional linear invariance via the Mori–Zwanzig formalism, which augments the linear backbone with non-Markovian memory terms to improve predictive accuracy. Practically, MERLIN employs discretization-invariant function encoders that map partial, irregular observations to observables, and resolution-free function decoders that reconstruct states at arbitrary query points. Training under linear constraints yields an interpretable, low-dimensional model that captures principal modes, supports reduced-order modeling, and—augmented with memory correction—delivers stable long-horizon rollouts even in ultra-low-dimensional latent spaces.

2026-04-23 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Surrogate models are essential for accelerating computationally expensive simulations in science and engineering, especially for parametric partial differential equations (PDEs) posed on geometrically complex and variable domains that are often represented as point clouds. We present the Point Cloud Neural Operator (PCNO), a neural-operator framework designed for such problems. Starting from the neural-operator paradigm and the Fourier Neural Operator (FNO), we discuss the limitations of FNO on nonuniform discretizations, irregular domains, and the extraction of local geometric information. PCNO addresses these challenges by representing the computational domain as a point cloud and constructing neural layers that couple a density-weighted global integral operator with a gradient-enhanced local differential operator. This design enables the model to capture long-range interactions while remaining sensitive to local geometric variation. Numerical experiments on the 1D Burgers equation, Darcy flow on variable domains, flow over airfoils, and vehicle surface pressure prediction demonstrate that PCNO provides accurate surrogate predictions for PDE problems with complex and changing geometries.

2026-04-30 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Two-stage orthogonalization is essential in numerical algorithms such as Krylov subspace methods. For this task we need to orthogonalize a matrix \(A\) against another matrix \(V\) with orthonormal columns. A common approach is to employ the block Gram--Schmidt algorithm. However, its stability largely depends on the condition number of \([V,A]\). While performing a Householder orthogonalization on \([V,A]\) is unconditionally stable, it does not utilize the knowledge that \(V\) has orthonormal columns. To address these issues, we propose a two-stage Householder orthogonalization algorithm based on the generalized Householder transformation. Instead of explicitly orthogonalizing the entire \(V\), our algorithm only needs to orthogonalizes a square submatrix of \(V\). Theoretical analysis and numerical experiments demonstrate that our method is also unconditionally stable.

2026-05-07 15:10:00 - 16:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Diffusion-based methods provide a flexible framework for approximating and sampling high-dimensional probability distributions, but their practical performance is often limited by the difficulty of learning high-dimensional score functions. In this talk, we consider target distributions with locality structure and develop a localized diffusion model that explicitly exploits this property. Under locality assumptions, the score function can be well approximated using only neighborhood information, leading to a substantial reduction in effective complexity. We discuss both the algorithmic construction and theoretic properties, with particular emphasis on the trade-off between reduced statistical complexity and the localization error introduced by truncating long-range dependencies. Under realistic sample size scaling, we show both theoretically and numerically that a moderate localization radius can balance the statistical and localization errors, yielding better overall performance. Localized structure also facilitates parallel training, making localized diffusion models potentially more efficient for large-scale applications.

2026-05-07 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Nuclear Magnetic Resonance (NMR) spectroscopy is one of the most powerful and widely used tools for molecular structure elucidation in organic chemistry. However, the interpretation of NMR spectra to determine unknown molecular structures remains a labor-intensive and expertise-dependent process. Here, we present NMR-Solver, a practical and interpretable framework for the automated determination of small organic molecule structures from 1D NMR spectra. Our method introduces an automated framework for molecular structure elucidation, integrating large-scale spectral matching with physics-guided molecular optimization that exploits atomic-level structure spectrum relationships in NMR. Evaluated on literature data and real-world experiments, NMR-Solver shows strong generalization, robustness, and practical utility in real-life scenarios. By integrating computational NMR analysis, deep learning, and interpretable chemical reasoning into a unified system, it facilitates a scalable, automated, and chemically meaningful solution for inverse problems in molecular science.

2026-05-14 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand This report presents a series of works focusing on first-order bilevel optimization, specifically examining their computability, lower complexity bounds, and the development of near-optimal algorithms across various settings. The content is organized into three primary contributions: (1) In the first part of this report, we start by discussing which classes of problems have computable stationary points even in infinite dimensions and which do not. Specifically, while finding a stationary point is widely known to be tractable for nonconvex-strongly-convex (NC-SC) problems, we demonstrate that it can become intractable in the absence of lower-level strong convexity. (2) In the second part, we establish lower complexity bounds for all first-order algorithms to find stationary points of NC-SC problems. (3) Finally, we introduce how to design efficient algorithms to reach these lower bounds under both the deterministic and stochastic settings, respectively.

2026-05-21 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this work, we develop a numerical homogenization approach for the fully nonlinear Landau-Lifshitz equation with rough coefficients, including non-periodicity and nonseparable scales. Direct fine-mesh resolution of such multiscale problems incurs prohibitive computational costs. To address this, we propose an efficient coarse-scale approximation using localized basis functions derived from energy minimization within the GRPS/LOD framework. These basis functions preserve critical multiscale features on a tractable coarse mesh. The nonlinear, vectorial, non-symmetric nature of the Landau-Lifshitz equation necessitates careful design of variational formulations for basis construction. We introduce several such formulations, each tailored to specific structural aspects. Systematic numerical experiments demonstrate that our approach achieves significant computational savings without compromising accuracy, offering a robust framework for simulating multiscale magnetic systems with complex microstructures. The theoretical foundation is established via systematic error analysis, decomposing the total error into temporal and spatial components. The spatial error analysis is conducted by formulating the LL equation within the LOD framework.