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One might equally plausibly try two values of λ, one giving χ2 =N + (2N )1/2 , the other N − (2N )1/2 .Zeroth-order regularization, though dominated by better methods, demonstratesmost of the basic ideas that are used in inverse problem theory. In general, there aretwo positive functionals, call them A and B. The first, A, measures something likethe agreement of a model to the data (e.g., χ2 ), or sometimes a related quantity likethe “sharpness” of the mapping between the solution and the underlying function.When A by itself is minimized, the agreement or sharpness becomes very good(often impossibly good), but the solution becomes unstable, wildly oscillating, or inother ways unrealistic, reflecting that A alone typically defines a highly degenerateminimization problem.That is where B comes in.
It measures something like the “smoothness” of thedesired solution, or sometimes a related quantity that parametrizes the stability ofthe solution with respect to variations in the data, or sometimes a quantity reflectinga priori judgments about the likelihood of a solution. B is called the stabilizingfunctional or regularizing operator. In any case, minimizing B by itself is supposedto give a solution that is “smooth” or “stable” or “likely” — and that has nothingat all to do with the measured data.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited.
To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).Better Agreement18.4 Inverse Problems and the Use of A Priori Information808Chapter 18.Integral Equations and Inverse TheoryThe single central idea in inverse theory is the prescriptionminimize:A + λB(18.4.12)CITED REFERENCES AND FURTHER READING:Craig, I.J.D., and Brown, J.C. 1986, Inverse Problems in Astronomy (Bristol, U.K.: Adam Hilger).Twomey, S.
1977, Introduction to the Mathematics of Inversion in Remote Sensing and IndirectMeasurements (Amsterdam: Elsevier).Tikhonov, A.N., and Arsenin, V.Y. 1977, Solutions of Ill-Posed Problems (New York: Wiley).Tikhonov, A.N., and Goncharsky, A.V. (eds.) 1987, Ill-Posed Problems in the Natural Sciences(Moscow: MIR).Parker, R.L. 1977, Annual Review of Earth and Planetary Science, vol.
5, pp. 35–64.Frieden, B.R. 1975, in Picture Processing and Digital Filtering, T.S. Huang, ed. (New York:Springer-Verlag).Tarantola, A. 1987, Inverse Problem Theory (Amsterdam: Elsevier).Baumeister, J. 1987, Stable Solution of Inverse Problems (Braunschweig, Germany: Friedr. Vieweg& Sohn) [mathematically oriented].Titterington, D.M. 1985, Astronomy and Astrophysics, vol. 144, pp. 381–387.Jeffrey, W., and Rosner, R. 1986, Astrophysical Journal, vol. 310, pp. 463–472.18.5 Linear Regularization MethodsWhat we will call linear regularization is also called the Phillips-Twomeymethod [1,2] , the constrained linear inversion method [3], the method of regularization [4], and Tikhonov-Miller regularization [5-7].
(It probably has other names also,Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).for various values of 0 < λ < ∞ along the so-called trade-off curve (see Figure18.4.1), and then to settle on a “best” value of λ by one or another criterion, rangingfrom fairly objective (e.g., making χ2 = N ) to entirely subjective. Successfulmethods, several of which we will now describe, differ as to their choices of A andB, as to whether the prescription (18.4.12) yields linear or nonlinear equations, asto their recommended method for selecting a final λ, and as to their practicality forcomputer-intensive two-dimensional problems like image processing.They also differ as to the philosophical baggage that they (or rather, theirproponents) carry.
We have thus far avoided the word “Bayesian.” (Courts haveconsistently held that academic license does not extend to shouting “Bayesian” in acrowded lecture hall.) But it is hard, nor have we any wish, to disguise the fact thatB has something to do with a priori expectation, or knowledge, of a solution, whileA has something to do with a posteriori knowledge. The constant λ adjudicates adelicate compromise between the two.
Some inverse methods have acquired a moreBayesian stamp than others, but we think that this is purely an accident of history.An outsider looking only at the equations that are actually solved, and not at theaccompanying philosophical justifications, would have a difficult time separating theso-called Bayesian methods from the so-called empirical ones, we think.The next three sections discuss three different approaches to the problem ofinversion, which have had considerable success in different fields.
All three fitwithin the general framework that we have outlined, but they are quite different indetail and in implementation..
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