2003/01/31 by A. G. Ramm, Ramm, A. G.
Mathematics · #34R30 #35R25 #35R30 #37C35 #37L05 #37N30 #47A52 #47J06 #65M30 #65N21 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:34R30 #msc:35R25 #msc:35R30 #msc:37C35 #msc:37L05 #msc:37N30 #msc:47A52 #msc:47J06 #msc:65M30 #msc:65N21
paper · pdf · doi:10.48550/arxiv.math/0302001
arxiv created 2003/01/31 · arxiv updated 2009/11/30
Assume that Au=f, (1) is a solvable linear equation in a Hilbert space, ||A||<∞, and R(A) is not closed, so problem (1) is ill-posed. Here R(A) is the range of the linear operator A. A DSM (dynamical systems method) for solving (1), consists of solving the following Cauchy problem: u= -u +(B+\ep(t))-1A^*f, u(0)=u0, (2) where B:=A^*A, u:=\frac dudt, u0 is arbitrary, and \ep(t)>0 is a continuously differentiable function, monotonically decaying to zero as t→ ∞. A.G.Ramm has proved that, for any u0, problem (2) has a unique solution for all t>0, there exists y:=w(∞):=limt→ ∞u(t), Ay=f, and y is the unique minimal-norm solution to (1). If f_\d is given, such that ||f-f_\d||≤ \d, then u_\d(t) is defined as the solution to (2) with f replaced by f_\d. The stopping time is defined as a number t_\dsuch that lim\d → 0||u_\d (t_\d)-y||=0, and lim\d → 0t_\d=∞. A discrepancy principle is proposed and proved in this paper. This principle yields t_\d as the unique solution to the equation: ||A(B+\ep(t))-1A^*f_\d -f_\d||=\d, (3) where it is assumed that ||f_\d||>\d and f_\d⊥ N(A^*). For nonlinear monotone A a discrepancy principle is formulated and justified.