Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10415160 | Communications in Nonlinear Science and Numerical Simulation | 2005 | 7 Pages |
Abstract
Assume thatAu=fis a solvable linear equation in a Hilbert space, â¥Aâ¥<â, and R(A) is not closed, so this problem is ill-posed. Here R(A) is the range of the linear operator A. A dynamical systems method for solving this problem, consists of solving the following Cauchy problem:uÌ=âu+(B+ϵ(t))â1Aâf,u(0)=u0,where B:=AâA, uÌ:=du/dt, u0 is arbitrary, and ϵ(t)>0 is a continuously differentiable function, monotonically decaying to zero as tââ. Ramm has proved [Commun Nonlin Sci Numer Simul 9(4) (2004) 383] that, for any u0, the Cauchy problem has a unique solution for all t>0, there exists y:=w(â):=limtââu(t), Ay=f, and y is the unique minimal-norm solution to Au=f. If fδ is given, such that â¥fâfδâ¥â©½Î´, then uδ(t) is defined as the solution to the Cauchy problem with f replaced by fδ. The stopping time is defined as a number tδ such that limδâ0â¥uδ(tδ)âyâ¥=0 and limδâ0tδ=â. A discrepancy principle is proposed and proved in this paper. This principle yields tδ as the unique solution to the equation:â¥A(B+ϵ(t))â1Aâfδâfδâ¥=δ,where it is assumed that â¥fδâ¥>δ and fδâ¥N(Aâ). The last assumption is removed, and if it does not hold, then the right-hand side of the above equation is replaced by Cδ, where C=const>1, and one assumes that â¥fδâ¥>Cδ. For nonlinear monotone A a discrepancy principle is formulated and justified.
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Mechanical Engineering
Authors
A.G. Ramm,