Applied Math
PhD Seminar

2023 Fall

The seminar of this semester is organized by Yuer Chen and Jiaxin Jiang, and co-organized by the graduate student union in the School of Mathematical Sciences at Fudan. This section is partially sponsored by Lei Shi.

Past Presentations

2023-09-07 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, I will give some results about the universal approximation property of deep neural networks. We focus on the continuous-time ResNet, and using tools from control theory. This enables us to discuss the expressive power of neural networks by its depth. I will also discuss the generalization on neural networks with symmetry.

2023-09-14 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand SpMV is a fundamental kernel in high-performance computing. It is a core workload in iterative solvers (one of the seven dwarfs of scientific computing and engineering), data analysis, and graph computing. Over the past 50 years, a lot of research has been conducted on this kernel, and a lot of sparse matrix formats and their corresponding SpMV algorithms have been proposed. However, it is not easy to design a high-performance SpMV program because of the diversity of sparse matrix features and the sensitivity of program design to sparse matrix features. For this reason, we propose AlphaSparse. AlphaSparse aims to bypass the limitations of human observation and practice, directly giving the sparse matrix format and SpMV algorithm designed by machine, which is suitable for this matrix. Driven by the new SpMV program design method, AlphaSparse achieves more than 200% improvement on NVIDIA GPUs, while the annual improvement of SpMV programs is less than 5%.

2023-09-21 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, I want to introduce a splitting Hamiltonian Monte Carlo (SHMC) algorithm, which can be computationally efficient when combined with the random mini-batch strategy. By splitting the potential energy into numerically nonstiff and stiff parts, one makes a proposal using the nonstiff part, followed by a Metropolis rejection step using the stiff part that is often easy to compute. The splitting allows efficient sampling from systems with singular potentials (or distributions with degenerate points) and/or with multiple potential barriers. We also use random batch strategies to reduce the computational cost in generating the proposals for problems arising from many-body systems and Bayesian inference, and estimate both the strong and the weak errors in the Hamiltonian induced by the random batch approximation.

2023-09-28 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The consistent splitting schemes for the Navier-Stokes equations decouple the computation of pressure and velocity, and do not suffer from the splitting error. However, only the first-order version of the consistent splitting schemes is proven to be unconditionally stable for the time dependent Stokes equations. We construct a new class of consistent splitting schemes of orders two to four for Navier-Stokes equations based on Taylor expansions at time $t_{n+k}$ where $k\ge 1$ is a tunable parameter. By choosing suitable $k$, we construct, for the very first time, unconditionally stable and totally decoupled schemes of orders two to four for the velocity and pressure, and provide rigorous optimal error estimates. We shall also present some numerical results to show the computational advantages of these schemes. This is a joint work with Jie Shen.

2023-10-12 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We consider Klein-Gordon equations with cubic nonlinearity in three spatial dimensions. It is assumed that the corresponding Klein-Gordon operator admits an arbitrary number of possibly degenerate eigenvalues in $(0, m)$, and hence the unperturbed linear equation has multiple time-periodic solutions known as bound states. In 1999, Soffer and Weinstein discovered a mechanism called Fermi's Golden Rule for this nonlinear system in the case of one simple but relatively large eigenvalue $\Omega \in (\frac{m}{3},m)$, by which energy is transferred from discrete to continuum modes and the solution still decays in time. Since then, many efforts have been made in the case of relatively small eigenvalue, in which Fermi's golden rule fails, and the case of general multiple eigenvalue. In 2022, we solved the general one simple eigenvalue case. In our recent work, we solved this problem in full generality: multiple and simple or degenerate eigenvalues in $(0,m)$. Indeed, we obtained the sharp rate of energy transfer from one discrete state to continuum modes in the most general case.

2023-10-19 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, based on a proximal gradient method, we present a Riemannian proximal quasi-Newton method, named ManPQN, to solve the composite optimization problems over the Stiefel manifold. The global convergence of the ManPQN method is proved and iteration complexity for obtaining an $\epsilon$-stationary point is analyzed. Under some mild conditions, we also establish the local linear convergence result of the ManPQN method. Numerical results are encouraging, which show that the proximal quasi-Newton technique can be used to accelerate the proximal gradient method.

2023-10-26 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand The aversion to noise and the discomfort stemming from its uncertainties are commonplace. However, it is worth noting that, in reality, noise can be harnessed as a valuable tool for guiding systems towards their desired states. Recent years have witnessed a burgeoning body of research that delves into the art of using noise to attain stability, synchronization, and an array of control objectives in systems. In this forthcoming presentation, we will explore the intricate landscape of employing diverse types of noise to achieve distinct degrees of control over stochastic dynamic systems under varying conditions.

2023-11-02 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Understanding the ecological mechanisms associated with the collapse and restoration is especially critical in promoting harmonious coexistence between humans and nature. So far, it remains challenging to elucidate the mechanisms of stochastic dynamical transitions for ecological systems. Using an example of plant-pollinator network, we quantified the energy landscape of ecological system. The landscape displays multiple attractors characterizing the high, low and intermediate abundance stable states. Interestingly, we detected the intermediate states under pollinator decline, and demonstrated the indispensable role of the intermediate state in state transitions. From the landscape, we define the barrier height (BH) as a global quantity to evaluate the transition feasibility. We propose that the BH can serve as a new early-warning signal (EWS) for upcoming catastrophic breakdown, which provides an earlier and more accurate warning signal than traditional metrics based on time series. Our results promote developing better management strategies to achieve environmental sustainability.

2023-11-09 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In the numerical simulation of highly-oscillatory wave fields, direct numerical methods, such as finite-difference or finite-element methods, may suffer from dispersion or pollution errors so that such methods require an enormous computational grid to resolve these oscillations. Alternative methods, such as geometrical-optics based asymptotic methods, have been sought to resolve these highly-oscillatory wave phenomena, where how to solve the caustics phenomenon in geometric optics becomes a challenging problem. We propose a novel Hadamard integrator for the self-adjoint time-dependent wave equation in an inhomogeneous medium. We create a new asymptotic series based on the Gelfand-Shilov function, dubbed Hadamard's ansatz, to approximate the Green's function of the wave equation. Incorporating the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation, we develop an original Hadamard integrator for the Cauchy problem of the time-dependent wave equation and derive the corresponding Lagrangian formulation in geodesic polar coordinates. Equipped with low-rank representations, we apply the Hadamard integrator to efficiently solve time-dependent wave equations with highly oscillatory initial conditions. By judiciously choosing the medium-dependent time step, our new Hadamard integrator can propagate wave field beyond caustics implicitly and advance spatially overturning waves in time naturally.

2023-11-16 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Cross-species comparative analyses of single-cell RNA sequencing (scRNA-seq) data allow us to explore, at single-cell resolution, the origins of the cellular diversity and evolutionary mechanisms that shape cellular form and function. Cell-type assignment is a crucial step to achieve that. However, the poorly annotated genome and limited known biomarkers hinder us from assigning cell identities for nonmodel species. Here, we design a heterogeneous graph neural network model, CAME, to learn aligned and interpretable cell and gene embeddings for cross-species cell-type assignment and gene module extraction from scRNA-seq data. CAME achieves significant improvements in cell-type characterization across distant species owing to the utilization of non-one-to-one homologous gene mapping ignored by early methods. Our large-scale benchmarking study shows that CAME significantly outperforms five classical methods in terms of cell-type assignment and model robustness to insufficiency and inconsistency of sequencing depths. CAME can transfer the major cell types and interneuron subtypes of human brains to mouse and discover shared cell-type-specific functions in homologous gene modules. CAME can align the trajectories of human and macaque spermatogenesis and reveal their conservative expression dynamics. In short, CAME can make accurate cross-species cell-type assignments even for nonmodel species and uncover shared and divergent characteristics between two species from scRNA-seq data.

2023-11-23 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we observe an interesting phenomenon for a hybridizable discontinuous Galerkin (HDG) method for eigenvalue problems. Specifically, using the same finite element method, we may achieve both upper and lower eigenvalue bounds simultaneously, simply by the fine tuning of the stabilization parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence rate at a lower computational cost. Meanwhile, we demonstrate that certain type of HDG methods can only provide upper bounds. As a by-product, the asymptotic upper bound property of the Brezzi-Douglas-Marini mixed finite element is also established. Numerical results supporting our theory are given. Extension to WG method will also be mentioned.

2023-11-30 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we introduce the convergence properties of the projected policy gradient (PPG) method. We demonstrate that this method indeed achieves the exact convergence in a finite number of iterations for any given constant step size. To establish this result, we first establish the sublinear convergence of PPG for an arbitrary fixed step size, which is also new, to the best of knowledge. The finite iteration convergence property is also applicable to a preconditioned version of PPG, namely the projected Q-ascent (PQA) method. Additionally, the linear convergence of PPG and its equivalence to PI are established under the non-adaptive increasing step sizes and the adaptive step sizes, respectively.

2023-12-07 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we introduce a contour integral-based algorithm for computing a few generalized singular values/vectors of a pair of matrices. Although SVD and GSVD are special cases of symmetric eigenvalue problems, additional care is required in order to apply the FEAST solver to SVD/GSVD. We propose an effective and robust projection scheme for the FEAST solver by analyzing several plausible candidates. Both theoretical analysis and numerical experiments confirm the effectiveness of our algorithm.

2023-12-14 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this presentation, we focus on the matrix computation problems in the power grid analysis of high-resolution flat panel displays (FPD). Throughout the simulation of FPD, we find that the challenge is how to efficiently solve the billion-order bordered multilevel block Toeplitz linear systems. To address this challenge, we first discuss the properties and the solver for multilevel block Toeplitz linear systems. Then, we extend the corresponding results to the case of bordered multilevel block Toeplitz linear systems. Finally, we will conclude all these findings and provide a comprehensive overview of the structured simulation process for FPD.

2023-12-28 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand While Mixed-integer linear programming (MILP) is NP-hard in general, practical MILP has received roughly 100-fold speedup in the past twenty years. Still, many classes of MILPs quickly become unsolvable as their sizes increase, motivating researchers to seek new acceleration techniques for MILPs. With deep learning, they have obtained strong empirical results, and many results were obtained by applying graph neural networks (GNNs) to making decisions in various stages of MILP solution processes. We study the theoretical foundation and discover a fundamental limitation: there exist feasible and infeasible MILPs that all GNNs will, however, treat equally, indicating GNN's lacking power to express general MILPs. Then we show that linear programs (LPs) without integer constraints do not suffer from this limitation and that, by restricting the MILPs to unfoldable ones or by adding random features, there exist GNNs that can reliably predict MILP feasibility, optimal objective values, and optimal solutions up to prescribed precision. We conduct small-scale numerical experiments to validate our theoretical findings. This talk is based on joint works with Jialin Liu, Xinshang Wang, Jianfeng Lu, and Wotao Yin.