模型參數(shù)估計的反問題理論與方法

前言

　　要使我國的數(shù)學(xué)事業(yè)更好地發(fā)展起來，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強對數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強數(shù)學(xué)的出版工作?！　某霭娣矫鎭碇v，除了較好較快地出版我們自己的成果外，引進(jìn)國外的先進(jìn)出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社?？茖W(xué)出版社影印一批他們出版的好的新書，使我國廣大數(shù)學(xué)家能以較低的價格購買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無疑是對推動我國數(shù)學(xué)的科研與教學(xué)十分有益的事?！　∵@次科學(xué)出版社購買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙?，都是由該領(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊。對從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學(xué)獎”。這些大數(shù)學(xué)家的著作無疑將會對我國的科研人員起到非常好的指導(dǎo)作用?！　‘?dāng)然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?！　】傊?，我對科學(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內(nèi)容概要

Prompted by recent developments in inverse theory, Inverse Problem Theory and Methods for Model Parameter Estimation is a completely rewritten version of a 1987 book by the same author. In this version there are many algorithmic details for Monte Carlo methods, leastsquares discrete problems, and least-squares problems involving functions. In addition, some notions are clarified, the role of optimization techniques is underplayed, and Monte Carlo methods are taken much more seriously. The first part of the book deals exclusively with discrete inverse problems with afinite number of parameters, while the second part of the book deals with general inverse problems. ...

書籍目錄

Preface1　 The General Discrete Inverse Problem　1.1　 Model Space and Data Space　1.2　 States of Information　1.3　 Forward Problem　1.4　 Measurements and A Priori Information　1.5　 Defining the Solution of the Inverse Problem　1.6　 Using the Solution of the Inverse Problem2　 Monte Carlo Methods　2.1　 Introduction　2.2　 The Movie Strategy for Inverse Problems　2.3　 Sampling Methods　2.4　 Monte Carlo Solution to Inverse Problems  2.5　 Simulated Annealing3　 The Least-Squares Criterion  3.1　 Preamble: The Mathematics of Linear Spaces  3.2　 The Least-Squares Problem  3.3　 Estimating Posterior Uncertainties  3.4　 Least-Squares Gradient and Hessian4　 Least-Absolute-Values Criterion and Minimax Criterion  4.1　 Introduction  4.2　 Preamble:ln-Norms  4.3　 The ln-Norm Problem  4.4　 The l1-Norm Criterion for Inverse Problems  4.5　 The ln-Norm Criterion for Inverse Problems5　 Functional Inverse Problems  5.1　 Random Functions  5.2　 Solution of General Inverse Problems  5.3　 Introduction to Functional Least Squares  5.4　 Derivative and Transpose Operators in Functional Spaces  5.5　 General Least-Squares Inversion  5.6　 Example: X-Ray Tomography as an Inverse Problem  5.7　 Example: Travel-Time Tomography  5.8　 Example: Nonlinear Inversion of Elastic Waveforms6　 Appendices  6.1　 Volumetric Probability and Probability Density  6.2　 Homogeneous Probability Distributions  6.3　 Homogeneous Distribution for Elastic Parameters  6.4　 Homogeneous Distribution for Second-Rank Tensors  6.5　 Central Estimators and Estimators of Dispersion  6.6　 Generalized Gaussian  6.7　 Log-Normal Probability Density  6.8　 Chi-Squared Probability Density  6.9　 Monte Carlo Method of Numerical Integration  6.10　Sequential Random Realization  6.11　Cascaded Metropolis Algorithm  6.12　Distance and Norm  6.13　The Different Meanings of the Word Kernel  6.14　Transpose and Adjoint of a Differential Operator  6.15　The Bayesian Viewpoint of Backus (1970)  6.16　The Method of Backus and Gilbert  6.17　Disjunction and Conjunction of Probabilities  6.18　Partition of Data into Subsets  6.19　Marginalizing in Linear Least Squares  6.20　Relative Information of Two Gaussians  6.21　Convolution of Two Gaussians  6.22　Gradient-Based Optimization Algorithms  6.23　Elements of Linear Programming  6.24　Spaces and Operators  6.25　Usual Functional Spaces  6.26　Maximum Entropy Probability Density  6.27　Two Properties of ln-Norms  6.28　Discrete Derivative Operator  6.29　Lagrange Parameters  6.30　Matrix Identities  6.31　Inverse of a Partitioned Matrix  6.32　Norm of the Generalized Gaussian7　 Problems  7.1　 Estimation of the Epicentral Coordinates of a Seismic Event  7.2　 Measuring the Acceleration of Gravity  7.3　 Elementary Approach to Tomography  7.4　 Linear Regression with Rounding Errors  7.5　 Usual Least-Squares Regression  7.6　 Least-Squares Regression with Uncertainties in Both Axes  7.7　 Linear Regression with an Outlier  7.8　 Condition Number and A Posteriori Uncertainties  7.9　 Conjunction of Two Probability Distributions  7.10　Adjoint of a Covariance Operator  7.11　Problem 7.1 Revisited  7.12　Problem 7.3 Revisited  7.13　An Example of Partial Derivatives  7.14　Shapes of the ln-Norm Misfit Functions  7.15　Using the Simplex Method  7.16　Problem 7.7 Revisited  7.17　Geodetic Adjustment with Outliers  7.18　Inversion of Acoustic Waveforms  7.19　Using the Backus and Gilbert Method  7.20　The Coefficients in the Backus and Gilbert Method  7.21　The Norm Associated with the 1D Exponential Covariance  7.22　The Norm Associated with the 1D Random Walk  7.23　The Norm Associated with the 3D Exponential CovarianceReferences and References for General ReadingIndex

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