A variational approach to modeling slow processes in stochastic dynamical systems

The slow processes of metastable stochastic dynamical systems are difficult to access by direct numerical simulation due the sampling problem. Here, we suggest an approach for modeling the slow parts of Markov processes by approximating the dominant eigenfunctions and eigenvalues of the propagator. To this end, a variational principle is derived that is based on the maximization of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be estimated from statistical observables that can be obtained from short distributed simulations starting from different parts of state space. The approach forms a basis for the development of adaptive and efficient computational algorithms for simulating and analyzing metastable Markov processes while avoiding the sampling problem. Since any stochastic process with finite memory can be transformed into a Markov process, the approach is applicable to a wide range of processes relevant for modeling complex real-world phenomena

Boundary conditions for augmented plane wave methods

The augmented plane wave method uses the Rayleigh-Ritz principle for basis functions that are continuous but with discontinuous derivatives and the kinetic energy is written as a pair of gradients rather than as a Laplacian. It is shown here that this procedure is fully justified from the mathematical point of view. The domain of the self-adjoint Hamiltonian, which does not contain functions with discontinuous derivatives, is extended to its form domain, which contains them, and this modifies the form of the kinetic energy. Moreover, it is argued that discontinuous basis functions should be avoided.Comment: 5 pages, no figur

Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc

We describe our software package Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) publicly released recently. BLOPEX is available as a stand-alone serial library, as an external package to PETSc (``Portable, Extensible Toolkit for Scientific Computation'', a general purpose suite of tools for the scalable solution of partial differential equations and related problems developed by Argonne National Laboratory), and is also built into {\it hypre} (``High Performance Preconditioners'', scalable linear solvers package developed by Lawrence Livermore National Laboratory). The present BLOPEX release includes only one solver--the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method for symmetric eigenvalue problems. {\it hypre} provides users with advanced high-quality parallel preconditioners for linear systems, in particular, with domain decomposition and multigrid preconditioners. With BLOPEX, the same preconditioners can now be efficiently used for symmetric eigenvalue problems. PETSc facilitates the integration of independently developed application modules with strict attention to component interoperability, and makes BLOPEX extremely easy to compile and use with preconditioners that are available via PETSc. We present the LOBPCG algorithm in BLOPEX for {\it hypre} and PETSc. We demonstrate numerically the scalability of BLOPEX by testing it on a number of distributed and shared memory parallel systems, including a Beowulf system, SUN Fire 880, an AMD dual-core Opteron workstation, and IBM BlueGene/L supercomputer, using PETSc domain decomposition and {\it hypre} multigrid preconditioning. We test BLOPEX on a model problem, the standard 7-point finite-difference approximation of the 3-D Laplacian, with the problem size in the range $10^5-10^8$.Comment: Submitted to SIAM Journal on Scientific Computin

Uplink Multiuser MIMO Detection Scheme with Reduced Computational Complexity

The wireless communication systems with multiple antennas have recently received significant attention due to their higher capacity and better immunity to fading channels as compared to single antenna systems. A fast antenna selection scheme has been introduced for the uplink multiuser multiple-input multiple-output (MIMO) detection to achieve diversity gains, but the computational complexity of the fast antenna selection scheme in multiuser systems is very high due to repetitive pseudo-inversion computations. In this paper, a new uplink multiuser detection scheme is proposed adopting a switch-and-examine combining (SEC) scheme and the Cholesky decomposition to solve the computational complexity problem. K users are considered that each users is equipped with two transmit antennas for Alamouti space-time block code (STBC) over wireless Rayleigh fading channels. Simulation results show that the computational complexity of the proposed scheme is much lower than the systems with exhaustive and fast antenna selection, while the proposed scheme does not experience the degradations of bit error rate (BER) performances

On Temple--Kato like inequalities and applications

We give both lower and upper estimates for eigenvalues of unbounded positive definite operators in an arbitrary Hilbert space. We show scaling robust relative eigenvalue estimates for these operators in analogy to such estimates of current interest in Numerical Linear Algebra. Only simple matrix theoretic tools like Schur complements have been used. As prototypes for the strength of our method we discuss a singularly perturbed Schroedinger operator and study convergence estimates for finite element approximations. The estimates can be viewed as a natural quadratic form version of the celebrated Temple--Kato inequality.Comment: submitted to SIAM Journal on Numerical Analysis (a major revision of the paper

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh–Ritz method using the ordinary and degenerated free–free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates

Simple one-dimensional quantum-mechanical model for a particle attached to a surface

We present a simple one-dimensional quantum-mechanical model for a particle attached to a surface. We solve the Schr\"odinger equation in terms of Weber functions and discuss the behavior of the eigenvalues and eigenfunctions. We derive the virial theorem and other exact relationships as well as the asymptotic behaviour of the eigenvalues. We calculate the zero-point energy for model parameters corresponding to H adsorbed on Pd(100) and also outline the application of the Rayleigh-Ritz variational method

Significance of norms and completeness in variational based methods

By means of a simple structural problem, an important requirement often overlooked in practice on the basis functions used in Rayleigh-Ritz-Galerkin type methods is brought into focus. The problem of the static deformation of a uniformly loaded beam is solved variationally by expanding the beam displacement in a Fourier Cosine series. The potential energy functional is rendered stationary subject to the geometric boundary conditions. It is demonstrated that the variational approach does not converge to the true solution. The object is to resolve this paradox, and in so doing, indicate the practical implications of norms and completeness in an appropriate inner product space