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

Zhen-Chen Guo, Xin Liang

Abstract

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.

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

Abstract

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.
Paper Structure (17 sections, 2 theorems, 44 equations, 2 figures, 2 algorithms)

This paper contains 17 sections, 2 theorems, 44 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Incorporation and RADI-type method
  4. Residual and incorporation
  5. Low-rank expression of a special residual
  6. Implementation aspects
  7. Storage and compression
  8. Time complexity reduction
  9. Practical algorithm and its complexity
  10. Shift strategy
  11. Deal with the original SCARE (1.1)
  12. Deal with the generalized SCARE
  13. Experiments and discussions
  14. Shift strategy
  15. Compression/Truncation
  16. ...and 2 more sections

Key Result

Theorem 2.1

Given $X\succeq 0$ and let $A_{X},B_{X},\widehat{A}_{X},\widehat{B}_{X}$ be as in eq:AB-Xi, and let $\mathscr{ C}_{X}(0)=\mathscr{ C}(X) \succeq 0$. Then

Figures (2)

  • Figure 4.1: Rail at $n=79841$
  • Figure 4.2: Lung2-: large truncation case

Theorems & Definitions (6)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Example 4.1: Rail
  • Example 4.2: Lung2$-$