Sparse Grid Adaptive Interpolation in Problems of Modeling Dynamic Systems with Interval Parameters

The paper is concerned with the issues of modeling dynamic systems with interval parameters. In previous works, the authors proposed an adaptive interpolation algorithm for solving interval problems; the essence of the algorithm is the dynamic construction of a piecewise polynomial function that interpolates the solution of the problem with a given accuracy. The main problem of applying the algorithm is related to the curse of dimension, i.e., exponential complexity relative to the number of interval uncertainties in parameters. The main objective of this work is to apply the previously proposed adaptive interpolation algorithm to dynamic systems with a large number of interval parameters. In order to reduce the computational complexity of the algorithm, the authors propose using adaptive sparse grids. This article introduces a novelty approach of applying sparse grids to problems with interval uncertainties. The efficiency of the proposed approach has been demonstrated on representative interval problems of nonlinear dynamics and computational materials science.

Two-Scale Network Dynamic Model for Energy Commodity Processes

In this work, we examine the relationship between different energy commodity spot prices. To do this, multivariate stochastic models with and without external random interventions describing the price of energy commodities are developed. Random intervention process is described by a continuous jump process. The developed mathematical model is utilized to examine the relationship between energy commodity prices. The time-varying parameters in the stochastic model are estimated using the recently developed parameter identification technique called local lagged adapted generalized method of moment (LLGMM). The LLGMM method provides an iterative scheme for updating statistic coefficients in a system of generalized method of moment/observation equations. The usefulness of the LLGMM approach is illustrated by applying to energy commodity data sets for state and parameter estimation problems. Moreover, the forecasting and confidence interval problems are also investigated (U.S. Patent Pending for the LLGMM method described in this manuscript).

Three-Stage Estimation of the Mean and Variance of the Normal Distribution with Application to an Inverse Coefficient of Variation with Computer Simulation

This paper considers sequentially two main problems. First, we estimate both the mean and the variance of the normal distribution under a unified one decision framework using Hall’s three-stage procedure. We consider a minimum risk point estimation problem for the variance considering a squared-error loss function with linear sampling cost. Then we construct a confidence interval for the mean with a preassigned width and coverage probability. Second, as an application, we develop Fortran codes that tackle both the point estimation and confidence interval problems for the inverse coefficient of variation using a Monte Carlo simulation. The simulation results show negative regret in the estimation of the inverse coefficient of variation, which indicates that the three-stage procedure provides better estimation than the optimal.

Uncertain Friction-Induced Vibration Study: Coupling of Fuzzy Logic, Fuzzy Sets, and Interval Theories

This paper presents a complete method to carry out a fuzzy study of a friction-induced vibration system and to analyze the effects of uncertainty on the output data of a stability problem. The proposed approach decomposes the fuzzy problem into interval problems and calculates interval output solutions by optimization. Next, each calculation of the stability problem output data, which is useful during the optimization process, is reanalyzed by integrating fuzzy logic controllers for the static step and homotopy development and projection techniques for the modal step. Finally, the obtained results are compared with Zadeh’s extension principle reference.

A Solution to the Fingering Problem of Brahms Cellos Sonata No. 1 Opus 38 and Shostakovich Cello Sonata Opus 40

The flexibility of fingers is one main factor of responsibility on their spread ability. Considering thecello playing, the spread ability of cellists’ fingers is very important since it has a direct correlation to theirability to play the octave interval. The short- fingers cellists would find difficulties to play the octave intervalsat the first, second, third, and fourth position. Nevertheless, short-fingers are not dead-end for cellist. Thespread ability of finger, up to some extent, might be increasing through practices and exercises. It is with thisspirit that the author proposes some stratagems, which may be used to increase the spread ability of a cellist’sfingers in this section - using the octave interval problems as a case of pint. The discussion on the types andstyles of problem show that even though the octave intervals in Brahms cellos sonata No. 1 opus 38 andShostakovich cello sonata opus 40 are different in details, but fundamentally they are similar: the problemscome from the difficulty of the first and the fourth fingers to reach the proper note when the size and lengthof the fingers are limited.Solusi dari Permasalahan Fingering dalam Brahms Cellos Sonata No.1 Opus 38 danShostakovich Cello Sonata Opus 40. Kemampuan penyebaran jari pemain cello sangat penting karenamemiliki korelasi langsung dengan kemampuan mereka untuk bermain interval oktaf. Jari-jari pendekpemain cello akan menemukan kesulitan untuk bermain interval-interval oktaf di posisi pertama,kedua, ketiga, dan keempat. Namun demikian, kemampuan penyebaran jari, bisa ditingkatkan melaluipraktik dan latihan. Hal tersebut menggugah penulis untuk mengusulkan beberapa siasat baru mengenaikemampuan teknis memainkan interval dan oktav dan persoalan jarak interval dalam konteks Sonatayang dapat digunakan untuk meningkatkan kemampuan penyebaran jari seorang pemain cello. Meskipuninterval oktaf di Brahms cello sonata No 1 opus 38 dan Shostakovich cello sonata opus 40 berbeda,tetapi pada dasarnya keduanya sama: masalah datang ketika kesulitan terjadi pada jari pertama dankeempat untuk mencapai nada yang tepat ketika ukuran dan panjang jari-jari terbatas.

Spectral Local Linearisation Approach for Natural Convection Boundary Layer Flow

The present work introduces a spectral local linearisation method (SLLM) to solve a natural convection boundary layer flow problem with domain transformation. It is customary to find solutions of semi-infinite interval problems by first truncating the interval and subsequently applying a suitable numerical method. However, this gives rise to increased error terms in the numerical solution. Carrying out a transformation of the semi-infinite interval problems into singular problems posed on a finite interval can avoid the domain truncation error and enables the efficient application of collocation methods. The SLLM is based on linearising and decoupling nonlinear systems of equations into a sequence or subsystems of differential equations which are then solved using spectral collocation methods. A comparative study between the SLLM and existing results in the literature was carried out to validate the results. The method has shown to be a promising efficient tool for nonlinear boundary value problems as it gives converging results after very few iterations.