A convergent nonlinear splitting via orthogonal projection

Jan Mandel

Aplikace matematiky (1984)

  • Volume: 29, Issue: 4, page 250-257
  • ISSN: 0862-7940

Abstract

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We study the convergence of the iterations in a Hilbert space V , x k + 1 = W ( P ) x k , W ( P ) z = w = T ( P w + ( 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 ( I - P ) W ( P ) < 1 . If an operator W ( P 1 ) satisfies these assumptions and P 2 is an orthogonal projection such that P 1 P 2 = P 2 P 1 = P 1 , then the operator W ( P 2 ) is defined and continuous in V and satisfies ( I - P 2 ) W ( P 2 ) ≤ ( I - P 1 ) W ( P 1 ) .

How to cite

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Mandel, Jan. "A convergent nonlinear splitting via orthogonal projection." Aplikace matematiky 29.4 (1984): 250-257. <http://eudml.org/doc/15355>.

@article{Mandel1984,
abstract = {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 $\left\Vert (I-P)W(P)\right\Vert <1$. 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\Vert (I-P_2)W(P_2)\right\Vert \le \left\Vert (I-P_1)W(P_1)\right\Vert $.},
author = {Mandel, Jan},
journal = {Aplikace matematiky},
keywords = {convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point; convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point},
language = {eng},
number = {4},
pages = {250-257},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A convergent nonlinear splitting via orthogonal projection},
url = {http://eudml.org/doc/15355},
volume = {29},
year = {1984},
}

TY - JOUR
AU - Mandel, Jan
TI - A convergent nonlinear splitting via orthogonal projection
JO - Aplikace matematiky
PY - 1984
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 29
IS - 4
SP - 250
EP - 257
AB - 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 $\left\Vert (I-P)W(P)\right\Vert <1$. 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\Vert (I-P_2)W(P_2)\right\Vert \le \left\Vert (I-P_1)W(P_1)\right\Vert $.
LA - eng
KW - convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point; convergent nonlinear splitting; orthogonal projection; iterations; Hilbert space; fixed point
UR - http://eudml.org/doc/15355
ER -

References

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  8. J. Mandel B. Sekerka, A local convergence proof for the iterative aggregation method, Linear Algebra Appl. 51 (1983), 163-172. (1983) MR0699731
  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. (1982) 
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