Adaptive Methods of Calculus Mathematics and Mechanics: Stochastic Variant

by Arsenev, D. G.; Ivanov, V. M.; Kulchitskii, O. Iu

• ISBN13: 9789810235017
• ISBN10: 9810235011
• Format: Hardcover
• Copyright: 1998-09-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This book describes adaptive methods of statistical numerical analysis using evaluation of integrals, solution of integral equations, boundary value problems of the theory of elasticity and heat conduction as examples.The results and approaches provided in this book are different from those available in the literature as detailed descriptions of the mechanisms of adaptation of statistical evaluation procedures, which accelerate their convergence, are given.

Table of Contents

Foreword	v
Part I. Evaluation of integrals and solution of integral equations	1	(204)
Fundamentals of the Monte-Carlo method	3	(18)
Idea of the Monte-Carlo method	3	(3)
Simulation of implementation of a scalar random variable	6	(8)
The transforming functions method	6	(3)
The superposition method	9	(2)
The selection method	11	(3)
Simulation of implementation of a vector random variable	14	(3)
Evaluation of definite integrals by means of Monte-Carlo method	17	(4)
Evaluation of integrals by means of statistic simulation employing adaptation	21	(28)
Adaptation idea in statistic methods of numerical analysis, based on the principles of importance sampling	21	(2)
Adaptive algorithm for evaluating one-dimensional integral	23	(9)
Selection of probability densities	23	(4)
Evaluation procedure	27	(1)
Results of numerical experiments	28	(1)
Report on the results	29	(3)
Adaptive algorithm of evaluation of two-dimensional and multi-dimensional integrals	32	(5)
Description of the algorithm	32	(4)
Results of numerical experiments	36	(1)
Some comments	36	(1)
Stochastic computing algorithms as an object of adaptive control	37	(12)
Introduction	37	(1)
Statement of a problem of control over the process of computation	38	(4)
Synthesis of the optimal control over the process of computation	42	(5)
Strategy of adaptive optimization of computation process	47	(2)
Semi-statistical method of numerical solving integral equations	49	(33)
Introduction	49	(1)
Basic relations of the method	50	(2)
Recurrent inversion formulae	52	(3)
Convergence of the method	55	(15)
Adaptive abilities of the algorithm	70	(2)
Qualitative considerations concerning connections between the semi-statistical and variational methods	72	(1)
Application of the method to singular integral equations	72	(10)
Description and application of the method	72	(4)
Recurrent inversion formulae	76	(1)
Analysis of the method's errors	77	(3)
Adaptive abilities of the algorithm	80	(2)
Projection-statistical method of numerical solution of integral equations	82	(34)
Introduction	82	(1)
Basic relations of the method	82	(4)
Formulae of recurrent inversion	86	(2)
The algorithm convergence	88	(11)
Merits of the method	99	(1)
Adaptive abilities	99	(2)
Peculiarities of numerical implementation	101	(2)
An alternative computing technique: approximate solutions should be averaged	103	(2)
Numerical experiments	105	(11)
A test problem	105	(5)
The problem on steady-state forced small transverse vibration of pinned string caused by harmonic force	110	(6)
The problem of vibration conductivity	116	(25)
The boundary-value problem of vibration conductivity	116	(2)
Integral equations of vibration conductivity	118	(6)
Regularization of the equations	124	(6)
Integral equations with improved asymptotic properties at small β	130	(4)
Numerical solution of the vibration conductivity problems	134	(7)
Solution of a test problem	134	(4)
Research of the influence of distortion of a sphere and character of an external load on the results of numerical solution	138	(3)
The first basic problem of the elasticity theory	141	(20)
Potentials and integral equations of the first basic problem of the elasticity theory	142	(4)
The force and pseudo-force tensors	142	(3)
Integral equations of the first basic problem	145	(1)
Solution of some space problems of elasticity theory using the method of potentials	146	(5)
Solution of the first basic problem for some centrally symmetrical spatial areas	146	(2)
Solution of the first basic problem for a ball	148	(1)
Solution of the first basic problem for an unlimited medium with a spherical cavity	149	(1)
Solution of the first basic problem for a hollow ball	149	(2)
Using semi-statistical method for solution of integral equations of the elasticity theory	151	(4)
Formulae for optimal density	155	(2)
Results of numerical simulation	157	(4)
The second basic problem of the elasticity theory	161	(10)
Fundamental solutions of the first and second kinds	161	(4)
Boussinesq potentials	165	(1)
Weyl tensor	166	(2)
Weyl force tensors	168	(2)
Arbitrary Liapunov surface	170	(1)
A way to solve non-stationary problems	171	(34)
The general scheme of solution of non-stationary integral equation	171	(4)
A way to integrate systems of linear differential equations	175	(30)
Setting the problem	175	(2)
Estimating an error of evaluation of matrix exponential	177	(4)
Optimization of the algorithm of approximate computation of matrix exponential	181	(7)
Analysis of condition of optimal parameter selection	188	(1)
Comparison of the proposed way to select method parameters with known analogues	189	(6)
Estimate of integration method error of differential equations linear systems with constant coefficient matrix	195	(5)
Algorithm of rigid linear systems numerical integration of differential equations with constant coefficient matrix	200	(2)
Advantages of the algorithm	202	(3)
Part II. The random walk method. Solution of boundary-value problems	205	(142)
Introduction to the random walk method (RWM)	207	(2)
Numerical solution of the heat conductivity problems by means of the random walk method	209	(44)
Basic boundary-value problems of flat stationary heat conductivity theory	209	(2)
Main ideas of RWM for solution of the heat conductivity problems	211	(3)
The process of random walking over circles and its properties	214	(5)
RWM algorithms for solution of the Dirichlet problem	219	(23)
RWM algorithms on circles with displaced centers	221	(9)
Traditional RWM algorithm	230	(1)
RWM algorithms with partial integration	231	(11)
RWM algorithms for solution of the Neumann problem	242	(8)
Integral representations of the solution	242	(3)
RWM algorithm with displaced centers	245	(5)
Numerical results	250	(3)
The Monte-Carlo method applied to problems of plate curving	253	(62)
Statement of the problem	253	(3)
Integral representations	256	(7)
Curving of supported plates with linear piece-wise boundary	263	(14)
Basic relations	263	(3)
Traditional RWM method	266	(2)
RWM algorithm with displaced centers	268	(5)
RWM algorithm with partial integration	273	(4)
Curving of supported plates with arbitrary contour	277	(23)
Designing traditional algorithm in a formal way	277	(2)
Designing of converging algorith (modulation method)	279	(6)
Convergence conditions	285	(2)
Justification of the results obtained	287	(13)
Curving of the fastened plates	300	(12)
Basic relations	300	(1)
Algorithm with partial integration	301	(8)
Some ways to improve efficiency of the algorithm	309	(3)
Results of numerical simulation	312	(3)
The Monte-Carlo method applied to the flat problems of elasticity theory	315	(7)
Statement of the problem	315	(1)
Description of RWM algorithm with partial integration	316	(1)
Finding displacements by the RWM	317	(3)
Solution of test problems	320	(2)
Application of Monte-Carlo method towards finding of tensions at dangerous points of cog-wheels	322	(9)
Selection of computation models and setting of boundary conditions	322	(2)
A rigid-rimmed cog-wheel	323	(1)
A flexible-rimmed cog-wheel	323	(1)
Results of numerical simulation	324	(6)
Rigid-rimmed cog-wheels	324	(3)
Flexible-rimmed cog-wheels	327	(1)
Problem of finding the best gear-cutting tool	328	(2)
Some conclusions	330	(1)
The spatial problem of heat conductivity	331	(16)
Defining relations	332	(1)
Solution of the statistical problem of heat conductivity with boundary conditions of the second kind for a parallelepiped and a cube	333	(3)
Construction of statistical estimates	336	(3)
Analysis of the algorithm convergence	339	(3)
Results of numerical simulation	342	(2)
Statement of the final algorithm	344	(3)
Part III. Optimization of an FEM grid	347	(48)
Introduction	349	(3)
Optimal distribution of nodes for the problem of tension of a balk of variable section	352	(10)
Statement of the problem	352	(1)
Solution of the problem by means of FEM	353	(9)
Optimization in general case	362	(6)
Asymptotically optimal density	362	(4)
Examples	366	(2)
Numerical simulation	368	(5)
Statement of the problem	368	(1)
Step approximation of distribution function	368	(5)
BEMM optimal nodes with variable section, linear with respect to the length	373	(8)
Determinate case	374	(4)
Optimal asymptotic density	378	(2)
Construction of an optimal density of nodes distribution	380	(1)
Connection between optimal determinate nodes and optimal density	381	(4)
Results of numerical experiments	385	(10)
Simulation of determinate FEM solutions	385	(3)
Simulation of stochastic FEM solutions	388	(7)
Afterword	395	(2)
Bibliography	397	(16)
Index	413