scholarly journals A sparse FFT approach for ODE with random coefficients

2020 ◽  
Vol 46 (5) ◽  
Author(s):  
Maximilian Bochmann ◽  
Lutz Kämmerer ◽  
Daniel Potts
Keyword(s):  
Random Variables ◽  
Approximate Solutions ◽  
Random Coefficients ◽  
Spatial Variable ◽  
General Strategy ◽  
Approximation Process ◽  
Suggested Approach ◽  
Dimensional Approximation ◽  
Sparse Fast Fourier Transform

Abstract The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimension-incremental sparse fast Fourier transform. Since such algorithms require periodic signals, we discuss periodization strategies and associated necessary deperiodization modifications within the occurring solution steps. The computed approximate solutions of the ODE depend on the spatial variable and on the random variables as well. Certainly, one of the crucial challenges of the high-dimensional approximation process is to rate the influence of each variable on the solution as well as the determination of the relations and couplings within the set of variables. The suggested approach meets these challenges in a full automatic manner with reasonable computational costs, i.e., in contrast to already existing approaches, one does not need to seriously restrict the used set of ansatz functions in advance.

scholarly journals Combined Games with Randomly Delayed Beginnings

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 534
Author(s):  
F. Thomas Bruss
Keyword(s):  
Discrete Time ◽  
Real World ◽  
Optimal Stopping ◽  
Random Variables ◽  
Approximate Solutions ◽  
Real World Problems

This paper presents two-person games involving optimal stopping. As far as we are aware, the type of problems we study are new. We confine our interest to such games in discrete time. Two players are to chose, with randomised choice-priority, between two games G1 and G2. Each game consists of two parts with well-defined targets. Each part consists of a sequence of random variables which determines when the decisive part of the game will begin. In each game, the horizon is bounded, and if the two parts are not finished within the horizon, the game is lost by definition. Otherwise the decisive part begins, on which each player is entitled to apply their or her strategy to reach the second target. If only one player achieves the two targets, this player is the winner. If both win or both lose, the outcome is seen as “deuce”. We motivate the interest of such problems in the context of real-world problems. A few representative problems are solved in detail. The main objective of this article is to serve as a preliminary manual to guide through possible approaches and to discuss under which circumstances we can obtain solutions, or approximate solutions.

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

scholarly journals A note on the implications of factorial invariance for common factor variable equivalence

2016 ◽  
Author(s):  
Michael Maraun ◽  
Moritz Heene
Keyword(s):  
Common Factor ◽  
Random Variables ◽  
Random Variable ◽  
Factorial Invariance ◽  
Technical Examination ◽  
The Common ◽  
Factor Variable

There has come to exist within the psychometric literature a generalized belief to the effect that a determination of the level of factorial invariance that holds over a set of k populations Δj, j = 1..s, is central to ascertaining whether or not the common factor random variables ξj, j = 1..s, are equivalent. In the current manuscript, a technical examination of this belief is undertaken. The chief conclusion of the work is that, as long as technical, statistical senses of random variable equivalence are adhered to, the belief is unfounded.

scholarly journals Accurate Determination of Human CPR Conformational Equilibrium by smFRET using Dual Orthogonal Non-Canonical Amino Acid Labeling

2018 ◽  
Author(s):  
Robert B. Quast ◽  
Fataneh Fatemi ◽  
Michel Kranendonk ◽  
Emmanuel Margeat ◽  
Gilles Truan
Keyword(s):  
Single Molecule ◽  
Conformational Equilibrium ◽  
Fluorescent Dyes ◽  
Photophysical Properties ◽  
Accurate Determination ◽  
Conformational Dynamics ◽  
Resonance Energy Transfer ◽  
Resonance Energy ◽  
General Strategy

ABSTRACTConjugation of fluorescent dyes to proteins - a prerequisite for the study of conformational dynamics by single molecule Förster resonance energy transfer (smFRET) - can lead to substantial changes of the dye’s photophysical properties, ultimately biasing the quantitative determination of inter-dye distances. In particular the popular cyanine dyes and their derivatives, which are by far the most used dyes in smFRET experiments, exhibit such behavior. To overcome this, a general strategy to site-specifically equip proteins with FRET pairs by chemo-selective reactions using two distinct non-canonical amino acids simultaneously incorporated through genetic code expansion in Escherichia coli was developed. Applied to human NADPH- cytochrome P450 reductase (CPR), the importance of homogenously labeled samples for accurate determination of FRET efficiencies was demonstrated. Furthermore, the effect of NADP+ on the ionic strength dependent modulation of the conformational equilibrium of CPR was unveiled. Given its generality and accuracy, the presented methodology establishes a new benchmark to decipher complex molecular dynamics on single molecules.

Imprecise Solutions of Ordinary Differential Equations for Boundary Value Problems Using Metaheuristic Algorithms

2016 ◽  
pp. 401-421
Author(s):  
Ali Sadollah ◽  
Joong Hoon Kim
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Water Cycle ◽  
Error Function ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Metaheuristic Algorithms ◽  
General Strategy ◽  
Optimization Task

In this chapter, a general strategy is recommended to solve variety of linear and nonlinear ordinary differential equations (ODEs) with boundary value conditions. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic algorithms, ODEs can be represented as an optimization problem. The purpose is to reduce the weighted residual error (error function) of the ODEs. Boundary values of ODEs are considered as constraints for the optimization model. Inverted generational distance metric is utilized for evaluation and assessment of approximate solutions versus exact solutions. Four ODEs having different orders and features are approximately solved and compared with their exact solutions. The optimization task is carried out using different optimizers including the particle swarm optimization and the water cycle algorithm. The optimization results obtained show that the proposed method equipped with metaheuristic algorithms can be successfully applied for approximate solving of different types of ODEs.

Self-tuning Kalman filter for compensation of transmission delay and loss in line-of-sight guidance

2018 ◽  
Vol 233 (11) ◽  
pp. 4191-4201
Author(s):  
Akram Nikfetrat ◽  
Reza Mahboobi Esfanjani
Keyword(s):  
Kalman Filter ◽  
State Estimation ◽  
Random Variables ◽  
Measurement Model ◽  
Line Of Sight ◽  
Uncertain Parameters ◽  
Suggested Approach ◽  
Self Tuning ◽  
Simulation Results ◽  
Sequence Simulation

A self-tuning Kalman filter is introduced to reduce the destructive effects of the delayed and lost measurements in the guidance systems employing command to line-of-sight strategy. A sequence of Bernoulli distributed random variables with uncertain probabilities are used to model the delayed and lost observations. Besides the state estimation, the uncertain parameters of the measurement model are identified online using the covariance of innovation sequence. Simulation results are given to demonstrate the merits of the suggested approach.

The Combined Closed Form-Perturbation Approach to the Analysis of Mistuned Bladed Disks

1993 ◽  
Vol 115 (4) ◽  
pp. 771-780 ◽  
Author(s):  
M. P. Mignolet ◽  
C.-C. Lin
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Approximate Solutions ◽  
Forced Response ◽  
Perturbation Approach ◽  
Perturbation Techniques ◽  
Step Method ◽  
Bladed Disk

A two-step method is presented for the determination of reliable approximations of the probability density function of the forced response of a randomly mistuned bladed disk. Under the assumption of linearity, an integral representation of the probability density function of the blade amplitude is first derived. Then, deterministic perturbation techniques are employed to produce simple approximations of this function. The adequacy of the method is demonstrated by comparing several approximate solutions with simulation results.

Random packing of an interval. II

1979 ◽  
Vol 11 (03) ◽  
pp. 591-602
Author(s):  
David Mannion
Keyword(s):  
Fine Particles ◽  
Random Variables ◽  
Random Variable ◽  
Difficult Problem ◽  
Random Packing ◽  
Point Of View ◽  
Initial Size ◽  
Moderate Size ◽  
The Individual

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x −c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y 1, y 2, ···, y k and if for each k x - y 1 - y 2 - ··· - y k (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψ n (x, B), of Z(x, B) satisfies Ψ n (x, B)x -c → κ n (B), as x → ∞, for some κ n (B). Thus, writing N = x c , Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.

Hydrogeological applications of the empirical approach to water resistivity - salinity relationships

1971 ◽  
Vol 2 (4) ◽  
pp. 35
Author(s):  
B.M. Haines ◽  
D.W. Emerson ◽  
M.J. Smith
Keyword(s):  
Direct Measurement ◽  
Empirical Relationship ◽  
Approximate Solutions ◽  
Difficult Problem ◽  
Well Logs ◽  
Quantitative Interpretation ◽  
Satisfactory Solution ◽  
Subsequent Determination ◽  
Water Resistivity

Hydrogeological evaluation of subsurface aquifers involves measurement of electrolyte resistivity and subsequent determination of solution salinity. Resistivity of the water may be evaluated by quantitative interpretation of electrical well logs or by direct measurement on recovered samples. Determination of a reliable relationship between electrolyte resistivity and salinity presents a more difficult problem. Approximate solutions have been attempted frequently on theoretical and experimental bases. An empirical relationship derived from previously collected data provides the most satisfactory solution for any particular situation (be it region, valley, basin or aquifer).

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