Multiscale Problems and Methods in Numerical Simulations

by Bramble, James H.; Cohen, Albert; Dahmen, Wolfgang; Canuto, C.; C. I. M. E. COURSE ON MULTISCALE PROBLEMS

• ISBN13: 9783540200994
• ISBN10: 3540200991
• Format: Paperback
• Copyright: 2003-10-01
• Publisher: Springer Nature

Summary

This volume aims to disseminate a number of new ideas that have emerged in the last few years in the field of numerical simulation, all bearing the common denominator of the "multiscale" or "multilevel" paradigm. This covers the presence of multiple relevant "scales" in a physical phenomenon; the detection and representation of "structures", localized in space or in frequency, in the solution of a mathematical model; the decomposition of a function into "details" that can be organized and accessed in decreasing order of importance; and the iterative solution of systems of linear algebraic equations using "multilevel" decompositions of finite dimensional spaces.

Table of Contents

Theoretical, Applied and Computational Aspects of Nonlinear Approximation	1	(30)
Albert Cohen
Introduction	1	(3)
A Simple Example	4	(3)
The Haar System and Thresholding	7	(2)
Linear Uniform Approximation	9	(6)
Nonlinear Adaptive Approximation	15	(3)
Data Compression	18	(3)
Statistical Estimation	21	(3)
Adaptive Numerical Simulation	24	(2)
The Curse of Dimensionality	26	(5)
References	28	(3)
Multiscale and Wavelet Methods for Operator Equations	31	(66)
Wolfgang Dahmen
Introduction	31	(1)
Examples, Motivation	32	(7)
Sparse Representations of Functions, an Example	32	(4)
(Quasi-) Sparse Representation of Operators	36	(1)
Preconditioning	37	(2)
Summary	39	(1)
Wavelet Bases -- Main Features	39	(6)
The General Format	39	(1)
Notational Conventions	40	(1)
Main Features	40	(5)
Criteria for (NE)	45	(6)
What Could Additional Conditions Look Like?	45	(1)
Fourier- and Basis-free Criteria	46	(5)
Multiscale Decompositions -- Construction and Analysis Principles	51	(6)
Multiresolution	51	(1)
Stability of Multiscale Transformations	52	(1)
Construction of Biorthogonal Bases -- Stable Completions	53	(1)
Refinement Relations	53	(2)
Structure of Multiscale Transformations	55	(1)
Parametrization of Stable Completions	56	(1)
Scope of Problems	57	(9)
Problem Setting	57	(2)
Scalar 2nd Order Elliptic Boundary Value Problem	59	(1)
Global Operators -- Boundary Integral Equations	59	(2)
Saddle Point Problems	61	(5)
An Equivalent l2-Problem	66	(2)
Connection with Preconditioning	67	(1)
There is always a Positive Definite Formulation -- Least Squares	68	(1)
Adaptive Wavelet Schemes	68	(20)
Introductory Comments	68	(2)
Adaptivity from Several Perspectives	70	(1)
The Basic Paradigm	70	(1)
(III) Convergent Iteration for the ∞-dimensional Problem	71	(3)
(IV) Adaptive Application of Operators	74	(1)
The Adaptive Algorithm	75	(1)
Ideal Bench Mark -- Best N-Term Approximation	76	(1)
Compressible Matrices	76	(1)
Fast Approximate Matrix/Vector Multiplication	77	(2)
Application Through Uzawa Iteration	79	(1)
Main Result -- Convergence/Complexity	79	(1)
Some Ingredients of the Proof of Theorem 8	80	(5)
Approximation Properties and Regularity	85	(3)
Further Issues, Applications	88	(2)
Nonlinear Problems	88	(2)
Time Dependent Problems	90	(1)
Appendix: Some Useful Facts	90	(7)
Function Spaces	90	(1)
Local Polynomial Approximation	91	(1)
Condition Numbers	92	(1)
References	93	(4)
Multilevel Methods in Finite Elements	97	(56)
James H. Bramble
Introduction	97	(9)
Sobolev Spaces	97	(1)
A Model Problem	98	(2)
Finite Element Approximation of the Model Problem	100	(1)
The Stiffness Matrix and its Condition Number	101	(1)
A Two-Level Multigrid Method	102	(4)
Multigrid I	106	(6)
An Abstract V-cycle Algorithm	107	(1)
The Multilevel Framework	107	(1)
The Abstract V-cycle Algorithm, I	108	(1)
The Two-level Error Recurrence	109	(1)
The Braess-Hackbusch Theorem	110	(2)
Multigrid II: V-cycle with Less Than Full Elliptic Regularity	112	(9)
Introduction and Preliminaries	112	(4)
The Multiplicative Error Representation	116	(1)
Some Technical Lemmas	117	(2)
Uniform Estimates	119	(2)
Non-nested Multigrid	121	(12)
Non-nested Spaces and Varying Forms	121	(1)
General Multigrid Algorithms	122	(3)
Multigrid V-cycle as a Reducer	125	(2)
Multigrid W-cycle as a Reducer	127	(4)
Multigrid V-cycle as a Preconditioner	131	(2)
Computational Scales of Sobolev Norms	133	(20)
Introduction	133	(2)
A Norm Equivalence Theorem	135	(3)
Development of Preconditioner	138	(1)
Preconditioning Sums of Operators	138	(1)
A Simple Approximation Operator Qk	139	(1)
Some Basic Approximation Properties	140	(1)
Approximation Properties: the Multilevel Case	141	(3)
The Coercivity Estimate	144	(1)
Applications	145	(1)
A Preconditioning Example	146	(1)
Two Examples Involving Sums of Operators	147	(1)
H1 (Ω) Bounded Extensions	148	(2)
References	150	(3)
List of Participants	153