Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind
Authors:
M. Z. Nashed and Grace Wahba
Journal:
Math. Comp. 28 (1974), 6980
MSC:
Primary 65J05; Secondary 47A50
MathSciNet review:
0461895
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider approximations obtained by moment discretization to (i) the minimal norm solution of where is a HilbertSchmidt integral operator on , and to (ii) the least squares solution of minimal norm of the same equation when y is not in the range of . In case (i), if , then , where is the generalized inverse of , and otherwise. Rates of convergence are given in this case if further , where is the adjoint of , and the HilbertSchmidt kernel of satisfies certain smoothness conditions. In case (ii), if , then , and otherwise. If further , then rates of convergence are given in terms of the smoothness properties of the HilbertSchmidt kernel of . Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.
 [1]
N. Aronszajn, "Theory of reproducing kernels," Trans. Amer. Math. Soc., v. 68, 1950, pp. 337404. MR 14, 479. MR 0051437 (14:479c)
 [2]
J. B. Diaz & F. T. Metcalf, "On iteration procedures for equations of the first kind, , and Picard's criterion for the existence of a solution," Math. Comp., v. 24, 1970, pp. 923935. MR 43 #7094. MR 0281376 (43:7094)
 [3]
R. J. Hanson, "A numerical method for solving Fredholm integral equations of the first kind using singular values," SIAM J. Numer. Anal., v. 8, 1971, pp. 616622, MR 45 #2943. MR 0293867 (45:2943)
 [4]
E. Hellinger & O. Toeplitz, "Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten," Enzyklopädie der Mathematischen Wissenschaften II C13, 1928, p. 1349.
 [5]
W. J. Kammerer & M. Z. Nashed, "Iterative methods for best approximate solutions of linear integral equations of the first and second kinds," MRC Technical Summary Report #1117, Mathematics Research Center, The University of Wisconsin, 1971; J. Math. Anal. Appl., v. 40, 1972, pp. 547573. MR 0320677 (47:9213)
 [6]
M. Z. Nashed, "Generalized inverses, normal solvability, and iteration for singular operator equations," in Nonlinear Functional Analysis and Applications (Proc. Advanced Semi., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970), L. B. Rall (editor), Academic Press, New York, 1971, pp. 311359. MR 43 #1003. MR 0275246 (43:1003)
 [7]
M. Z. Nashed & G. Wahba, Approximate Regularized Solutions to Linear Operator Equations When the DataVector is Not in the Range of the Operator, MRC Technical Summary Report #1265, Mathematics Research Center, The University of Wisconsin, 1973.
 [8]
B. Noble, A Bibliography on: "Methods for Solving Integral Equations"Subject Listing, MRC Technical Summary Report #1177, Mathematics Research Center, The University of Wisconsin, 1971.
 [9]
W. V. Petryshyn, "On generalized inverses and uniform convergence of with applications to iterative methods," J. Math. Anal. Appl., v. 18, 1967, pp. 417439. MR 34 #8191. MR 0208381 (34:8191)
 [10]
H. L. Shapiro, Topics in Approximation Theory, Lecture Notes in Math., vol. 187, SpringerVerlag, Berlin, 1970.
 [11]
D. W. Showalter & A. BenIsrael, "Representation and computation of the generalized inverse of a bounded linear operator between two Hilbert spaces," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8), v. 48, 1970, pp. 184194. MR 42 #8310. MR 0273432 (42:8310)
 [12]
F. Smithies, Integral Equations, Cambridge Tracts in Math and Math. Phys., no. 49, Cambridge Univ. Press, New York, 1958. MR 21 #3738. MR 0104991 (21:3738)
 [13]
O. N. Strand, Theory and Methods for Operator Equations of the First Kind, Ph.D. Thesis, Colorado State University, Fort Collins, Colorado, 1972, 86 pp.
 [14]
G. Wahba, "Convergence rates for certain approximate solutions to Fredholm integral equations of the first kind," J. Approximation Theory, v. 7, 1973, pp. 167185. MR 0346453 (49:11178)
 [15]
G. Wahba, "A class of approximate solutions to linear operator equations," J. Approximation Theory (To appear.) MR 0386284 (52:7142)
 [16]
G. Wahba, Convergence Properties of the Method of Regularization for Noisy Linear Operator Equations, MRC Technical Summary Report #1132, Mathematics Research Center, The University of Wisconsin, 1973.
 [1]
 N. Aronszajn, "Theory of reproducing kernels," Trans. Amer. Math. Soc., v. 68, 1950, pp. 337404. MR 14, 479. MR 0051437 (14:479c)
 [2]
 J. B. Diaz & F. T. Metcalf, "On iteration procedures for equations of the first kind, , and Picard's criterion for the existence of a solution," Math. Comp., v. 24, 1970, pp. 923935. MR 43 #7094. MR 0281376 (43:7094)
 [3]
 R. J. Hanson, "A numerical method for solving Fredholm integral equations of the first kind using singular values," SIAM J. Numer. Anal., v. 8, 1971, pp. 616622, MR 45 #2943. MR 0293867 (45:2943)
 [4]
 E. Hellinger & O. Toeplitz, "Integralgleichungen und Gleichungen mit unendlich vielen Unbekannten," Enzyklopädie der Mathematischen Wissenschaften II C13, 1928, p. 1349.
 [5]
 W. J. Kammerer & M. Z. Nashed, "Iterative methods for best approximate solutions of linear integral equations of the first and second kinds," MRC Technical Summary Report #1117, Mathematics Research Center, The University of Wisconsin, 1971; J. Math. Anal. Appl., v. 40, 1972, pp. 547573. MR 0320677 (47:9213)
 [6]
 M. Z. Nashed, "Generalized inverses, normal solvability, and iteration for singular operator equations," in Nonlinear Functional Analysis and Applications (Proc. Advanced Semi., Math. Res. Center, Univ. of Wisconsin, Madison, Wis., 1970), L. B. Rall (editor), Academic Press, New York, 1971, pp. 311359. MR 43 #1003. MR 0275246 (43:1003)
 [7]
 M. Z. Nashed & G. Wahba, Approximate Regularized Solutions to Linear Operator Equations When the DataVector is Not in the Range of the Operator, MRC Technical Summary Report #1265, Mathematics Research Center, The University of Wisconsin, 1973.
 [8]
 B. Noble, A Bibliography on: "Methods for Solving Integral Equations"Subject Listing, MRC Technical Summary Report #1177, Mathematics Research Center, The University of Wisconsin, 1971.
 [9]
 W. V. Petryshyn, "On generalized inverses and uniform convergence of with applications to iterative methods," J. Math. Anal. Appl., v. 18, 1967, pp. 417439. MR 34 #8191. MR 0208381 (34:8191)
 [10]
 H. L. Shapiro, Topics in Approximation Theory, Lecture Notes in Math., vol. 187, SpringerVerlag, Berlin, 1970.
 [11]
 D. W. Showalter & A. BenIsrael, "Representation and computation of the generalized inverse of a bounded linear operator between two Hilbert spaces," Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8), v. 48, 1970, pp. 184194. MR 42 #8310. MR 0273432 (42:8310)
 [12]
 F. Smithies, Integral Equations, Cambridge Tracts in Math and Math. Phys., no. 49, Cambridge Univ. Press, New York, 1958. MR 21 #3738. MR 0104991 (21:3738)
 [13]
 O. N. Strand, Theory and Methods for Operator Equations of the First Kind, Ph.D. Thesis, Colorado State University, Fort Collins, Colorado, 1972, 86 pp.
 [14]
 G. Wahba, "Convergence rates for certain approximate solutions to Fredholm integral equations of the first kind," J. Approximation Theory, v. 7, 1973, pp. 167185. MR 0346453 (49:11178)
 [15]
 G. Wahba, "A class of approximate solutions to linear operator equations," J. Approximation Theory (To appear.) MR 0386284 (52:7142)
 [16]
 G. Wahba, Convergence Properties of the Method of Regularization for Noisy Linear Operator Equations, MRC Technical Summary Report #1132, Mathematics Research Center, The University of Wisconsin, 1973.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65J05,
47A50
Retrieve articles in all journals
with MSC:
65J05,
47A50
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197404618951
PII:
S 00255718(1974)04618951
Keywords:
First kind integral equations,
linear operator equations,
moment discretization,
convergence rates,
least squares solutions,
generalized inverses,
reproducing kernel Hilbert spaces,
Picard criterion
Article copyright:
© Copyright 1974
American Mathematical Society
