Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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), 69-80
MSC: Primary 65J05; Secondary 47A50
MathSciNet review: 0461895
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider approximations $ \{ {x_n}\} $ obtained by moment discretization to (i) the minimal $ {\mathcal{L}_2}$-norm solution of $ \mathcal{K}x = y$ where $ \mathcal{K}$ is a Hilbert-Schmidt integral operator on $ {\mathcal{L}_2}$, and to (ii) the least squares solution of minimal $ {\mathcal{L}_2}$-norm of the same equation when y is not in the range $ \mathcal{R}(\mathcal{K})$ of $ \mathcal{K}$. In case (i), if $ y \in \mathcal{R}(\mathcal{K})$, then $ {x_n} \to {\mathcal{K}^\dag }y$, where $ {\mathcal{K}^\dag }$ is the generalized inverse of $ \mathcal{K}$, and $ \left\Vert {{x_n}} \right\Vert \to \infty $ otherwise. Rates of convergence are given in this case if further $ {\mathcal{K}^\dag }y \in {\mathcal{K}^\ast}({\mathcal{L}_2})$, where $ {\mathcal{K}^\ast}$ is the adjoint of $ \mathcal{K}$, and the Hilbert-Schmidt kernel of $ \mathcal{K}{\mathcal{K}^\ast}$ satisfies certain smoothness conditions. In case (ii), if $ y \in \mathcal{R}(\mathcal{K}) \oplus \mathcal{R}{(\mathcal{K})^ \bot }$, then $ {x_n} \to {\mathcal{K}^\dag }y$, and $ \left\Vert {{x_n}} \right\Vert \to \infty $ otherwise. If further $ {\mathcal{K}^\dag }y \in {\mathcal{K}^\ast}\mathcal{K}({\mathcal{L}_2})$, then rates of convergence are given in terms of the smoothness properties of the Hilbert-Schmidt kernel of $ {(\mathcal{K}{\mathcal{K}^\ast})^2}$.

Some of these results are generalized to a class of linear operator equations on abstract Hilbert spaces.

References [Enhancements On Off] (What's this?)

  • [1] N. Aronszajn, "Theory of reproducing kernels," Trans. Amer. Math. Soc., v. 68, 1950, pp. 337-404. MR 14, 479. MR 0051437 (14:479c)
  • [2] J. B. Diaz & F. T. Metcalf, "On iteration procedures for equations of the first kind, $ Ax = y$, and Picard's criterion for the existence of a solution," Math. Comp., v. 24, 1970, pp. 923-935. 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. 616-622, 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. 547-573. 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. 311-359. MR 43 #1003. MR 0275246 (43:1003)
  • [7] M. Z. Nashed & G. Wahba, Approximate Regularized Solutions to Linear Operator Equations When the Data-Vector 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 $ {(I - \beta K)^{\text{n}}}$ with applications to iterative methods," J. Math. Anal. Appl., v. 18, 1967, pp. 417-439. MR 34 #8191. MR 0208381 (34:8191)
  • [10] H. L. Shapiro, Topics in Approximation Theory, Lecture Notes in Math., vol. 187, Springer-Verlag, Berlin, 1970.
  • [11] D. W. Showalter & A. Ben-Israel, "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. 184-194. 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. 167-185. 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

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

American Mathematical Society