Abstract : Structured eigenvalue problems and nonlinear equations arise from many applications including 3D phononic crystals, medical image processing and phase retrieval. Their structure reflects certain physical properties that need to be preserved during calculations, posing a huge challenge for computational scientists. In recent years, many new methods and algorithms have been achieved, such as Jacobi–Davidson type algorithms, Newton-Noda iteration, GPR parameter prediction method, and multi-symplectic block-Lanczos methods. And theoretical analysis receives new progress on generalized orthogonal flow, nonlinear energy minimization and Davis-Kahan theorem. In this minisymposia, the invited speakers will present their recent advances about such interesting subjects.
Organizer(s) : Zhigang Jia, Yanfei Jing, Yuan Lei, Tiexiang Li
[01505] Eigen-decomposition and Fast Solvers for Maxwell's Equations for 3D Photonic Crystals
Format : Talk at Waseda University
Author(s) :
Tiexiang Li (Southeast University)
Heng Tian (Sichuan University of Science and Engineering)
Xing-Long Lyu (Southeast University)
Wen-Wei Lin (National Yang Ming Chiao Tung University)
Abstract : In this article, we propose the Fast Algorithms for Maxwell's Equations FAME package for solving Maxwell's equations for modeling three-dimensional photonic crystals. FAME combines the null-space free method with fast Fourier transform FFT-based matrix-vector multiplications to solve the generalized eigenvalue problems GEPs arising from the oblique Yee's discretization. Numerical results demonstrate the potential of our proposed package to enable large-scale numerical simulations for novel physical discoveries and engineering applications of photonic crystals.
[01246] Projected Gradient Method for Volume-Measure-Preserving Optimal Mass Transportation Problems
Format : Talk at Waseda University
Author(s) :
Tsung-Ming Huang (National Taiwan Normal University)
Wei-Hung Liao (National Yang Ming Chiao Tung University)
Wen-Wei Lin (National Yang Ming Chiao Tung University)
Mei-Heng Yueh (National Taiwan Normal University)
Shing-Tung Yau (Tsinghua University)
Abstract : Volumetric stretch energy minimization (VSEM) has been widely applied to the computation of volume-/mass-preserving parameterizations of simply connected tetrahedral mesh models. In this talk, based on the VSEM algorithm, we propose a projected gradient method for the computation of the volume/mass-preserving optimal mass transport map with a guaranteed convergence rate of O(1/m). Numerical experiments are presented to justify the theoretical convergence behavior for various examples drawn from known benchmark models. Moreover, these numerical experiments show the effectiveness and accuracy of the proposed algorithm, particularly in the processing of 3D medical MRI brain images.
[01095] Multitask kernel-learning Gaussian process regression parameter prediction method and its application in matrix splitting iteration methods
Format : Talk at Waseda University
Author(s) :
Juan Zhang (Xiangtan University)
Abstract : In this talk, we present a multitask kernel-learning Gaussian process regression (GPR) parameter prediction method, and apply it in matrix splitting iteration methods. The first application is developing a general alternating direction implicit (ADI) framework, which can put most existing ADI methods into a unified framework and offer more new ADI approaches. Using the GPR method, the splitting parameter selection of this framework can be solved. The second application is for solving time-dependent linear systems, we give a new matrix splitting Kronecker product method, which can unify the existing Kronecker product schemes. Using the Multitask kernel-learning method, simultaneous multiple splitting parameters prediction and data-driven kernel learning can all be achieved. Based on Bayesian inference, the GPR method only requires a small training data set for learning the regression prediction mapping, and has sufficient accuracy and high generalization capability. We apply our developed methods to solving (time-dependent) convection-diffusion equations and (differential) Sylvester matrix equations. Numerical results illustrate our methods can solve large sparse linear systems more efficiently compared with existing methods.
[01322] Breakdown Avoidance Structure-Preserving Doubling Algorithms for Nonlinear Matrix Equations
Format : Talk at Waseda University
Author(s) :
Yueh-Cheng Kuo (National university of Kaohsiung)
Abstract : Structure-Preserving Doubling Algorithms (SDAs) are efficient algorithms for solving Riccati-type matrix equations. However, the breakdown may happen in the SDA. To avoid the breakdown, we first introduce the class of $\Omega$-symplectic forms, $\Omega$-SF, consisting of symplectic matrix pairs with Hermitian parametric matrix $\Omega$. Based on the $\Omega$-SF, we developed modified SDAs, called MSDAs, for solving the Riccati-type equations. The MSDA generates a sequence of symplectic matrix pairs in $\Omega$-SF and can avoid the breakdown by employing a suitable Hermitian matrix $\Omega$. In addition, we show that the Hermitian matrix $\Omega$ in MSDAs can be chosen as a real diagonal matrix which can reduce the computational complexity. In this case, MSDA and SDA have the same computational complexity.
[01293] Newton-Noda iteration for nonlinear eigenvalue problems
Format : Talk at Waseda University
Author(s) :
Ching-Sung Liu (National University of Kaohsiung)
Abstract : In this talk, we will introduce nonlinear eigenvalue problems, including tensor eigenvalue problems and nonlinear Schrödinger equations. We present a Newton-Noda iteration (NNI) to find positive eigenvectors of nonlinear problems. A great advantage of this method is that it converges quadratically and maintains positivity, i.e., the vector approximating the Perron vector (or ground state vector) is strictly positive in each iteration.
[01584] Phase Retrieval of Quaternion Signal via Wirtinger Flow
Format : Talk at Waseda University
Author(s) :
Junren Chen (University of Hong Kong)
Michael Kwok-Po Ng (University of Hong Kong)
Abstract : Quaternion phase retrieval (QPR) is concerned with the recovery of quaternion signal from magnitude of quaternion
linear measurements. We develop the scalable algorithm quaternion Wirtinger flow (QWF) for solving QPR, and establish its linear convergence guarantee. We develop a variant of QWF that can effectively utilize a pure quaternion priori by incorporating a quaternion phase factor estimate into QWF iterations.
[01454] Numerical Methods for the Complete Solution of Multiparameter Eigenvalue Problems
Format : Online Talk on Zoom
Author(s) :
Bo Dong (Dalian University of Technology)
Abstract : The linear/nonlinear multiparameter eigenvalue problems appear frequently in the study of multiparameter Sturm-Liouville problems and delay differential equations with some delays. In this talk, I will introduce two classes of numerical methods for solving the multiparameter eigenvalue problems: the method of transforming to silmultaneous eigenvalue problems and the homotopy method. The former is suitable for small-scale and medium-scale problems while the latter tends to be more effective in terms of speed, accuracy and memory storage as the problem size grows. Numerical experiments and applications are presented to show the efficiency of these two classes of methods.
[02763] Perturbation theory for the symmetry eigenvalue problem and singular value decomposition followed by deflation techniques
Format : Online Talk on Zoom
Author(s) :
Xiang Wang (Nanchang University)
Hongjia Chen (Nanchang University)
Abstract : The calculation of the dominant eigenvalues of a symmetric matrix $A$ together with its eigenvectors, followed by the calculation of the deflation of $A_1 = A - \rho U_kU_k^T$ corresponds to one step of the Wielandt deflation technique, where $\rho$ is a shift and $U_k$ are eigenvectors of $A$. In this paper, we investigate how the eigenspace of $A_1$ changes when $A_1$ is perturbed to $\tilde{A}_1 = A - \rho \tilde{U}_k\tilde{U}_k^T$, $\tilde{U}_k$ are approximate eigenvectors of $U_k$. We establish the bounds
for the angle of eigenspaces of $A_1$ and $\tilde{A}_1$ based on the Davis-Kahan theorem.
Moreover, in the practical implementation for singular value decomposition, once one or several singular triplets converge to a preset accuracy, they should be deflated by $B_1 = B - \gamma W_kV_k^H$ with $\gamma$ is a shift and $W_k$ and $V_k$ are singular vectors so that they will not be re-computed.
We investigate how the singular subspaces of $B_1 = B - \gamma WV^H$ changes when $B_1$ is perturbed to $\tilde{B}_1 = B - \gamma \widetilde{W}\widetilde{V}^H$, $\widetilde{W}$ and $\widetilde{V}^H$ are approximate singular vectors.
We also establish the bounds
for the angle of singular subspaces of $B_1$ and $\tilde{B}_1$ based on the Wedin theorem.
We show, by numerical experiment, the angles of eigenspaces
and the angles of singular subspaces are well--predicted by
these bounds with an appropriable shift.
[01112] Nonlinear Energy Minimization for Simplicial Manifold Parameterizations
Format : Talk at Waseda University
Author(s) :
Mei-Heng Yueh (National Taiwan Normal University)
Abstract : The parameterization problem aims to efficiently compute a diffeomorphism between a simplicial manifold and a specified canonical domain. In this talk, I will introduce the theoretical and computational aspects of nonlinear energy minimization for the computation of parameterizations of simplicial manifolds that preserve the specified geometry property. Applications to manifold registration and image processing will be demonstrated to show the effectiveness of parameterizations.
[01534] Two harmonic Jacobi–Davidson methods for computing a partial GSVD of a large matrix pair
Format : Talk at Waseda University
Author(s) :
Jinzhi Huang (Soochow University)
Zhongxiao Jia (Tsinghua University)
Abstract : Two harmonic extraction-based Jacobi–Davidson (JD) type algorithms are proposed to compute a partial generalized singular value decomposition (GSVD) of a large regular matrix pair. They are called cross product-free (CPF) and inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and IF-HJDGSVD, respectively. Compared with the standard extraction-based JDGSVD algorithm, the harmonic extraction-based algorithms converge more regularly and suit better for computing GSVD components corresponding to interior generalized singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some deflation and purgation techniques are developed to compute more than one GSVD components. Numerical experiments confirm the superiority of CPF-HJDGSVD and IF-HJDGSVD to the standard extraction-based JDGSVD algorithm.
[01473] Harmonic multi-symplectic Lanczos algorithm for quaternion singular triplets
Format : Talk at Waseda University
Author(s) :
XUAN LIU (University of Macau)
Zhigang Jia (Jiangsu Normal University)
Jingfei Zhu (Jiangsu Normal University)
Meixiang Zhao (Jiangsu Normal University)
Abstract : The computation of quaternion singular triplets has become one of the core targets of color image processing. A novel harmonic multi-symplectic Lanczos algorithm is presented for approximating extreme quaternion singular triplets. The underlying theory is to preserve an algebraic structure during the partial bidiagonalization, the argumentation, and the restarted bidiagonalization. The proposed algorithm is successfully applied to color video semantic segmentation.
[01319] The asymptotic analysis of generalized orthogonal flows
Format : Talk at Waseda University
Author(s) :
Shih-Feng Shieh (National Taiwan Normal University)
Abstract : This talk is concerned with the matrix differential equation approximating the k-dimensional dominant eigenspace of a matrix. The solution of the matrix differential equation is orthogonal and is called generalized orthogonal flow. The existence and uniqueness of the generalized orthogonal flow are guaranteed for all time tЄ R. An orthogonal flow, which has the shortest arc-length, has been constructed and is called the best path of generalized orthogonal flows. We show that the best path is Oja's flow. We also analyze the asymptotic behaviors and the convergence rate of the best path.
