arXiv subjects

Mu-Fa Chen

Publications and source records attributed to Mu-Fa Chen.

At least 19 recordsLinked to original sources

arXiv2017

Trilogy on Computing Maximal Eigenpair

The eigenpair here means the twins consist of eigenvalue and its eigenvector. This paper introduces the three steps of our study on computing the maximal eigenpair. In the first two steps, we construct efficient initials for a known but dangerous algorithm, first for tridiagonal matrices and then for irreducible matrices, having nonnegative off-diagonal elements. In the third step, we present two global algorithms which are still efficient and work well for a quite large class of matrices, even complex for instance.

math.GM↗

arXiv2017

Global algorithms for maximal eigenpair

This paper is a continuation of \ct{cmf16} where an efficient algorithm for computing the maximal eigenpair was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements. This paper introduces two global algorithms for computing the maximal eigenpair in a rather general setup, including even a class of real (with some negative off-diagonal elements) or complex matrices.

math.PR↗

arXiv2017

The Charming Leading Eigenpair

The leading eigenpair (the couple of eigenvalue and its eigenvector) or the first nontrivial one has different names in different contexts. It is the maximal one in the matrix theory. The talk starts from our new results on computing the maximal eigenpair of matrices. For the unexpected results, our contribution is the efficient initial value for a known algorithm. The initial value comes from our recent theoretic study on the estimation of the leading eigenvalues. To which we have luckily obtained unified estimates which consist of the second part of the talk. In the third part of the talk, the original motivation of the study along this direction is explained in terms of a specific model. The paper is concluded by a brief overview of our study on the leading eigenvalue, or more generally on the speed of various stabilities.

math.PR↗

arXiv2016

Efficient initials for computing maximal eigenpair

This paper introduces some efficient initials for a well-known algorithm (an inverse iteration) for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.

math.PR↗

arXiv2015

Practical Criterion for Uniqueness of $Q$-processes

The note begins with a short story on seeking for a practical sufficiency theorem for the uniqueness of time-continuous Markov jump processes, starting around 1977. The general result was obtained in 1985 for the processes with general state spaces. To see the sufficient conditions are sharp, a dual criterion for non-uniqueness was obtained in 1991. This note is restricted however to the discrete state space (then the processes are called $Q$-processes or Markov chains), for which the sufficient conditions just mentioned are showing at the end of the note to be necessary. Some examples are included to illustrate that the sufficient conditions either for uniqueness or for non-uniqueness are not only powerful but also sharp.

math.PR↗

arXiv2015

Mixed Eigenvalues of {\LARGE $\pmb p$\,}-Laplacian

The mixed principal eigenvalue of $p\,$-Laplacian (equivalently, the optimal constant of weighted Hardy inequality in $L^p$ space) is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.

math.SP↗

arXiv2015

The optimal constant in Hardy-type inequalities

To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined.

math.PR↗

arXiv2015

Criteria for Discrete Spectrum of 1D Operators

For discrete spectrum of 1D second-order differential/difference operators (with or without potential (killing), with the maximal/minimal domain), a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators. Interes\-tingly, these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier, simply replacing the `finite supremum' by `vanishing at infinity'. Except a dual technique, the main tool used here is a transform in terms of the harmonic function, to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context. All of them are illustrated by examples. The main body of the paper is devoted to the hard part of the story, the easier part but powerful one is delayed to the end of the paper.

math.PR↗

arXiv2014

A Comment on the Book "Continuous-Time Markov Chains" by W.J. Anderson

The book "Continuous-Time Markov Chains" by W. J. Anderson collects a large part of the development in the past thirty years. It is now a popular reference for the researchers on this subject or related fields. Unfortunately, due to a misunderstanding of the approximating methods, several results in the book are incorrectly stated or proved. Since the results are related to the present author's work, to whom it may be a duty to correct the mistakes in order to avoid further confusion. We emphasize the approximating methods because they are useful in many situations.

math.PR↗

arXiv2014

Progress on Hardy-type Inequalities

This paper surveys some of our recent progress on Hardy-type inequa\-lities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in terms of the $L^2$-theory. A crucial application of a result by Fukushima and Uemura (2003) is included. In the second section, the non-linear case (a general Hardy-type inequality) is handled with a direct and analytic proof. In the last section, it is illustrated that the basic estimates presented in the first two sections can still be improved considerably.

math.PR↗

arXiv2014

Isospectral operators

For a large class of integral operators or second order differential operators, their isospectral (or cospectral) operators are constructed explicitly in terms of $h$-transform (duality). This provides us a simple way to extend the known knowledge on the spectrum (or the estimation of the principal eigenvalue) from a smaller class of operators to a much larger one. In particular, an open problem about the positivity of the principal eigenvalue for birth--death processes is solved in the paper.

math.AP↗

arXiv2014

Mixed Eigenvalues of Discrete {\LARGE $\pmb p$\,}-Laplacian

This paper deals with the principal eigenvalue of discrete $p$-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation: the difference form, the single summation form and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity (which was known years ago), as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at infinity is also studied. Two examples are presented at the end of Section 2 to illustrate the value of the investigation.

math.SP↗

arXiv2014

Unified representation of formulas for single birth processes

Based on a new explicit representation of the solution to the Poisson equation with respect to single birth processes, the unified treatment for various criteria on classical problems (including uniqueness, recurrence, ergodicity, exponential ergodicity, strong ergodicity, as well as extinction probability etc.) for the processes are presented.

math.PR↗

arXiv2014

On the Optimality in General Sense for Odd-Block Search

In his classical article[3](1953), J.Kiefer introduced the Fibonacci search as a direct optimal method. The optimality was proved under the restriction: the total number of tests is given in advance and fixed. To avoid this restriction, some different concepts of optimality were proposed and some corresponding optimal me\-thods were obtained in [1], [2], [5] and [6]. In particular, the even-block search was treated in [1]. This paper deals with the odd-block search. The main result is Theorem (1.15).

math.PR↗

arXiv2012

Mixed principal eigenvalues in dimension one

This is one of a series of papers exploring the stability speed of one-dimensional stochastic processes. The present paper emphasizes on the principal eigenvalues of elliptic operators. The eigenvalue is just the best constant in the $L^{2}$-Poincar\'e inequality and describes the decay rate of the corresponding diffusion process. We present some variational formulas for the mixed principal eigenvalues of the operators. As applications of these formulas, we obtain case by case explicit estimates, a criterion for positivity, and an approximating procedure for the eigenvalue.

math.PR↗

arXiv2012

Lower bounds of principal eigenvalue in dimension one

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth--death processes). Possible generalization of the results studied here is discussed at the end of the paper.

math.PR↗

arXiv2012

Bilateral Hardy-type inequalities

This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new in the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.

math.PR↗

arXiv2011

Basic Estimates of Stability Rate for One-dimensional Diffusions

In the context of one-dimensional diffusions, we present basic estimates (having the same lower and upper bounds with a factor of 4 only) for four Poincar\'e-type (or Hardy-type) inequalities. The derivation of two estimates have been open problems for quite some time. The bounds provide exponentially ergodic or decay rates. We refine the bounds and illustrate them with typical examples.

math.PR↗