Applied Math
PhD Seminar

2022 Spring

The seminar of this semester is 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

2022-03-03 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand We propose a new method to calculate band structure of quantum systems. It constructs the quasi-Hamiltonian matrix using several eigenvalues and corresponding eigenvectors at the uniform k-point grids, then performing Fourier interpolation to obtain the quasi-Hamiltonian at any k-point. We find some inversible transformations make the quasi-Hamiltonian matrix decays faster in real space, which generates more accurate band structures. The decay properties of Hamiltonian transformation associated with both sparse and dense Hamiltonian matrices can be described by an inequality using approximation theory and matrix analysis. In the sparse matrix case, we can simplify the inequality and estimate the decay property analytically. In the dense matrix case, the inequality is more complicated and is studied numerically.

2022-03-10 16:10:00 - 17:00:00 @ Rm 1801, Guanghua East Tower [poster]

Abstract: Click to expand Domain decomposition (DD) methods are very popular for solving linear systems. Recently, some DD methods for eigenvalue problems are proposed. In this talk, we will revisit Newton-Schur (NS) method, an algebraic DD method for symmetric eigenvalue problems, and study it in Hilbert space. As a Newton method, we will present some sufficient conditions for the quadratic convergence of NS method. For symmetric elliptic eigenvalue problems discretized by the standard finite element method and non-overlapping DD method, we will show that the rate of convergence is $$\epsilon_{N}\leq C \epsilon^{2},$$ where $C$ is a constant independent of mesh sizes, $\epsilon_{N}$ and $\epsilon$ are errors of the approximated eigenvalue after and before one iteration, respectively. Furthermore, we will also introduce some first order iterative methods for symmetric elliptic eigenvalue problems based on DD methods. Estimations for the rate of convergence will also be covered.

2022-04-07 16:10:00 - 17:00:00 @ Tencent Meeting ID 860-509-894 | Passcode 200433 [poster]

Abstract: Click to expand This talk is concerned with the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. Our most general result is a perturbation bound of the signal space in terms of the minimum singular value of Fourier matrices. When the point sources are located in several separated clumps, we provide an explicit upper bound of the noise-space correlation perturbation error in MUSIC and the support error in ESPRIT in terms of a Super-Resolution Factor (SRF). The upper bound for ESPRIT is then compared with a new Cramér-Rao lower bound for the clumps model. As a result, we show that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems. Joint work with Weilin Li, Weiguo Gao and Wenjing Liao.

2022-04-14 16:10:00 - 17:00:00 @ Tencent Meeting ID 474-696-334 [poster]

Abstract: Click to expand The distance to uncontrollability is an important measure in classical control theory. For a linear control system $(A,B)$, the distance to uncontrollability is \begin{equation*} \tau(A,B)=\min\limits_{z\in\mathbb{C}}\sigma_{\min}\bigl([A-z I,B]\bigr). \end{equation*} Historical methods of this problem take much time and have numerical problems when the system is nearly uncontrollable. We give a new method to compute the distance to uncontrollability. Our method combines Boyd-Balakrishnan method, shift and BFGS method. The key point of our method is to shift the parameter by eigenvalue perturbation theory, which reduces the computation cost and provides reliable result. Numerical experiments show that our method is much faster than the latest method, and in some cases, our computed global minimum is smaller than his computation result, i.e., our algorithm is more reliable.