Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems

We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD, we can approximate its termination probabilities (i.e., its $G$ matrix) to within $i$ bits of precision (i.e., within additive error $1/2^i$), in time polynomial in \underline{both} the encoding size of the QBD and in $i$, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results) that for the more general TL-QBDs such a polynomial upper bound on Newton's method fails badly. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for strongly connected monotone systems of polynomial equations. We show that the quantitative termination decision problem for QBDs (namely, ``is $G_{u,v} \geq 1/2$?'') is at least as hard as long standing open problems in the complexity of exact numerical computation, specifically the square-root sum problem. On the other hand, it follows from our earlier results for RMCs that any non-trivial approximation of termination probabilities for TL-QBDs is sqrt-root-sum-hard

Interior point methods : current status and future directions

Cover title.Includes bibliographical references (leaves 23-24).Robert Freund and Shinji Mizuno

Interior point methods : current status and future directions

Cover title.Includes bibliographical references (leaves 23-24).Robert Freund and Shinji Mizuno

Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata and Pushdown Systems

to appear in QEST 2008We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to (discrete-time) probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD (even a null-recurrent one), we can approximate its termination probabilities (i.e., its $G$ matrix) to within $i$ bits of precision (i.e., within additive error $1/2^i$), in time polynomial in \underline{both} the encoding size of the QBD and in $i$, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results for pPDSs) that for the more general TL-QBDs this bound fails badly. Specifically, in the worst case Newton's method ``converges linearly'' to the termination probabilities for TL-QBDs, but requires exponentially many iterations in the encoding size of the TL-QBD to approximate these probabilities within any non-trivial constant error $c < 1$. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for monotone systems of polynomial equations

Quasi-Birth-Death Processes, Tree-Like QBDs, Probabilistic 1-Counter Automata, and Pushdown Systems

We begin by observing that (discrete-time) Quasi-Birth-Death Processes (QBDs) are equivalent, in a precise sense, to probabilistic 1-Counter Automata (p1CAs), and both Tree-Like QBDs (TL-QBDs) and Tree-Structured QBDs (TS-QBDs) are equivalent to both probabilistic Pushdown Systems (pPDSs) and Recursive Markov Chains (RMCs). We then proceed to exploit these connections to obtain a number of new algorithmic upper and lower bounds for central computational problems about these models. Our main result is this: for an arbitrary QBD, we can approximate its termination probabilities (i.e., its $G$ matrix) to within $i$ bits of precision (i.e., within additive error $1/2^i$), in time polynomial in \underline{both} the encoding size of the QBD and in $i$, in the unit-cost rational arithmetic RAM model of computation. Specifically, we show that a decomposed Newton's method can be used to achieve this. We emphasize that this bound is very different from the well-known ``linear/quadratic convergence'' of numerical analysis, known for QBDs and TL-QBDs, which typically gives no constructive bound in terms of the encoding size of the system being solved. In fact, we observe (based on recent results) that for the more general TL-QBDs such a polynomial upper bound on Newton's method fails badly. Our upper bound proof for QBDs combines several ingredients: a detailed analysis of the structure of 1-counter automata, an iterative application of a classic condition number bound for errors in linear systems, and a very recent constructive bound on the performance of Newton's method for strongly connected monotone systems of polynomial equations. We show that the quantitative termination decision problem for QBDs (namely, ``is $G_{u,v} \geq 1/2$?'') is at least as hard as long standing open problems in the complexity of exact numerical computation, specifically the square-root sum problem. On the other hand, it follows from our earlier results for RMCs that any non-trivial approximation of termination probabilities for TL-QBDs is sqrt-root-sum-hard

Upper Bounds for Newton's Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata

A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multi-type branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of {\em termination probabilities}, and computing these probabilities in turn boils down to computing the {\em least fixed point} (LFP) solution of a corresponding {\em monotone polynomial system} (MPS) of equations, denoted x=P(x). It was shown by Etessami & Yannakakis that a decomposed variant of Newton's method converges monotonically to the LFP solution for any MPS that has a non-negative solution. Subsequently, Esparza, Kiefer, & Luttenberger obtained upper bounds on the convergence rate of Newton's method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs. However, prior to this paper, for arbitrary (not necessarily strongly-connected) MPSs, no upper bounds at all were known on the convergence rate of Newton's method as a function of the encoding size |P| of the input MPS, x=P(x). In this paper we provide worst-case upper bounds, as a function of both the input encoding size |P|, and epsilon > 0, on the number of iterations required for decomposed Newton's method (even with rounding) to converge within additive error epsilon > 0 of q^*, for any MPS with LFP solution q^*. Our upper bounds are essentially optimal in terms of several important parameters. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) omega-regular or LTL property

Energy Models for One-Carrier Transport in Semiconductor Devices

Moment models of carrier transport, derived from the Boltzmann equation, made possible the simulation of certain key effects through such realistic assumptions as energy dependent mobility functions. This type of global dependence permits the observation of velocity overshoot in the vicinity of device junctions, not discerned via classical drift-diffusion models, which are primarily local in nature. It was found that a critical role is played in the hydrodynamic model by the heat conduction term. When ignored, the overshoot is inappropriately damped. When the standard choice of the Wiedemann-Franz law is made for the conductivity, spurious overshoot is observed. Agreement with Monte-Carlo simulation in this regime required empirical modification of this law, or nonstandard choices. Simulations of the hydrodynamic model in one and two dimensions, as well as simulations of a newly developed energy model, the RT model, are presented. The RT model, intermediate between the hydrodynamic and drift-diffusion model, was developed to eliminate the parabolic energy band and Maxwellian distribution assumptions, and to reduce the spurious overshoot with physically consistent assumptions. The algorithms employed for both models are the essentially non-oscillatory shock capturing algorithms. Some mathematical results are presented and contrasted with the highly developed state of the drift-diffusion model

BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function

We discuss <code>R</code> package <b>BB</b>, in particular, its capabilities for solving a nonlinear system of equations. The function <code>BBsolve</code> in <b>BB</b> can be used for this purpose. We demonstrate the utility of these functions for solving: (a) large systems of nonlinear equations, (b) smooth, nonlinear estimating equations in statistical modeling, and (c) non-smooth estimating equations arising in rank-based regression modeling of censored failure time data. The function <code>BBoptim</code> can be used to solve smooth, box-constrained optimization problems. A main strength of <b>BB</b> is that, due to its low memory and storage requirements, it is ideally suited for solving high-dimensional problems with thousands of variables

An O(log sup 2 N) parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix

An O(log sup 2 N) parallel algorithm is presented for computing the eigenvalues of a symmetric tridiagonal matrix using a parallel algorithm for computing the zeros of the characteristic polynomial. The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros. The exact behavior of the polynomials at the interval endpoints is used to eliminate the usual problems induced by finite precision arithmetic

Following a "balanced" trajectory from an infeasible point to an optimal linear programming solution with a polynomial-time algorithm

Includes bibliographical references.Supported by NSF, AFOSR and ONR through NSF grant. DMS-8920550 Supported by the MIT-NTU Collaboration Research Fund.Robert M. Freund