| Title:
|
A convergent nonlinear splitting via orthogonal projection (English) |
| Author:
|
Mandel, Jan |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
29 |
| Issue:
|
4 |
| Year:
|
1984 |
| Pages:
|
250-257 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| Summary lang:
|
Russian |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the convergence of the iterations in a Hilbert space $V,x_{k+1}=W(P)x_k, W(P)z=w=T(Pw+(I-P)z)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W(P)$ is continuous and the Lipschitz constant $\left\|(I-P)W(P)\right\|<1$. If an operator $W(P_1)$ satisfies these assumptions and $P_2$ is an orthogonal projection such that $P_1P_2=P_2P_1=P_1$, then the operator $W(P_2)$ is defined and continuous in $V$ and satisfies $\left\|(I-P_2)W(P_2)\right\|\leq \left\|(I-P_1)W(P_1)\right\|$. (English) |
| Keyword:
|
convergent nonlinear splitting |
| Keyword:
|
orthogonal projection |
| Keyword:
|
iterations |
| Keyword:
|
Hilbert space |
| Keyword:
|
fixed point |
| MSC:
|
47H10 |
| MSC:
|
47H17 |
| MSC:
|
47J25 |
| MSC:
|
65J15 |
| idZBL:
|
Zbl 0613.65060 |
| idMR:
|
MR0754077 |
| DOI:
|
10.21136/AM.1984.104093 |
| . |
| Date available:
|
2008-05-20T18:25:11Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104093 |
| . |
| Reference:
|
[1] M. Fiedler V. Pták: On aggregation in matrix theory and its applications to numerical inverting of large matrices.Bull. Acad. Pol. Sci. Math. Astr. Phys. 11 (1963) 757-759. MR 0166911 |
| Reference:
|
[2] N. S. Kurpeľ: Projection-iterative Methods of Solution of Operator Equations.(Russian). Naukova Dumka, Kiev 1968. MR 0254691 |
| Reference:
|
[3] D. P. Looze N. R. Sandell, Jr.: Analysis of decomposition algorithms via nonlinear splitting functions.J. Optim. Theory Appl. 34 (1981) 371-382. MR 0628203, 10.1007/BF00934678 |
| Reference:
|
[4] A. Ju. Lučka: Projection-iterative Methods of Solution of Differential and Integral Equations.(Russian). Naukova Dumka, Kiev 1980. MR 0598991 |
| Reference:
|
[5] A. Ju. Lučka: Convergence criteria of the projection-iterative method for nonlinear equations.(Russian). Preprint 82.24, Institute of Mathematics AN Ukrain. SSR, Kiev 1982. |
| Reference:
|
[6] J. Mandel: Convergence of some two-level iterative methods.(Czech). PhD Thesis, Charles University, Prague 1982. |
| Reference:
|
[7] J. Mandel: On some two-level iterative methods.In: Defect Correction - Theory and Applications (K. Böhmer, H. J. Stetter. editors), Computing Supplementum Vol. 5, Springer-Verlag, Wien, to appear. Zbl 0552.65049, MR 0782691 |
| Reference:
|
[8] J. Mandel B. Sekerka: A local convergence proof for the iterative aggregation method.Linear Algebra Appl. 51 (1983), 163-172. MR 0699731 |
| Reference:
|
[9] O. Pokorná I. Prágerová: Approximate matrix invertion by aggregation.In: Numerical Methods of Approximation Theory, Vol. 6 (L. Collatz, G. Meinhardus, H. Werner, editors), Birghäuser Verlag, Basel-Boston-Stuttgart 1982. |
| Reference:
|
[10] A. E. Taylor: Introduction to Functional Analysis.J. Wiley Publ., New York 1958. Zbl 0081.10202, MR 0098966 |
| Reference:
|
[11] R. S. Varga: Matrix Iterative Analysis.Prentice Hall Inc., Englewood Cliffs, New Jersey 1962. MR 0158502 |
| Reference:
|
[12] T. Wazewski: Sur une procédé de prouver la convergence des approximations successives sans utilisation des séries de comparaison.Bull. Acad. Pol. Sci. Math. Astr. Phys. 8 (1960) 47-52. MR 0126109 |
| . |
