Efficient solution of parabolic equations by Krylov approximation methods

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms

Some fast elliptic solvers on parallel architectures and their complexities

The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR

On the parallel solution of parabolic equations

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented

Computing symmetric nonnegative rank factorizations

An algorithm is described for the nonnegative rank factorization (NRF) of some completely positive (CP) matrices whose rank is equal to their CP-rank. The algorithm can compute the symmetric NRF of any nonnegative symmetric rank-r matrix that contains a diagonal principal submatrix of that rank and size with leading cost O(rm2) operations in the dense case. The algorithm is based on geometric considerations and is easy to implement. The implications for matrix graphs are also discussed

Dialkyldithiophosphate Acids (HDDPs) as Effective Lubricants of Sol–Gel Titania Coatings in Technical Dry Friction Conditions

The goal of this study was the investigation of the effectiveness of dialkyldithiophosphate acids (HDDPs) films in improving the tribological properties of thin, sol– gel derived titania coatings. Amorphous, anatase, and rutile titania coatings were obtained using sol–gel dip–coating deposition after treatment at 100, 500, and 1,000 C, respectively. Titania coatings were then modified from the liquid phase by HDDPs acids having dodecyl-(C12), tetradecyl-(C14), and hexadecyl-(C16) alkyl chains deposited by dip–coating (DC) and Langmuir–Blodgett (LB) methods. The influence of the deposition procedure, the length of the HDDPs alkyl chain and the type of titania substrate on the surface morphology and tribological properties were studied. It was found, using wetting contact angle measurements, that these modifications of titania coatings decrease the surface free energy and increase its hydrophobicity. The surface topography imaged by Atomic force microscopy (AFM), exhibit island-like or agglomerate features for the DC deposition method, while smooth topographies were observed for LB depositions. Tribological tests were conducted by means of a microtribometer operating in the normal load range 30–100 mN. An enhancement of tribological properties was observed upon modification, as compared to unmodified titania

Solving the chemical master equation using sliding windows

<p>Abstract</p> <p>Background</p> <p>The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.</p> <p>Results</p> <p>In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.</p> <p>Conclusions</p> <p>The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.</p

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

We implement an exponential integrator for large and sparse systems of ODEs, generated by FE (Finite Element) discretization with mass-lumping of advection-diffusion equations. The relevant exponential-like matrix function is approximated by polynomial interpolation, at a sequence of real Leja points related to the spectrum of the FE matrix (ReLPM, Real Leja Points Method). Application to 2D and 3D advection-dispersion models shows speed-ups of one order of magnitude with respect to a classical variable step-size Crank-Nicolson solver