arXiv subjects

Zhen-Chen Guo

Publications and source records attributed to Zhen-Chen Guo.

13 recordsLinked to original sources

arXiv2025

Flexible fixed-point iteration and its applications for nonsymmetric algebraic Riccati equations

In this paper, we reveal the intrinsic Toeplitz structure in the unique stabilizing solution for nonsymmetric algebraic Riccati equations by employing a shift-involved fixed-point iteration, and propose an RADI-type method for computing this solution for large-scale equations of this type with sparse and low-rank structure by incorporating flexible shifts into the fixed-point iteration. We present a shift-selection strategy, termed Leja shifts, based on rational approximation theory, which is incorporated into the RADI-type method. We further discuss important implementation aspects for the method, such as low-rank factorization of residuals, implicit update of large-scale sparse matrices, real arithmetics with complex shifts, and related equations of other type. Numerical experiments demonstrate the efficiency of both the proposed method and the introduced shift-selection strategy.

math.NA↗

arXiv2024

An RADI-type method for stochastic continuous-time algebraic Riccati equations

In this paper, we propose an RADI-type method for large-scale stochastic continuous-time algebraic Riccati equations with sparse and low-rank matrices. This new variant of RADI-type methods is developed by integrating the core concept of the original RADI method with the implicit appearance of the left semi-tensor product in stochastic continuous-time algebraic Riccati equations.The method employs different shifts to accelerate convergence and uses compression techniques to reduce storage requirements and computational complexity.Unlike many existing methods for large-scale problems such as Newton-type methods and homotopy method, it calculates the residual at a low cost and does not require a stabilizing initial approximation, which can often be challenging to find. Numerical experiments are provided to demonstrate its efficiency.

math.NA↗

arXiv2023

Stochastic algebraic Riccati equations are almost as easy as deterministic ones theoretically

Stochastic algebraic Riccati equations, also known as rational algebraic Riccati equations, arising in linear-quadratic optimal control for stochastic linear time-invariant systems, were considered to be not easy to solve. The-state-of-art numerical methods most rely on differentiability or continuity, such as Newton-type method, LMI method, or homotopy method. In this paper, we will build a novel theoretical framework and reveal the intrinsic algebraic structure appearing in this kind of algebraic Riccati equations. This structure guarantees that to solve them is almost as easy as to solve deterministic/classical ones, which will shed light on the theoretical analysis and numerical algorithm design for this topic.

math.OC↗

arXiv2022

The intrinsic Toeplitz structure and its applications in algebraic Riccati equations

In this paper we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the found form and fast Fourier transform, we propose a new algorithm for solving both discrete-time and continuous-time large-scale algebraic Riccati equations with low-rank structure. It works without unnecessary assumptions, complicated shift selection strategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting.

math.NA↗

arXiv2022

Nearly Optimal Stochastic Approximation for Online Principal Subspace Estimation

Principal component analysis (PCA) has been widely used in analyzing high-dimensional data. It converts a set of observed data points of possibly correlated variables into a set of linearly uncorrelated variables via an orthogonal transformation. To handle streaming data and reduce the complexities of PCA, (subspace) online PCA iterations were proposed to iteratively update the orthogonal transformation by taking one observed data point at a time. Existing works on the convergence of (subspace) online PCA iterations mostly focus on the case where sample are almost surely uniformly bounded. In this paper, we analyze the convergence of a subspace online PCA iteration under more practical assumption and obtain a nearly optimal finite-sample error bound. Our convergence rate almost matches the minimax information lower bound. We prove that the convergence is nearly global in the sense that the subspace online PCA iteration is convergent with high probability for random initial guesses. This work also leads to a simpler proof of the recent work on analyzing online PCA for the first principal component only.

math.OC↗

arXiv2020

Highly accurate decoupled doubling algorithm for large-scale M-matrix algebraic Riccati equations

We consider the numerical solution of large-scale M-matrix algebraic Riccati equations with low-rank structures. We derive a new doubling iteration, decoupling the four original iteration formulae in the alternating-directional doubling algorithm. We prove that the kernels in the decoupled algorithm are small M-matrices. Illumined by the highly accurate algorithm proposed by Xue and Li in 2017, we construct the triplet representations for the small M-matrix kernels in a highly accurate doubling algorithm. Illustrative numerical examples will be presented on the efficiency of our algorithm.

math.NA↗

arXiv2020

Bifurcation Analysis of the Eigenstructure of the Discrete Single-curl Operator in Three-dimensional Maxwell's Equations with Pasteur Media

This paper focuses on studying the bifurcation analysis of the eigenstructure of the $γ$-parameterized generalized eigenvalue problem ($γ$-GEP) arising in three-dimensional (3D) source-free Maxwell's equations with Pasteur media, where $γ$ is the magnetoelectric chirality parameter. For the weakly coupled case, namely, $γ< γ_{*} \equiv$ critical value, the $γ$-GEP is positive definite, which has been well-studied by Chern et.\ al, 2015. For the strongly coupled case, namely, $γ> γ_{*}$, the $γ$-GEP is no longer positive definite, introducing a totally different and complicated structure. For the critical strongly coupled case, numerical computations for electromagnetic fields have been presented by Huang et.\ al, 2018. In this paper, we build several theoretical results on the eigenstructure behavior of the $γ$-GEPs. We prove that the $γ$-GEP is regular for any $γ> 0$, and the $γ$-GEP has $2 \times 2$ Jordan blocks of infinite eigenvalues at the critical value $γ_{*}$. Then, we show that the $2 \times 2$ Jordan block will split into a complex conjugate eigenvalue pair that rapidly goes down and up and then collides at some real point near the origin. Next, it will bifurcate into two real eigenvalues, with one moving toward the left and the other to the right along the real axis as $γ$ increases. A newly formed state whose energy is smaller than the ground state can be created as $γ$ is larger than the critical value. This stunning feature of the physical phenomenon would be very helpful in practical applications. Therefore, the purpose of this paper is to clarify the corresponding theoretical eigenstructure of 3D Maxwell's equations with Pasteur media.

math.NA↗

arXiv2020

Decoupled Structure-Preserving Doubling Algorithm with Truncation for Large-Scale Algebraic Riccati Equations

In \emph{Guo et al, arXiv:2005.08288}, we propose a decoupled form of the structure-preserving doubling algorithm (dSDA). The method decouples the original two to four coupled recursions, enabling it to solve large-scale algebraic Riccati equations and other related problems. In this paper, we consider the numerical computations of the novel dSDA for solving large-scale continuous-time algebraic Riccati equations with low-rank structures (thus possessing numerically low-rank solutions). With the help of a new truncation strategy, the rank of the approximate solution is controlled. Consequently, large-scale problems can be treated efficiently. Illustrative numerical examples are presented to demonstrate and confirm our claims.

math.NA↗

arXiv2020

A decoupled form of the structure-preserving doubling algorithm with low-rank structures

The structure-preserving doubling algorithm (SDA) is a fairly efficient method for solving problems closely related to Hamiltonian (or Hamiltonian-like) matrices, such as computing the required solutions to algebraic Riccati equations. However, for large-scale problems in $\mathbb{C}^n$ (also $\mathbb{R}^n$), the SDA with an $O(n^3)$ computational complexity does not work well. In this paper, we propose a new decoupled form of the SDA (we name it as dSDA), building on the associated Krylov subspaces thus leading to the inherent low-rank structures. Importantly, the approach decouples the original two to four iteration formulae. The resulting dSDA is much more efficient since only one quantity (instead of the original two to four) is computed iteratively. For large-scale problems, further efficiency is gained from the low-rank structures. This paper presents the theoretical aspects of the dSDA. A practical algorithm dSDA t with truncation and many illustrative numerical results will appear in a second paper.

math.NA↗

arXiv2018

Structure-Preserving ΓQR and Γ-Lanczos Algorithms for Bethe-Salpeter Eigenvalue Problems

To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called ΓQR algorithm and Γ-Lanczos algorithm, are proposed in this paper. Both algorithms preserve the special structure of the initial matrix $H=\begin{bmatrix}A & B-\overline{B} & -\overline{A}\end{bmatrix}$, resulting the computed eigenvalues and the associated eigenvectors still hold the properties similar to those of $H$. Theorems are given to demonstrate the validity of the proposed two algorithms in theory. Numerical results are presented to illustrate the superiorities of our methods.

math.NA↗

arXiv2018

Doubling algorithm for the discretized Bethe-Salpeter eigenvalue problem

The discretized Bethe-Salpeter eigenvalue problem arises in the Green's function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for $H \in \mathbb{C}^{2n\times 2n}$ with a Hamiltonian-like structure. After an appropriate transformation of $H$ to a standard symplectic form, the structure-preserving doubling algorithm, originally for algebraic Riccati equations, is extended for the discretized Bethe-Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due to the ill condition or singularity of certain matrices, can be avoided with a double-Cayley transform or a three-recursion remedy. A detailed convergence analysis is conducted for the proposed algorithm, especially on the benign effects of the double-Cayley transform. Numerical results are presented to demonstrate the efficiency and structure-preserving nature of the algorithm.

math.NA↗

arXiv2016

Schur Method for Robust Pole Assignment of Descriptor Systems via Proportional Plus Derivative State Feedback

The pole assignment problem for descriptor systems is a classical inverse algebraic eigenvalue problem, which has attracted attention for decades. In this paper, we propose a direct method to solve the problem with the application of the proportional plus derivative state feedback, intending to obtain a robust closed-loop system. Theorems on the feasibility of our method will be presented. Numerical examples show that our method yields poles of high relative accuracy.

math.OC↗

arXiv2016

Refined Schur Method for Robust Pole Assignment with Repeated Poles

Schur-type methods in \cite{Chu2} and \cite{GCQX} solve the robust pole assignment problem by employing the departure from normality of the closed-loop system matrix as the measure of robustness. They work well generally when all poles to be assigned are simple. However, when some poles are close or even repeated, the eigenvalues of the computed closed-loop system matrix might be inaccurate. In this paper, we present a refined Schur method, which is able to deal with the case when some or all of the poles to be assigned are repeated. More importantly, the refined Schur method can still be applied when \verb|place| \cite{KNV} and \verb|robpole| \cite{Tits} fail to output a solution when the multiplicity of some repeated poles is greater than the input freedom.

math.NA↗