Applied Math
PhD Seminar

2022 Fall

Due to the pandemic outbreak, most of the talks in the preceding semester were canceled. The seminar of this semester is again organized by Xiang Li and Nian Shao, and co-organized by the graduate student union in the School of Mathematical Sciences at Fudan.

Past Presentations

2022-09-22 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Accurate spatio-temporal traffic data is crucial to intelligent transportation systems. Missing traffic data is an important problem to solve. Low-rank matrix completion provides an effective way to find the missing data. The completion aims to obtain a low-rank matrix that can approximate the known entries as far as possible. Meanwhile, some linear constraint marginal information of the matrix can also be observed in the real application. In this paper, we utilize such marginal information to largely improve the performance of common matrix completion algorithms and propose an alternating direction method of multipliers (ADMM) and conjugate gradient descent method (CGD) based SoftImpute alternative least square (ALS) algorithm. We analyze their convergence rates and prove that the model can always converge to a first-order stationary point. We also utilize ADMM and CGD to largely accelerate the subproblem and make its complexity of each iteration at the same level as the popular SoftImpute-ALS matrix completion algorithm. Furthermore, this algorithm can be used in distributed computation, suitable for large-scale problems. In the numerical experiments, we demonstrate its outstanding matrix completion performance and high speed in several traffic matrix datasets.

2022-09-29 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand To solve the linear Hermitian extreme eigenpairs problem arising from the Full Configuration Interaction (FCI) method in electronic structure calculation, an unconstrained optimization model called weighted trace-penalty minimization is proposed. The main results that the corresponding global minimizers are the blocks of eigenvectors instead of invariant subspace indicates that the cost of orthonormalization or Rayleigh-Ritz process could be saved. Due to the characteristics of extremely large scale and "sparsity" of eigenvectors, solving the minimization problem by the coordinate descent algorithm accelerates the convergence and saves the storage of data in comparison with the gradient methods, making the cost affordable.

2022-10-13 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand High-order numerical schemes and adaptive time strategies are widely used to solve PDEs. In this paper, a BDF2 scheme with variable time steps is proposed and analyzed for the Cahn-Hilliard equation with a logarithmic Flory-Huggins energy potential. To guarantee the unique solvability and positivity-preserving property, the lumped mass method is adopted in the finite element space discretization. For energy stability analysis, two modified energy dissipation laws are derived and compared if the maximum step ratio has a finite upper bound. To estimate the spatial and temporal errors separately, a spatially semi-discrete scheme is proposed, and a new elliptic projection is introduced, and the super-closeness between this projection and the Ritz projection of the exact solution is attained. Based on the strict separation property of the numerical solution obtained by using the technique of combining the rough and refined error estimates, the convergence analysis in $\ell^{\infty}(0,T;L_h^2(\Omega))$ norm is established when $\tau\le Ch$ by using the technique of the DOC kernels. Finally, several numerical experiments are carried out to validate the theoretical results.

2022-10-20 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Sparse optimization problems arise from a remarkable variety of applications including machine learning, signal and image processing, data science, medical imaging, compressed sensing, computer vision, statistical regression, computational biology, and so on. In this paper, we mainly focus on the numerical methods for solving the sparse least squares regression problem with probabilistic simplex constraint in various applications such as computational biology and hyperspectral unmixing and the sparse stochastic matrix factorization in document clustering. We reformulate the least squares regression problem as a nonconvex and nonsmooth $\ell_1$ regularized minimization problem over the unit sphere and present a geometric proximal gradient method for solving the regularized problem. We provide explicit expression of the global solutions to the involved subproblems of our method and give the global convergence analysis. We also consider a special nonnegative matrix factorization, the sparse stochastic matrix factorization. We directly apply the $\ell_0$ constraint to measure the sparseness in the sparse stochastic matrix factorization. Based on the given factorization rank and the prescribed sparsity level, the considered sparse stochastic matrix factorization is reformulated as an unconstrained nonvonvex-nonsmooth minimization problem and a column-wise update algorithm is introduced. We combine the alternating minimization method with the proximal alternating linearized minimization method to update the factorization factors. Numerical experiments on both synthetic and real data sets are given to demonstrate the proposed algorithm is effective.

2022-10-27 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation in the solitonic regions. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem in the new scale $(y,t)$. Further using the $\overline\partial$ generalization of Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(y,t)$ in different space-time solitonic regions of $\xi=y/t$. We divide the half-plane $\{ (y,t): -\infty \lt y \lt \infty, \ t \gt 0\}$ into four asymptotic regions: The phase function $\theta(z)$ has no stationary phase point on the jump contour in the space-time solitonic regions $\xi \in(-\infty,-1/4)\cup(2,+\infty)$, corresponding asymptotic approximations can be characterized with an $N(\Lambda)$-solitons with diverse residual error order $\mathcal{O}(t^{-1+2\rho})$; The phase function $\theta(z)$ has four phase points and eight phase points on the jump contour in the space-time solitonic regions $\xi \in(0,2)$ and $\xi \in(-1/4,0)$, respectively. The corresponding asymptotic approximations can be characterized with an $N(\Lambda)$-soliton and an interaction term between soliton solutions and the dispersion term with diverse residual error order $\mathcal{O}(t^{-3/4})$. Our results also confirm the soliton resolution conjecture and asymptotically stability of the N-soliton solutions for the mCH equation.

2022-11-10 16:10:00 - 17:00:00 @ Tencent Meeting @ 526-3788-4054 [poster]

Abstract: Click to expand We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schr\"odinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schr\"odinger equation. Comparison between the Schr\"odinger and the Liouville framework will also be made.

2022-11-24 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Radiation magnetohydrodynamics (RMHD) systems couple the ideal magnetohydrodynamics equations with a gray radiation transfer equation. The main challenge is that the radiation travels at the speed of light while the magnetohydrodynamics changes with the time scale of the fluid. The time scales of these two processes can vary dramatically. In order to use mesh sizes and time steps that are independent of the speed of light, some asymptotic preserving (AP) schemes in both space and time are desired. We develop an AP scheme in both space and time for the RMHD system. The performance of the semi-implicit method is presented, for both optically thin and thick regions, as well as for the radiative shock problem. Comparisons with the semi-analytic solutions are given to verify the accuracy and asymptotic properties of the method.

2022-12-01 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We establish a sharp uniform-in-time error estimate for the Stochastic Gradient Langevin Dynamics (SGLD), which is a popular sampling algorithm. Under mild assumptions, we obtain a uniform-in-time $O(\eta^2)$ bound for the KL-divergence between the SGLD iteration and the Langevin diffusion, where $\eta$ is the step size (or learning rate). Our analysis is also valid for varying step sizes. Based on this, we are able to obtain an $O(\eta)$ bound for the distance between the SGLD iteration and the target distribution, in terms of Wasserstein or total variation distances. Moreover, via the technique of reflection coupling, we prove the geometric ergodicity of the SGLD algorithm under $W_1$ distance without global convexity.

2022-12-08 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand In this talk, we focus on Lipschitz bandit problems with batched feedback, where the expected reward is Lipschitz and the reward observations are communicated to the player in batches. We introduce a novel landscape-aware algorithm, called Batched Lipschitz Narrowing (BLiN), that optimally solves this problem. Specifically, we show that for a $T$-step problem with Lipschitz reward of zooming dimension $d_z$, our algorithm achieves theoretically optimal (up to logarithmic factors) regret rate $\widetilde{\mathcal{O}}\left(T^{\frac{d_z+1}{d_z+2}}\right)$ using only $ \mathcal{O} \left( \log\log T\right) $ batches. We also provide complexity analysis for this problem. Our theoretical lower bound implies that $\Omega(\log\log T)$ batches are necessary for any algorithm to achieve the optimal regret. Thus, BLiN achieves optimal regret rate using minimal communication.

2022-12-15 16:10:00 - 17:00:00 @ Tencent Meeting @ 526-3788-4054 [poster]

Abstract: Click to expand Many problems in data and imaging sciences can be formulated as composite optimization problems, for example, graph-guided Lasso, Computed Tomography (CT) reconstruction, etc. The primal dual fixed point (PDFP) method is one of the most popular algorithms for solving this class of problems. The paper is devoted to extending the PDFP from two perspectives. Firstly, to solve large-scale problems, by combining the stochastic gradient and stochastic variance reduced gradient (SVRG) with PDFP, we propose stochastic PDFP (SPDFP) and SVRG-PDFP. Secondly, to accelerate the convergence speed of solving medium-size problems, we import the inertial term into the update of PDFP and propose inertial PDFP (iPDFP). The convergence and convergence rate of the algorithms are provided under some standard assumption. The effectiveness of the algorithms is validated by several examples, such as graph-guided logistic regression, 2D and 3D CT reconstruction. Furthermore, the proposed three algorithms can be seen as generalizations of Proximal Stochastic Gradient Descent (Prox-SGD), Proximal Stochastic Variance Reduced Gradient (Prox-SVRG), and Fast Iterative Shrinkage-Thresholding Algorithm (FISTA).

2022-12-22 16:10:00 - 17:00:00 @ Tencent Meeting @ 526-3788-4054 [poster]

Abstract: Click to expand With the development of machine learning, neural network in solving PDEs, such as DeepRitz and PINN, is becoming rather popular. However (on physical problems less than 3 dimensions), its performance is worrying: in practice, there are few examples comparable to traditional numerical methods; in theory, there is a lack of stability and convergence proofs. In this report, we discuss the performance of methods such as PINN on problems with discontinuities coefficients. First, for elliptic PDEs with discontinuous coefficients, we show experimentally that PINN cannot successfully approximate the true solution of the equation. By introducing auxiliary equations, we prove that the fitting of PINN to the solutions of elliptic equations with discontinuous coefficients is invalid. Furthermore, we point out that there are still a series of patterns in this failure, which means that the PINN method has implicit regularization on such problems. Finally, we extend all results to quasilinear case.