Download Inverse and Ill-Posed Problems by Heinz W. Engl, C. W. Groetsch PDF

By Heinz W. Engl, C. W. Groetsch

ISBN-10: 0122390407

ISBN-13: 9780122390401

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2) results in the problem of finding c, d, Θ to minim­ ize 1 n \\y -Xc -Td-SQ\\ +Xc'Zc. 2) where X, T, S, Σ are given matrices. -ΓΓ(/-Λ(λ)) n where A (λ) is the influence or hat matrix. 4) G. Wahba 46 where g λ i s t h e minimizer. (For properties of the GCV estimate of λ, see Wahba (1985a)). This problem can be put in a standard form for which transportable code has been developed. 8) choose λ by GCV. Transportable code Bates et al. 3). The code uses Cholesky decomposition and a truncated singular value decomposition.

The diameter of S p is d (S p e ) = sup || x x - x 2 || , where the sup is taken over all x-, and Xo in S p . If an element of S p is known, we can define its uncertainty. (9) Reliability of Information from Approximately-Solved Problems D e f i n i t i o n 3. Given an element x 33 i n Sp , i t s u n c e r t a i n t y is defined as U(x) = sup || x - x | | , (10) where the sup is over all x in S p . In principle, the diameter of Sp £ and the uncertainty of x are computable without recourse to unverifiable assumptions.

S with particu­ lar application to the resevoir modelling problem. One has the equation (for example) Ì7 = S ar ( a ( *)Ìr ) + * ( * , ° ' *=(*ι^3)εΩ,ίε[0,7] (U) where the forcing function q is known exactly and u is measured with noise at a finite set of points in x and t. The problem is to estimate the transmittivity a, which is not neces­ sarily a constant. Recently Kravaris and Seinfeld (1985) proposed a penalized likelihood approach to the estimation of a. This problem is both nonlinear and in general ill posed.

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