Numerical results for the Metropolis algorithm. (English) Zbl 1058.65010

Summary: This paper deals with the spectrum of an operator associated with a special kind of random walk. The operator is related to the Metropolis algorithm, an important tool of large-scale scientific computing. The spectrum of this operator has both discrete and continuous parts. There is an interesting challenge due to the fact that any finite-dimensional approximation has only eigenvalues. Patterns are presented which give an idea of the full spectrum of this operator.


65C40 Numerical analysis or methods applied to Markov chains
65C05 Monte Carlo methods
60J22 Computational methods in Markov chains
60G50 Sums of independent random variables; random walks
68W30 Symbolic computation and algebraic computation
Full Text: DOI Euclid EuDML


[1] Billera L., Stat. Sci. 20 pp 1– (2001)
[2] DOI: 10.1006/jcss.1998.1576 · Zbl 0920.68054
[3] Diaconis P., ”On the Spectrum of a Simple Metropolis Algorithm (2003)
[4] Edmunds D. E., Spectral Theory and Differential Operators. (1987) · Zbl 0628.47017
[5] Kienetz J., PhD diss., in: ”Convergence of Markov Chains via Analytic and Isoperimetric Inequalities.” (2000)
[6] DOI: 10.1142/S0218127496000680 · Zbl 0920.73165
[7] DOI: 10.1214/aos/1033066201 · Zbl 0854.60065
[8] DOI: 10.1063/1.1699114
[9] DOI: 10.1007/BFb0103812
[10] DOI: 10.1002/jcc.540080425
[11] Neuberger J. W., Exp. Math. 8 pp 301– (1999)
[12] Sullivan F., Computing in Science and Engineering 2 (2000)
[13] DOI: 10.1214/aos/1176325750 · Zbl 0829.62080
[14] Wilkinson J. H., The Algebraic Eigenvalue Problem. (1965) · Zbl 0258.65037
[15] DOI: 10.1016/S0304-4149(99)00101-5 · Zbl 1045.60073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.