Square Root Approximation Research Articles

Tensor-SqRA: Modeling the transition rates of interacting molecular systems in terms of potential energies.

Estimating the rate of rare conformational changes in molecular systems is one of the goals of molecular dynamics simulations. In the past few decades, a lot of progress has been done in data-based approaches toward this problem. In contrast, model-based methods, such as the Square Root Approximation (SqRA), directly derive these quantities from the potential energy functions. In this article, we demonstrate how the SqRA formalism naturally blends with the tensor structure obtained by coupling multiple systems, resulting in the tensor-based Square Root Approximation (tSqRA). It enables efficient treatment of high-dimensional systems using the SqRA and provides an algebraic expression of the impact of coupling energies between molecular subsystems. Based on the tSqRA, we also develop the projected rate estimation, a hybrid data-model-based algorithm that efficiently estimates the slowest rates for coupled systems. In addition, we investigate the possibility of integrating low-rank approximations within this framework to maximize the potential of the tSqRA.

Consistency and convergence for a family of finite volume discretizations of the Fokker–Planck operator

We introduce a family of various finite volume discretization schemes for the Fokker–Planck operator, which are characterized by different Stolarsky weight functions on the edges. This family particularly includes the well-established Scharfetter–Gummel discretization as well as the recently developed square-root approximation (SQRA) scheme. We motivate this family of discretizations both from the numerical and the modeling point of view and provide a uniform consistency and error analysis. Our main results state that the convergence order primarily depends on the quality of the mesh and in second place on the choice of the Stolarsky weights. We show that the Scharfetter–Gummel scheme has the analytically best convergence properties but also that there exists a whole branch of Stolarsky means with the same convergence quality. We show by numerical experiments that for small convection the choice of the optimal representative of the discretization family is highly non-trivial, while for large gradients the Scharfetter–Gummel scheme stands out compared to the others.

Markov models from the square root approximation of the Fokker–Planck equation: calculating the grid-dependent flux

Molecular dynamics (MD) are extremely complex, yet understanding the slow components of their dynamics is essential to understanding their macroscopic properties. To achieve this, one models the MD as a stochastic process and analyses the dominant eigenfunctions of the associated Fokker–Planck operator, or of closely related transfer operators. So far, the calculation of the discretized operators requires extensive MD simulations. The square-root approximation of the Fokker–Planck equation is a method to calculate transition rates as a ratio of the Boltzmann densities of neighboring grid cells times a flux, and can in principle be calculated without a simulation. In a previous work we still used MD simulations to determine the flux. Here, we propose several methods to calculate the exact or approximate flux for various grid types, and thus estimate the rate matrix without a simulation. Using model potentials we test computational efficiency of the methods, and the accuracy with which they reproduce the dominant eigenfunctions and eigenvalues. For these model potentials, rate matrices with up to states can be obtained within seconds on a single high-performance compute server if regular grids are used.

The Mathematics of the Viereckschanzen of the La Tène Culture

The development of script-using ancient civilizations and their achievements in sciences such as astronomy and mathematics have been well researched. Illiterate cultures were able to attain an adequate level of knowledge and transferred this knowledge to succeeding generations via oral tradition and mnemonic artefacts. The constructions of the square ditched enclosures (Viereckschanzen) of the last phase of the La Tene period (LT D, 150-1 BC) are documentations of the mathematical knowledge of this culture and served as a kind of mnemonic artefact. The constructions comprise the application of Pythagorean theorem (in terms of Pythagorean triangles) and square-root approximation triangles. Via these types of triangles, convex quadrangles are constructed (i.e., trapezium, parallelogram, rectangle, kite, lozenge, and square). The constructions were laid out in the terrain on the basis of a consistent ‘Babylonian/Egyptian’ metrology.

A Maximum Likelihood Ensemble Filter via a Modified Cholesky Decomposition for Non-Gaussian Data Assimilation

This paper proposes an efficient and practical implementation of the Maximum Likelihood Ensemble Filter via a Modified Cholesky decomposition (MLEF-MC). The method works as follows: via an ensemble of model realizations, a well-conditioned and full-rank square-root approximation of the background error covariance matrix is obtained. This square-root approximation serves as a control space onto which analysis increments can be computed. These are calculated via Line-Search (LS) optimization. We theoretically prove the convergence of the MLEF-MC. Experimental simulations were performed using an Atmospheric General Circulation Model (AT-GCM) and a highly nonlinear observation operator. The results reveal that the proposed method can obtain posterior error estimates within reasonable accuracies in terms of error norms. Furthermore, our analysis estimates are similar to those of the MLEF with large ensemble sizes and full observational networks.

Convergences of the squareroot approximation scheme to the Fokker–Planck operator

We study the qualitative convergence behavior of a novel FV-discretization scheme of the Fokker–Planck equation, the squareroot approximation scheme (SQRA), that recently was proposed by Lie, Fackeldey and Weber [A square root approximation of transition rates for a markov state model, SIAM J. Matrix Anal. Appl. 34 (2013) 738–756] in the context of conformation dynamics. We show that SQRA has a natural gradient structure and that solutions to the SQRA equation converge to solutions of the Fokker–Planck equation using a discrete notion of G-convergence for the underlying discrete elliptic operator. The SQRA does not need to account for the volumes of cells and interfaces and is tailored for high-dimensional spaces. However, based on FV-discretizations of the Laplacian it can also be used in lower dimensions taking into account the volumes of the cells. As an example, in the special case of stationary Voronoi tessellations, we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property.

Convergence of a new FV‐discretization of the Fokker–Planck equation

AbstractWe present a new finite volume discretization of the Fokker–Planck equation. The so called Squareroot Approximation (SQRA) scheme has recently been derived by several authors and shows a gradient structure that is asymptotically compatible with the Wasserstein gradient structure studied by Jordan, Kinderlehrer and Otto using methods from stochastic homogenization. We show that the SQRA scheme converges for Voronoi discretizations. In case the Voronoi cells are regular enough, the scheme converges even for unknown cell volumes, which makes it computationally feasible for high dimensional problems.

Estimation of the infinitesimal generator by square-root approximation

In recent years, for the analysis of molecular processes, the estimation of time-scales and transition rates has become fundamental. Estimating the transition rates between molecular conformations is—from a mathematical point of view—an invariant subspace projection problem. We present a method to project the infinitesimal generator acting on function space to a low-dimensional rate matrix. This projection can be performed in two steps. First, we discretize the conformational space in a Voronoi tessellation, then the transition rates between adjacent cells is approximated by the geometric average of the Boltzmann weights of the Voronoi cells. This method demonstrates that there is a direct relation between the potential energy surface of molecular structures and the transition rates of conformational changes. We will show also that this approximation is correct and converges to the generator of the Smoluchowski equation in the limit of infinitely small Voronoi cells. We present results for a two dimensional diffusion process and alanine dipeptide as a high-dimensional system.

Fault diagnosis and minimum entropy fault tolerant control for non-Gaussian singular stochastic distribution systems using square-root approximation

The fault diagnosis (FD) and minimum entropy fault tolerant control (FTC) algorithms are proposed for non-Gaussian singular stochastic distribution control (SDC) systems in this paper. Different from general SDC systems, in singular SDC systems, the relationship between the weight vector and the control input is expressed by a singular state space model, which increases the difficulty in the design of fault diagnosis and fault tolerant control. A non-singular state transformation is made to transform the singular dynamic system into a differential-algebraic system. An iterative learning observer (ILO) is designed for fault estimation. The concept of entropy is introduced to fault tolerant control of the non-Gaussian stochastic distribution system when the target probability density function (PDF) is not known in advance. Based on the estimated fault information, the controller is reconfigured by minimising the performance function with regard to the entropy subjected to mean constraint. The reconfigured controller can make the output of the post-fault SDC system still have minimum uncertainty, leading to minimum entropy FTC of the non-Gaussian singular SDC system. Computer simulations are given to demonstrate the validity of the fault diagnosis and minimum entropy FTC algorithms.

Robust fault diagnosis for non-Gaussian stochastic systems based on the rational square-root approximation model

The task of robust fault detection and diagnosis of stochastic distribution control (SDC) systems with uncertainties is to use the measured input and the system output PDFs to still obtain possible faults information of the system. Using the rational square-root B-spline model to represent the dynamics between the output PDF and the input, in this paper, a robust nonlinear adaptive observer-based fault diagnosis algorithm is presented to diagnose the fault in the dynamic part of such systems with model uncertainties. When certain conditions are satisfied, the weight vector of the rational square-root B-spline model proves to be bounded. Convergency analysis is performed for the error dynamic system raised from robust fault detection and fault diagnosis phase. Computer simulations are given to demonstrate the effectiveness of the proposed algorithm.

Fault detection, diagnosis and tolerant control for non-Gaussian stochastic distribution systems using a rational square-root approximation model

Stochastic Distribution Control (SDC) systems are a group of systems where the outputs considered are the measured Probability Density Functions (PDFs) of the system output whilst subjected to a normal crisp input. The purpose of the control algorithm design of such systems is to choose a control input such that the PDF of the system output can follow a prespecified PDF as close as possible. Using the rational square-root B-spline model to represent the dynamics between the output PDF and the input, in this paper a non-linear adaptive observer-based fault diagnosis algorithm is proposed to diagnose the fault in the dynamic part of such systems. Using the estimation to the unknown fault, a fault tolerant control via a controller reconfiguration is proposed, where it has been shown that a good output PDF tracking can still be realised when fault occurs in the system. A simulated example is given to illustrate the use of the proposed algorithm.

Propagation of nonparaxial beams with a modified Arnoldi method

We have developed a modified Arnoldi method that includes a complex square-root approximation, which excels at modeling the propagation of highly diverging beams in various media. Simulations of one transverse dimensional beam with an ultranarrow width and of cylindrical Gaussian beams with various divergence angles reveal the strength of this nonparaxial-beam propagation method.

Rational square-root approximations for parabolic equation algorithms

In this article, stable Padé approximations to the function 1+z are derived by choosing a branch cut in the negative half-plane. The Padé coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z<−1, I(z)=0 [I(z) denotes the imaginary part of z]. The characteristics of the corresponding square-root approximation are illustrated for various segments of the complex plane. In particular, for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case. An elastic parabolic equation algorithm is used to illustrate the application of the new Padé approximations to a realistic ocean environment, including elasticity in the ocean bottom.

One-way wave equations for seismoacoustic propagation in elastic waveguides

One-way or parabolic wave equations for time-harmonic propagation in two-dimensional elastic waveguides are considered. It is shown that the direct application of a rational linear approximation with real coefficients to the elastic wave propagation case results in exponential growth in the numerical solutions. Elementary analysis demonstrates that this kind of approximation does not treat properly the modes with complex wavenumber which can exist in elastic waveguides. A new bilinear square-root approximation with complex coefficients is introduced that accommodates all mode types and leads to stable numerical solutions. In the case of thick elastic layers (such as sea-bottom sediments), this new approximation gives accurate total field prediction. When thin elastic layers (such as ice on the sea surface) are present, however, the method introduces excessive damping to modes with wavenumbers significantly different from a reference wavenumber.

Optimality in human motor performance: ideal control of rapid aimed movements.

A stochastic optimized-submovement model is proposed for Pitts' law, the classic logarithmic tradeoff between the duration and spatial precision of rapid aimed movements. According to the model, an aimed movement toward a specified target region involves a primary submovement and an optional secondary corrective submovement. The submovements are assumed to be programmed such that they minimize average total movement time while maintaining a high frequency of target hits. The programming process achieves this minimization by optimally adjusting the average magnitudes and durations of noisy neuromotor force pulses used to generate the submovements. Numerous results from the literature on human motor performance may be explained in these terms. Two new experiments on rapid wrist rotations yield additional support for the stochastic optimizedsubmovement model. Experiment 1 revealed that the mean durations of primary submovements and of secondary submovements, not just average total movement times, conform to a square-root approximation of Pitts' law derived from the model. Also, the spatial endpoints of primary submovements have standard deviations that increase linearly with average primary-submovement velocity, and the average primary-submovement velocity influences the relative frequencies of secondary submovements, as predicted by the model. During Experiment 2, these results were replicated and extended under conditions in which subjects made movements without concurrent visual feedback. This replication suggests that submovement optimization may be a pervasive property of movement production. The present conceptual framework provides insights into principles of motor performance, and it links the study of physical action to research on sensation, perception, and cognition, where psychologists have been concerned for some time about the degree to which mental processes incorporate rational and normative rules. An enduring issue in the study of the human mind concerns of mathematical probability theory and statistical decision thethe rationality and optimality of the mental processes that guide ory (e.g., see Edwards, 1961; Edwards, Lindman, & Savage,

Damage calculations for unreliable warheads

AbstractFor large numbers of perfectly reliable, optimally targeted warheads the square‐root law approximates the expected fraction damage achieved on an area target. In this paper a more exact expression is derived for this damage fraction which Holds for all numbers of warheads. This expression is shown to converge to the square‐root law when a large number of warheads are fired. The more exact expression is used in a procedure to calculate expected damage when warheads are unreliable, and this procedure is shown to be superior to a modified square‐root approximation which has been used previously.

APPROXIMATE SQUARE AND CUBE ROOTS

APPROXIMATE SQUARE AND CUBE ROOTS

TO FIND APPROXIMATE SQUARE ROOTS

School Science and MathematicsVolume 15, Issue 3 p. 245-246 TO FIND APPROXIMATE SQUARE ROOTS Norman Anning, Norman Anning Public School, Clayburn, B. CSearch for more papers by this author Norman Anning, Norman Anning Public School, Clayburn, B. CSearch for more papers by this author First published: March 1915 https://doi.org/10.1111/j.1949-8594.1915.tb10261.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article. Volume15, Issue3March 1915Pages 245-246 RelatedInformation