Browsing by Subject "Projection methods"

Now showing 1 - 6 of 6

Open Access
Block SOR preconditioned projection methods for Kronecker structured Markovian representations
(SIAM, 2005) Buchholz, Peter; Dayar, Tuğrul
Kronecker structured representations are used to cope with the state space explosion problem in Markovian modeling and analysis. Currently, an open research problem is that of devising strong preconditioners to be used with projection methods for the computation of the stationary vector of Markov chains (MCs) underlying such representations. This paper proposes a block successive overrelaxation (BSOR) preconditioner for hierarchical Markovian models (HMMs1) that are composed of multiple low-level models and a high-level model that defines the interaction among low-level models. The Kronecker structure of an HMM yields nested block partitionings in its underlying continuous-time MC which may be used in the BSOR preconditioner. The computation of the BSOR preconditioned residual in each iteration of a preconditioned projection method becomes the problem of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of the chosen partitioning. The proposed BSOR preconditioner solves these systems using sparse LU or real Schur factors of diagonal blocks. The fill-in of sparse LU factorized diagonal blocks is reduced using the column approximate minimum degree (COLAMD) ordering. A set of numerical experiments is presented to show the merits of the proposed BSOR preconditioner.
Open Access
Do CAPM results hold in a dynamic economy? a numerical analysis
(Elsevier, 1997) Akdeniz, L.; Dechert, W. D.
In this research we use the projection method (reported by Judd) to find numerical solutions to the Euler equations of a stochastic dynamic growth model. The model that we solve is Brock's asset pricing model for a variety of parameterizations of the production functions. Using simulated data from the model, conjectures (which are not analytically tractable) can be verified. We show that the market portfolio is mean-variance efficient in this dynamic context. We also show a result that is not available from the static CAPM theory: the efficient frontier shifts up and down over the business cycle.
Open Access
The equity premium in Brock's asset pricing model
(Elsevier, 2007) Akdeniz, L.; Dechert, W. D.
In this paper we combine dynamic programming methods with projection methods for solving stochastic growth models. As an application of these methods, we solve Brock’s asset pricing model with a variety of parameterizations. We focused on finding parameterizations that result in an equity premium that is high relative to the variation in consumption. We show (both analytically and numerically) that the equity premium can be higher in a production based asset pricing model than it is in the consumption based asset pricing model, even when the real output level is the same in both models.
Open Access
The equity premium in consumption and production models?
(Cambridge University Press, 2012-02-27) Akdeniz, L.; Dechert, W. D.
In this paper we use a simple model with a single Cobb–Douglas firm and a consumer with a CRRA utility function to show the difference between the equity premia in the production-based Brock model and the consumption-based Lucas model. With this simple example we show that the equity premium in the production-based model exceeds that of the consumption-based model with probability 1.
Open Access
Experiments with two-stage iterative solvers and preconditioned Krylov subspace methods on nearly completely decomposable Markov chains
(1997) Gueaieb, Wail
Preconditioned Krylov subspace methods are state-of-the-art iterative solvers developed mostly in the last fifteen years that may be used, among other things, to solve for the stationary distribution of Markov chains. Assuming Markov chains of interest are irreducible, the ¡problem amounts to computing a positive solution vector to a homogeneous system of linear algebraic equations with a singular coefficient matrix under a normalization constraint. That is, the (n X 1) unknown stationary vector x in Ax = 0, ||a:||^ = 1 (0.1 ) is sought. Here A = I — , an n x n singular M-matrix, and P is the one-step stochastic transition probability matrix. Albeit the recent advances, practicing performance analysts still widely prefer iterative methods based on splittings when they want to compare the performance of newly devised algorithms against existing ones, or when they need candidate solvers to evaluate the performance of a system model at hand. In fact, experimental results with Krylov subspace methods on Markov chains, especially the ill-conditioned nearly completely decomposable (NCD) ones, are few. We believe there is room for research in this area siDecifically to help us understand the effect of the degree of coupling of NCD Markov chains and their nonzero structure on the convergence characteristics and space requirements of preconditioned Krylov subspace methods. The work of several researchers have raised important and interesting questions that led to research in another, yet related direction. These questions are the following: “How must one go about partitioning the global coefficient matrix A in equation (0.1) into blocks if the system is NCD and a two-stage iterative solver (such as block successive overrelaxation— SOR) is to be employed? Are block partitionings dictated by the NCD normal form of F necessarily superior to others? Is it worth investing alternative partitionings? Better yet, for a fixed labelling and partitioning of the states, how does the performance of block SOR (or even that of point SOR) compare to the performance of the iterative aggregation-disaggregation (lAD) algorithm? Finally, is there any merit in using two-stage iterative solvers when preconditioned Krylov subspace methods are available?” Experimental results show that in most of the test cases two-stage iterative solvers are superior to Krylov subspace methods with the chosen preconditioners, on NCD Markov chains. For two-stage iterative solvers, there are cases in which a straightforward partitioning of the coefficient matrix gives a faster solution than can be obtained using the NCD normal form.
Open Access
The parallel surrogate constraint approach to the linear feasibility problem
(Springer, 1996) Özaktaş, Hakan; Akgül, Mustafa; Pınar, Mustafa Ç.
The linear feasibility problem arises in several areas of applied mathematics and medical science, in several forms of image reconstruction problems. The surrogate constraint algorithm of Yang and Murty for the linear feasibility problem is implemented and analyzed. The sequential approach considers projections one at a time. In the parallel approach, several projections are made simultaneously and their convex combination is taken to be used at the next iteration. The sequential method is compared with the parallel method for varied numbers of processors. Two improvement schemes for the parallel method are proposed and tested.