By Henryk Gzyl
The ebook describes a useful gizmo for fixing linear inverse difficulties topic to convex constraints. the tactic of utmost entropy within the suggest immediately looks after the limitations. It comprises a method for reworking a wide dimensional inverse challenge right into a small dimensional non-linear variational challenge. numerous mathematical points of the utmost entropy process are explored in addition.
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Extra info for Linear Inverse Problems: The Maximum Entropy Connection
L. “Flying Buttresses, Entropy and O. Rings”. Harvard Univ. Press, Cambridge, 1991.  Shepp, L. A. and Kruskal, J. B. “Computerized Tomography: the new medical X-ray technology”. Ann. Math. Monthly, Vol. 85 (1978) pp. 420-439.  Panton, D. “Mathematical reconstruction techniques in computer axial tomography”. Math. Scientist, Vol. 6 (1981), pp. 87-102. , Hermann, A. and Lackner, K. “Two-dimensional reconstruction of the radiation power density in ASDEX upgrade”. 21st. S. Conf. on Controlled Fusion and Plasma Physics.
9. Given a regularization scheme Rα for an operator A, a choice of regularization parameter α = α(ε) depending on the error level ε is called regular if for any y ∈ A(V ) and any y(ε) with y(ε) − y ≤ ε we have Rα(ε) y(ε) → A−1 y as ε ↓ 0. Comment: In other words, for a regular scheme, the approximate solution is near the true solution when the approximate data is near the true data. 7, and we do not demand that A−1 be continuous (if it exists). 10. , if x1 , x2 ∈ V1 and Ax1 = Ax2 = y, then x1 = x2 .
This happens when for example, f (t) is the response of a linear system. As started above, the problem belongs to the class of generalized moment problems. , f˜(sn ). First approach: Just solve for f (t) given the ﬁnite data. , n; then invert the Laplace transform assuming that f˜l (s) is the right Laplace transform of the unknown f. A collection of linear inverse problems 11 We shall have more to say about both approaches. , n. 5 Fourier reconstruction from partial data Let f (t) be a real-valued, function deﬁned on an interval [0, T ].
Linear Inverse Problems: The Maximum Entropy Connection by Henryk Gzyl