Dynamical systems method (DSM) for selfadjoint operators

Abstract

Let A be a selfadjoint linear operator in a Hilbert space H. The DSM (dynamical systems method) for solving equation A v = f consists of solving the Cauchy problem u ˙ = Φ ( t , u ) , u ( 0 ) = u 0 , where Φ is a suitable operator, and proving that (i) ∃ u ( t ) ∀ t > 0 , (ii) ∃ u ( ∞ ) , and (iii) A ( u ( ∞ ) ) = f . It is proved that if equation A v = f is solvable and u solves the problem u ˙ = i ( A + i a ) u − i f , u ( 0 ) = u 0 , where a > 0 is a parameter and u 0 is arbitrary, then lim a → 0 lim t → ∞ u ( t , a ) = y , where y is the unique minimal-norm solution of the equation A v = f . Stable solution of the equation A v = f is constructed when the data are noisy, i.e., f δ is given in place of f, ‖ f δ − f ‖ ⩽ δ . The case when a = a ( t ) > 0 , ∫ 0 ∞ a ( t ) d t = ∞ , a ( t ) ↘ 0 as t → ∞ is considered. It is proved that in this case lim t → ∞ u ( t ) = y and if f δ is given in place of f, then lim t → ∞ u ( t δ ) = y , where t δ is properly chosen.