Theory and Methods Related to the Singular-Function Expansion and Landweber's Iteration for Integral Equations of the First Kind
For the Fredholm integral equation of the first kind, written notationally as Kf = g, g ∈ L2 [ 0, 1 ], we study the behavior of the iteration $\hat{f}_k = \hat{f}_{k - 1} + DK^\ast(g - K \hat{f}_{k - 1}),\quad k = 1, 2, \cdots,$ both with respect to its convergence properties and its response to sin...
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| Vydáno v: | SIAM journal on numerical analysis Ročník 11; číslo 4; s. 798 - 825 |
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| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Philadelphia
Society for Industrial and Applied Mathematics
01.09.1974
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| ISSN: | 0036-1429, 1095-7170 |
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| Abstract | For the Fredholm integral equation of the first kind, written notationally as Kf = g, g ∈ L2 [ 0, 1 ], we study the behavior of the iteration $\hat{f}_k = \hat{f}_{k - 1} + DK^\ast(g - K \hat{f}_{k - 1}),\quad k = 1, 2, \cdots,$ both with respect to its convergence properties and its response to singular functions of K. Here K* is the adjoint of K, f̂0 is a suitable starting function and D is a fixed linear operator to be chosen. The general theory of singular functions is interpreted and extended for the study of the iteration, and a quantitative method of choosing D to shape the response to singular functions of K is derived. Several specific matrix and integral-equation examples are presented. |
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| AbstractList | For the Fredholm integral equation of the first kind, written notationally as Kf = g, g ∈ L2 [ 0, 1 ], we study the behavior of the iteration $\hat{f}_k = \hat{f}_{k - 1} + DK^\ast(g - K \hat{f}_{k - 1}),\quad k = 1, 2, \cdots,$ both with respect to its convergence properties and its response to singular functions of K. Here K* is the adjoint of K, f̂0 is a suitable starting function and D is a fixed linear operator to be chosen. The general theory of singular functions is interpreted and extended for the study of the iteration, and a quantitative method of choosing D to shape the response to singular functions of K is derived. Several specific matrix and integral-equation examples are presented. For the Fredholm integral equation of the first kind, written notationally as $Kf = g$, $g \in L_2 [0,1]$, we study the behavior of the iteration \[ \hat f_k = \hat f_{k - 1} + DK^ * (g - K\hat f_{k - 1} ),\quad k = 1,2, \cdots ,\] both with respect to its convergence properties and its response to singular functions of $K$. Here $K^ * $ is the adjoint of $K$, $\hat f_0 $ is a suitable starting function and $D$ is a fixed linear operator to be chosen. The general theory of singular functions is interpreted and extended for the study of the iteration, and a quantitative method of choosing $D$ to shape the response to singular functions of $K$ is derived. Several specific matrix and integral-equation examples are presented. |
| Author | Strand, Otto Neall |
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| Cites_doi | 10.1175/1520-0469(1965)022<0370:OTICOU>2.0.CO;2 10.1090/S0025-5718-1963-0163426-5 10.1093/comjnl/7.2.141 10.1137/0708058 10.1364/AO.10.002035 10.1016/0022-247X(72)90259-4 10.2307/2372313 10.1364/AO.9.001993 10.1145/321105.321114 10.1016/0016-0032(65)90209-7 10.1175/1520-0493(1972)100<0015:GDOLAT>2.3.CO;2 10.1137/0705025 10.1145/321150.321157 10.1175/1520-0493(1966)094<0363:IMOATP>2.3.CO;2 |
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| References | Bakušinski A. B. (R2) 1967; 7 R5 Picard E. (R16) 1910; 29 Riesz Frigyes (R18) 1955 Smithies F. (R22) 1958 R10 R21 Ralston Anthony (R17) 1965 R20 R31 R12 R14 R13 R24 Diaz J. B. (R6) 1970; 24 R27 R15 Tricomi F. G. (R26) 1957 R29 R28 R1 Fridman V. M. (R9) 1956; 11 Tikhonov A. N. (R25) 1963; 4 |
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| Snippet | For the Fredholm integral equation of the first kind, written notationally as Kf = g, g ∈ L2 [ 0, 1 ], we study the behavior of the iteration $\hat{f}_k =... For the Fredholm integral equation of the first kind, written notationally as $Kf = g$, $g \in L_2 [0,1]$, we study the behavior of the iteration \[ \hat f_k =... |
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| SubjectTerms | Ambiguity Approximation Differential equations Eigenfunctions Error function Error rates Integral equations Iterative solutions Mathematics Methods Perceptron convergence procedure Polynomials |
| Title | Theory and Methods Related to the Singular-Function Expansion and Landweber's Iteration for Integral Equations of the First Kind |
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