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On the discrepancy principle for the dynamical systems method
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Discrepancy principle for the dynamical systems
method
Key words: ill-posed problems, dynamical systems method (DSM), discrepancy principle, evolution
equations.
Math subject classification: 34R30, 35R25, 35R30, 37C35, 37L05, 37N30, 47A52, 47J06, 65M30, 65N21;
PACS 02.30.-f, 02.30.Tb, 02.30.Zz,02.60Lj, 02.60.Nm, 02.70.Pt, 05.45.-a
A.G. Ramm
Mathematics Department, Kansas State University,
Manhattan, KS 66506-2602, USA
ramm@math.ksu.edu
Abstract
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 + ?(t))?1A?f, u(0) = u0, (2)
where B := A?A, u? := du
dt
, u0 is arbitrary, and ?(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 fδ is given,
such that ||f ? fδ|| ≤ δ, then uδ(t) is defined as the solution to (2) with f replaced
by fδ. The stopping time is defined as a number tδ such that limδ→0 ||uδ(tδ) ? y|| = 0,
and limδ→0 tδ = ∞. A discrepancy principle is proposed and proved in this paper. This
principle yields tδ as the unique solution to the equation:
||A(B + ?(t))?1A?fδ ? fδ|| = δ, (3)
where it is assumed that ||fδ|| δ and fδ ⊥ N(A
?). For nonlinear monotone A a
discrepancy principle is formulated and justified.
1
1 Introduction and statement of the result.
Assume that
Au = f, (1.1)
is a solvable linear equation in a Hilbert space, ||A|| ∞, and R(A) is not closed, so
problem (1.1) is ill-posed. Here R(A) is the range of the linear operator A. Without loss
of generality, assume that ||A|| ≤ 1. Let y be the unique mini
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