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Rates of convergence for iterative solutions of equations involving set-valued accretive operators [article]

Ulrich Kohlenbach, Thomas Powell
2019 arXiv   pre-print
This paper studies proofs of strong convergence of various iterative algorithms for computing the unique zeros of set-valued accretive operators that also satisfy some weak form of uniform accretivity  ...  More precisely, we extract explicit rates of convergence from these proofs which depend on a modulus of uniform accretivity at zero, a concept first introduced by A.  ...  Then whenever the system of operator equations f ∈ A 1 q f ∈ A 2 q has some solution q ∈ D, then (x n ) converges strongly to q.  ... 
arXiv:1908.06734v1 fatcat:4xrb64474vdn5e4gmc6fdunvgi

Page 4202 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews  
(WAN-NIGR; Nsukka) Iterative solution of nonlinear equations involving set-valued uniformly accretive operators. (English summary) Comput. Math. Appl. 42 (2001), no. 1-2, 131-140.  ...  (English summary) Set-Valued Anal. 8 (2000), no. 4, 361-374. The authors present a scheme for finding approximate solutions of m-accretive operators A: E — 2° in a Banach space E.  ... 

Page 5275 of Mathematical Reviews Vol. , Issue 2003g [page]

2003 Mathematical Reviews  
This gives rise to new sufficient conditions for the existence of solutions of this type of equations. For the use of this type of iterative schemes using various contractive conditions refer to [B.  ...  It is proved that the Ishikawa and Mann iterative processes with mixed errors converge strongly to the unique solution of the equation z € Sx +4Ax for given z € X and 42> 0.” 2003g:47121 475325 45G10 47H10  ... 

Page 6114 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
Gerd Ponisch (Dresden) 2003h:47130 47525 47H06 65J15 Zeng, Liu-chuan (PRC-SNO; Shanghai) On the convergence rate estimates of iteration methods for equations involving accretive operators in Banach spaces  ...  Summary: “The convergence of the Ishikawa iteration sequences with errors for constructing solutions of m-accretive operator equations is characterized.  ... 

Page 6766 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
The authors apply the notion of p-regularity to inclusions in- volving operators of the form T+ C, where T is m-accretive (i.e., for every 4 >0 the mapping (J + AT)~' is single-valued, ev- ’ 47 OPERATOR  ...  ; Harbin); Sun, Xue Si (PRC-HIET; Harbin)- The construction and convergence of a type of Mann iterative sequence for a class of operators with boundary condition.  ... 

Page 4991 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
[Kang, Shin Min] (KR-GYSG; Chinju) Convergence of steepest descent approximation for set-valued quasi-accretive mapping equations. (English summary) Math. Comput.  ...  Summary: “In this paper, we obtain results on convergence of new steepest descent approximation methods for set-valued quasi- accretive mappings in reflexive Banach spaces and uniformly smooth Banach spaces  ... 

Page 4379 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
The author describes the rate of convergence and gives an example of applying this method for solving quasilinear elliptic boundary value problems. Alexei I.  ...  Related results deal with the iterative approximation of fixed points of strongly pseudocontractive oper- ators, and the solution of the equation x + 7x = f, f € E, when T: E > E is m-accretive.” 98g:47055  ... 

Page 6380 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
for set-valued mappings.  ...  of a regularly ordered pair of lower and upper solutions for equation (*) implies that some monotone sequences converge to the minimal and maximal solutions of this equation.  ... 

Iterative approximation of solutions to nonlinear equations involving m-accretive operators in Banach spaces

Lu-Chuan Zeng
2002 Journal of Mathematical Analysis and Applications  
Moreover, this result provides a convergence rate estimate for some special cases of such a sequence.  ...  Utilizing this result, we imply that if T : E → E is a Lipschitz continuous strongly accretive operator then the Ishikawa iterative sequence with errors converges strongly to the unique solution of the  ...  It is well known that Mann and Ishikawa iteration processes have been employed by many authors for iterative approximation of either solutions of nonlinear operator equations or fixed points of nonlinear  ... 
doi:10.1016/s0022-247x(02)00053-7 fatcat:a5zqxy6kjvggxg6opxdb7to2li

A universal preconditioner for linear systems [article]

Tom Vettenburg, Ivo M. Vellekoop
2022 arXiv   pre-print
We demonstrate and evaluate our approach for wave problems, diffusion problems, and the pantograph delay differential equation.  ...  Its unique form permits the elimination of the forward problem from the preconditioned system, often halving the time required per iteration.  ...  Figure 1a shows the solution of the 2-D Helmholtz equation after 6026 iterations of the preconditioned Richardson iteration.  ... 
arXiv:2207.14222v1 fatcat:bjwybp2aybe3pjddz4azhiaqri

Page 5836 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
Sourcewise representation of solutions to nonlinear operator equations in a Banach space and convergence rate estimates for a regularized Newton’s method. (English summary) J.  ...  The authors study averaged convergence of projection and prox- imal iterations for solving a class of variational-like inequality problems with d-accretive operators defined on closed subsets of Banach  ... 

Page 2999 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
Yi Hong Du (S-NENG; Armidale) 98e:47099 47HI15 47HI7 65)10 Ding, Xieping (PRC-SCHN; Chengdu) Iterative solutions of nonlinear equations involving accretive and dissipative operators.  ...  Dan Tiba (R-AOS; Bucharest) 98e:47100 47H15 47H06 47HI7 65)10 Ding, Xie Ping (PRC-SCHN; Chengdu) Iterative solution of equation f © x + 7x for an accretive operator T in uniformly smooth Banach spaces.  ... 

Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics

Scott C. Noble, Charles F. Gammie, Jonathan C. McKinney, Luca Del Zanna
2006 Astrophysical Journal  
The forward transformation (primitive to conserved) has a closed-form solution, but the inverse transformation (conserved to primitive) requires the solution of a set of five nonlinear equations.  ...  The first method solves the full set of five nonlinear equations directly using a Newton-Raphson scheme and a guess from the previous timestep.  ...  A version of this scheme is available for download on the web at http://rainman.astro.uiuc.edu/codelib/codes/pvs.tgz This work was supported by NSF grants PHY 02-05155 and AST 00-93091.  ... 
doi:10.1086/500349 fatcat:4cenewuosjgq5lwqt6aodssd5e

Page 1397 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
The convergence rates for the regularized solutions x,, Xan are given.  ...  Regularization properties of the iterated Lavrent/ev method (ap- plied a fixed number of times) on finite-dimensional subspaces are investigated for ill-posed problems involving accretive linear opera-  ... 

Nonlinear accretive and pseudo-contractive operator equations in Banach spaces

C.E. Chidume, M.O. Osilike
1998 Nonlinear Analysis  
As a consequence of this result, it is proved that both the Mann and Ishikawa iteration methods converge strongly to the unique solutions of the operator equations Tx = f and x + Tx = f for a given / 6  ...  It is proved that the Ishikawa iteration scheme converges strongly to the unique fixed point of T.  ...  schemes converge strongly to the unique solution of the operator equation Tx = /, for a given / G E, where T : E ->• E is Lipschitzian and strongly accretive.  ... 
doi:10.1016/s0362-546x(97)00439-2 fatcat:k53p3fhd75gfxo53uvwagcjley
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