Improvements on the hybrid Monte Carlo algorithms for matrix computations


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Fathi-Vajargah B., Hassanzadeh Z.

Sadhana - Academy Proceedings in Engineering Sciences, vol.44, no.1, 2019 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 44 Issue: 1
  • Publication Date: 2019
  • Doi Number: 10.1007/s12046-018-0983-y
  • Journal Name: Sadhana - Academy Proceedings in Engineering Sciences
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: convergence analysis, Markov chain Monte Carlo, matrix inversion, System of linear algebraic equations, transition probability
  • Istanbul Gelisim University Affiliated: No

Abstract

In this paper, we present some improvements on the Markov chain Monte Carlo and hybrid Markov chain Monte Carlo algorithms for matrix computations. We discuss the convergence of the Monte Carlo method using the Ulam–von Neumann approach related to selecting the transition probability matrix. Specifically, we show that if the norm of the iteration matrix T is less than 1 then the Monte Carlo Almost Optimal method is convergent. Moreover, we suggest a new technique to approximate the inverse of the strictly diagonally dominant matrix and we exert some modifications and corrections on the hybrid Monte Carlo algorithm to obtain the inverse matrix in general. Finally, numerical experiments are discussed to illustrate the efficiency of the theoretical results.