Dynamical Systems Gradient Method for Solving Nonlinear Equations with Monotone Operators
- PDF / 599,004 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 101 Downloads / 227 Views
Dynamical Systems Gradient Method for Solving Nonlinear Equations with Monotone Operators N.S. Hoang · A.G. Ramm
Received: 28 June 2008 / Accepted: 26 August 2008 / Published online: 18 September 2008 © Springer Science+Business Media B.V. 2008
Abstract A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications. Keywords Dynamical systems method (DSM) · Nonlinear operator equations · Monotone operators · Discrepancy principle Mathematics Subject Classification (2000) 47J05 · 47J06 · 47J35 · 65R30
1 Introduction In this paper we study a version of the Dynamical Systems Method (DSM) (see [10]) for solving the equation F (u) = f,
(1)
where F is a nonlinear, twice Fréchet differentiable, monotone operator in a real Hilbert space H , and (1) is assumed solvable, possibly nonuniquely. Monotonicity means that F (u) − F (v), u − v ≥ 0,
∀u, v ∈ H.
(2)
Equations with monotone operators are important in many applications and were studied extensively, see, for example, [5, 7, 21, 24], and references therein. One encounters many N.S. Hoang · A.G. Ramm () Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA e-mail: [email protected] N.S. Hoang e-mail: [email protected]
474
N.S. Hoang, A.G. Ramm
technical and physical problems with such operators in the cases where dissipation of energy occurs. For example, in [9] and [8], Chap. 3, pp. 156–189, a wide class of nonlinear dissipative systems is studied, and the basic equations of such systems can be reduced to (1) with monotone operators. Numerous examples of equations with monotone operators can be found in [5] and references mentioned above. In [19] and [20] it is proved that any solvable linear operator equation with a closed, densely defined operator in a Hilbert space H can be reduced to an equation with a monotone operator and solved by a convergent iterative process. In this paper, apparently for the first time, the convergence of the Dynamical Systems Gradient method is proved under natural assumptions and convergence of a corresponding iterative method is established. No special assumptions of smallness of the nonlinearity or other special properties of the nonlinearity are imposed. No source-type assumptions are used. Consequently, our result is quite general and widely applicable. It is well known, that without extra assumptions, usually, source-type assumption about the right-hand side, or some assumption concerning the smoothness of the solution, one cannot get a specific rate of convergence even for linear ill-posed equations (see, for example, [10], where one can find a proof of this statement). On the other hand, such assumptions are often difficult to verify and often they do not hold. By
Data Loading...
