The numerical solution of a large scale linear response eigenvalue problem is often accomplished by computing a pair of deflating subspaces associated with the interested part of the spectrum. This paper is concerned with the backward perturbation analysis for a given pair of approximate deflating subspaces or an approximate eigenquaternary. Various optimal backward perturbation bounds are obta...

The eikonal equation is instrumental in many applications in several fields ranging from computer vision to geoscience. This equation can be efficiently solved using the iterative Fast Sweeping (FS) methods and the direct Fast Marching (FM) methods. However, when used for a point source, the original eikonal equation is known to yield inaccurate numerical solutions, because of a singularity at ...

Approximate linear programming (ALP) has emerged recently as one of the most promising methods for solving complex factored MDPs with finite state spaces. In this work we show that ALP solutions are not limited only to MDPs with finite state spaces, but that they can also be applied successfully to factored continuous-state MDPs (CMDPs). We show how one can build an ALP-based approximation for ...

Perfect maps, factors and multifactors can be viewed as higher dimensional analogues of de Bruijn cycles and factored versions of these cycles. We present a unified framework for two basic techniques, concatenation and integration (also called the inverse of Lempel’s homomorphism), used to construct perfect multifactors. This framework simplifies proofs of known results and allows for extension...

Markov decision processes (MDPs) with discrete and continuous state and action components can be solved efficiently by hybrid approximate linear programming (HALP). The main idea of the approach is to approximate the optimal value function by a set of basis functions and optimize their weights by linear programming. It is known that the solution to this convex optimization problem minimizes the...

A partially observable Markov decision process (POMDP) provides an elegant model for problems of planning under uncertainty. Solving POMDPs is very computationally challenging, however, and improving the scalability of POMDP algorithms is an important research problem. One way to reduce the computational complexity of planning using POMDPs is by using state aggregation to reduce the (effective)...

Abstract Models in which the covariance matrix has structure of a sparse plus low rank perturbation are ubiquitous data science applications. It is often desirable for algorithms to take advantage such structures, avoiding costly computations that require cubic time and quadratic storage. This accomplished by performing operations maintain example, inversion via Sherman–Morrison–Woodbury formul...

Approximate linear programming (ALP) offers a promising framework for solving large factored Markov decision processes (MDPs) with both discrete and continuous states. Successful application of the approach depends on the choice of an appropriate set of feature functions defining the value function, and efficient methods for generating constraints that determine the convex space of the solution...

In this paper, we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse. For this purpose, considering usual continued fraction, we define backward continued fraction for a real number and give some basic results on backward continued fraction. We give the relationships between the usual and backward continued fractions. Then we reobtain the LU fact...

ABSTRACT. We extend a recent result on third and fourth-order Cauchy-Euler equations by establishing the Hyers-Ulam stability of higher-order linear non-homogeneous Cauchy-Euler dynamic equations on time scales. That is, if an approximate solution of a higher-order Cauchy-Euler equation exists, then there exists an exact solution to that dynamic equation that is close to the approximate one. We...