Regularization for thorder linear boundary value problems using thorder differential operators
Authors:
D. A. Kouba and John Locker
Journal:
Trans. Amer. Math. Soc. 293 (1986), 229255
MSC:
Primary 47E05; Secondary 34B05, 47A50
MathSciNet review:
814921
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let and denote real Hilbert spaces, and let be a closed denselydefined linear operator having closed range. Given an element , we determine least squares solutions of the linear equation by using the method of regularization. Let be a third Hilbert space, and let be a linear operator with . Under suitable conditions on and and for each , we show that there exists a unique element which minimizes the functional , and the converge to a least squares solution of as . We apply our results to the special case where is an thorder differential operator in , and we regularize using for an thorder differential operator in with . Using an approximating space of Hermite splines, we construct numerical solutions to by the method of continuous least squares and the method of discrete least squares.
 [1]
Uri
Ascher, Discrete least squares approximations for ordinary
differential equations, SIAM J. Numer. Anal. 15
(1978), no. 3, 478–496. MR 491701
(81e:65043), http://dx.doi.org/10.1137/0715031
 [2]
Richard
Bellman, A note on an inequality of E.
Schmidt, Bull. Amer. Math. Soc. 50 (1944), 734–736. MR 0010732
(6,61g), http://dx.doi.org/10.1090/S000299041944082281
 [3]
Frederick
J. Beutler and William
L. Root, The operator pseudoinverse in control and systems
identification, Generalized inverses and applications (Proc. Sem.,
Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1973) Academic Press,
New York, 1976, pp. 397–494. Publ. Math. Res. Center Univ.
Wisconsin, No. 32. MR 0490311
(58 #9658)
 [4]
Lamberto
Cesari, Nonlinear analysis, Nonlinear mechanics (Centro
Internaz. Mat. Estivo (C.I.M.E.), I Ciclo, Bressanone, 1972) Edizioni
Cremonese, Rome, 1973, pp. 1–95. MR 0407672
(53 #11444)
 [5]
Lamberto
Cesari, Functional analysis, nonlinear differential equations, and
the alternative method, Nonlinear functional analysis and differential
equations (Proc. Conf., Mich. State Univ., East Lansing, Mich., 1975)
Marcel Dekker, New York, 1976, pp. 1–197. Lecture Notes in Pure
and Appl. Math., Vol. 19. MR 0487630
(58 #7249)
 [6]
S. D. Conte and C. deBoor, Elementary numerical analysis, 2nd ed., McGrawHill, New York, 1972.
 [7]
Philip
J. Davis, Interpolation and approximation, Dover Publications,
Inc., New York, 1975. Republication, with minor corrections, of the 1963
original, with a new preface and bibliography. MR 0380189
(52 #1089)
 [8]
Carl
de Boor and Blâir
Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal.
10 (1973), 582–606. MR 0373328
(51 #9528)
 [9]
Jim
Douglas Jr. and Todd
Dupont, Galerkin approximations for the two point boundary problem
using continuous, piecewise polynomial spaces, Numer. Math.
22 (1974), 99–109. MR 0362922
(50 #15360)
 [10]
J. K. Hale, Applications of alternative problems, Lecture Notes, Brown Univ., Providence, R.I., 1971.
 [11]
Eugene
Isaacson and Herbert
Bishop Keller, Analysis of numerical methods, John Wiley &
Sons, Inc., New YorkLondonSydney, 1966. MR 0201039
(34 #924)
 [12]
V.
V. Ivanov, The theory of approximate methods and their application
to the numerical solution of singular integral equations, Noordhoff
International Publishing, Leyden, 1976. Translated from the Russian by A.
Ideh; Edited by R. S. Anderssen and D. Elliott; Monographs and Textbooks on
Mechanics of Solids and Fluids, Mechanics: Analysis, No. 2. MR 0405045
(53 #8841)
 [13]
R.
Kannan and John
Locker, Continuous dependence of least squares
solutions of linear boundary value problems, Proc. Amer. Math. Soc. 59 (1976), no. 1, 107–110. MR 0409947
(53 #13699), http://dx.doi.org/10.1090/S0002993919760409947X
 [14]
Tosio
Kato, Perturbation theory for linear operators, 2nd ed.,
SpringerVerlag, BerlinNew York, 1976. Grundlehren der Mathematischen
Wissenschaften, Band 132. MR 0407617
(53 #11389)
 [15]
D. Kouba, Regularization for thorder linear boundary value problems using thorder differential operators, Doctoral Dissertation, Colorado State Univ., 1982.
 [16]
John
Locker, The generalized Green’s function
for an 𝑛th order linear differential operator, Trans. Amer. Math. Soc. 228 (1977), 243–268. MR 0481204
(58 #1340), http://dx.doi.org/10.1090/S00029947197704812040
 [17]
, Functional analysis and twopoint differential operators, Lecture Notes, Colorado State Univ., Fort Collins, 1982.
 [18]
John
Locker and P.
M. Prenter, Optimal 𝐿² and 𝐿^{∞} error
estimates for continuous and discrete least squares methods for boundary
value problems, SIAM J. Numer. Anal. 15 (1978),
no. 6, 1151–1160. MR 512688
(80d:65097), http://dx.doi.org/10.1137/0715076
 [19]
John
Locker and P.
M. Prenter, Regularization with differential operators. I. General
theory, J. Math. Anal. Appl. 74 (1980), no. 2,
504–529. MR
572669 (83j:65062a), http://dx.doi.org/10.1016/0022247X(80)901456
 [20]
John
Locker and P.
M. Prenter, Regularization with differential operators. II. Weak
least squares finite element solutions to first kind integral
equations, SIAM J. Numer. Anal. 17 (1980),
no. 2, 247–267. MR 567272
(83j:65062b), http://dx.doi.org/10.1137/0717022
 [21]
, Regularization and linear boundary value problems (to appear).
 [22]
, Representors and superconvergence of least squares finite element approximates, Applicable Anal. (to appear).
 [23]
M.
Zuhair Nashed, Approximate regularized solutions to improperly
posed linear integral and operator equations, Constructive and
computational methods for differential and integral equations (Sympos.,
Indiana Univ., Bloomington, Ind., 1974) Springer, Berlin, 1974,
pp. 289–332. Lecture Notes in Math., Vol. 430. MR 0501883
(58 #19118)
 [24]
David
L. Phillips, A technique for the numerical solution of certain
integral equations of the first kind, J. Assoc. Comput. Mach.
9 (1962), 84–97. MR 0134481
(24 #B534)
 [25]
P.
M. Prenter, Splines and variational methods,
WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1975.
Pure and Applied Mathematics. MR 0483270
(58 #3287)
 [26]
Peter
H. Sammon, A discrete least squares
method, Math. Comp. 31
(1977), no. 137, 60–65. MR 0431699
(55 #4694), http://dx.doi.org/10.1090/S00255718197704316997
 [27]
Martin
H. Schultz, Error bounds for polynomial spline
interpolation, Math. Comp. 24 (1970), 507–515. MR 0275025
(43 #783), http://dx.doi.org/10.1090/S00255718197002750259
 [28]
A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Math. Dokl. 4 (1963), 10351038.
 [29]
, Regularization of incorrectly posed problems, Soviet Math. Dokl. 4 (1963), 16241627.
 [1]
 U. Ascher, Discrete least squares approximations for ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), 478496. MR 491701 (81e:65043)
 [2]
 R. Bellman, A note on an inequality of E. Schmidt, Bull. Amer. Math. Soc. 50 (1944), 734736. MR 0010732 (6:61g)
 [3]
 F. J. Beutler and W. L. Root, The operator pseudoinverse in control and systems identification, Generalized Inverses and Applications (M. Z. Nashed, ed.), Academic Press, New York, 1976, pp. 397494. MR 0490311 (58:9658)
 [4]
 L. Cesari, Nonlinear analysis, Lecture Notes, C.I.M.E., Bressanone, 1972. MR 0407672 (53:11444)
 [5]
 , Functional analysis, nonlinear differential equations, and the alternative method, Nonlinear Functional Analysis and Differential Equations (L. Cesari, R. Kannan and J. D. Schur, eds.), Marcel Dekker, New York, 1976, pp. 1197. MR 0487630 (58:7249)
 [6]
 S. D. Conte and C. deBoor, Elementary numerical analysis, 2nd ed., McGrawHill, New York, 1972.
 [7]
 P. J. Davis, Interpolation and approximation, Dover, New York, 1975. MR 0380189 (52:1089)
 [8]
 C. deBoor and B. Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582606. MR 0373328 (51:9528)
 [9]
 J. Douglas, Jr. and T. Dupont, Galerkin approximations for the two point boundary value problem using continuous, piecewise polynomial spaces, Numer. Math. 22 (1974), 99109. MR 0362922 (50:15360)
 [10]
 J. K. Hale, Applications of alternative problems, Lecture Notes, Brown Univ., Providence, R.I., 1971.
 [11]
 E. Isaacson and H. B. Keller, Analysis of numerical methods, Wiley, New York, 1966. MR 0201039 (34:924)
 [12]
 V. V. Ivanov, The theory of approximate methods and their application to the numerical solution of singular integral equations, Noordhoff, Leyden, 1976. MR 0405045 (53:8841)
 [13]
 R. Kannan and J. Locker, Continuous dependence of least squares solutions of linear boundary value problems, Proc. Amer. Math. Soc. 59 (1976), 107110. MR 0409947 (53:13699)
 [14]
 T. Kato, Perturbation theory for linear operators, 2nd ed., SpringerVerlag, Berlin, Heidelberg and New York, 1976. MR 0407617 (53:11389)
 [15]
 D. Kouba, Regularization for thorder linear boundary value problems using thorder differential operators, Doctoral Dissertation, Colorado State Univ., 1982.
 [16]
 J. Locker, The generalized Green's function for an thorder linear differential operator, Trans. Amer. Math. Soc. 288 (1977), 243268. MR 0481204 (58:1340)
 [17]
 , Functional analysis and twopoint differential operators, Lecture Notes, Colorado State Univ., Fort Collins, 1982.
 [18]
 J. Locker and P. M. Prenter, Optimal and error estimates for continuous and discrete least squares methods for boundary value problems, SIAM J. Numer. Anal. 15 (1978), 11511160. MR 512688 (80d:65097)
 [19]
 , Regularization with differential operators. I. General theory, J. Math. Anal. Appl. 74 (1980), 504529. MR 572669 (83j:65062a)
 [20]
 , Regularization with differential operators. II. Weak least squares finite element solutions to first kind integral equations, SIAM J. Numer. Anal. 17 (1980), 247267. MR 567272 (83j:65062b)
 [21]
 , Regularization and linear boundary value problems (to appear).
 [22]
 , Representors and superconvergence of least squares finite element approximates, Applicable Anal. (to appear).
 [23]
 M. Z. Nashed, Approximate regularized solutions to improperly posed linear integral and operator equations, Constructive and Computational Methods for Differential and Integral Equations (D. Colton and R. P. Gilbert, eds.), Lecture Notes in Math., vol. 430, SpringerVerlag, Berlin, Heidelberg and New York, 1974, pp. 289332. MR 0501883 (58:19118)
 [24]
 D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach. 9 (1962), 8497. MR 0134481 (24:B534)
 [25]
 P. M. Prenter, Splines and variational methods, Wiley, New York, 1975. MR 0483270 (58:3287)
 [26]
 P. Sammon, The discrete least squares method, Math. Comp. 31 (1977), 6065. MR 0431699 (55:4694)
 [27]
 M. H. Schultz, Error bounds for polynomial spline interpolation, Math. Comp. 24 (1970), 507515. MR 0275025 (43:783)
 [28]
 A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Math. Dokl. 4 (1963), 10351038.
 [29]
 , Regularization of incorrectly posed problems, Soviet Math. Dokl. 4 (1963), 16241627.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
47E05,
34B05,
47A50
Retrieve articles in all journals
with MSC:
47E05,
34B05,
47A50
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198608149216
PII:
S 00029947(1986)08149216
Article copyright:
© Copyright 1986
American Mathematical Society
