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Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces
Let be an operator equation in a Banach space ,
, where ,
, if , is strictly growing on
. Denote , where is the Fr\'{e}chet derivative
of , and Assume that (*) , , , . Here may be a complex number,
and is a smooth path on the complex -plane, joining the origin and some
point on the complex plane, is a
small fixed number, such that for any estimate (*) holds. It is proved
that the DSM (Dynamical Systems Method) \bee
\dot{u}(t)=-A^{-1}_{a(t)}(u(t))[F(u(t))+a(t)u(t)-f],\quad u(0)=u_0,\
\dot{u}=\frac{d u}{dt}, \eee converges to as , where , , and , where are
some suitably chosen constants, Existence of a solution to the
equation is assumed. It is also assumed that the equation
is uniquely solvable for any , , and
$\lim_{|a|\to 0,a\in L}\|w_a-y\|=0.
The Dynamical Systems Method for solving nonlinear equations with monotone operators
A review of the authors's results is given. Several methods are discussed for
solving nonlinear equations , where is a monotone operator in a
Hilbert space, and noisy data are given in place of the exact data. A
discrepancy principle for solving the equation is formulated and justified.
Various versions of the Dynamical Systems Method (DSM) for solving the equation
are formulated. These methods consist of a regularized Newton-type method, a
gradient-type method, and a simple iteration method. A priori and a posteriori
choices of stopping rules for these methods are proposed and justified.
Convergence of the solutions, obtained by these methods, to the minimal norm
solution to the equation is proved. Iterative schemes with a
posteriori choices of stopping rule corresponding to the proposed DSM are
formulated. Convergence of these iterative schemes to a solution to equation
is justified. New nonlinear differential inequalities are derived and
applied to a study of large-time behavior of solutions to evolution equations.
Discrete versions of these inequalities are established.Comment: 50p
Dynamical Systems Gradient method for solving nonlinear equations with monotone operators
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.Comment: 2 figure
Existence of solution to an evolution equation and a justification of the DSM for equations with monotone operators
An evolution equation, arising in the study of the Dynamical Systems Method
(DSM) for solving equations with monotone operators, is studied in this paper.
The evolution equation is a continuous analog of the regularized Newton method
for solving ill-posed problems with monotone nonlinear operators . Local and
global existence of the unique solution to this evolution equation are proved,
apparently for the firs time, under the only assumption that exists and
is continuous with respect to . The earlier published results required more
smoothness of . The Dynamical Systems method (DSM) for solving equations
with monotone Fr\'echet differentiable operator is justified under
the above assumption apparently for the first time
Dynamical systems method for solving operator equations
Consider an operator equation in a real Hilbert space.
The problem of solving this equation is ill-posed if the operator is
not boundedly invertible, and well-posed otherwise.
A general method, dynamical systems method (DSM) for solving linear and
nonlinear ill-posed problems in a Hilbert space is presented.
This method consists of the construction of a nonlinear dynamical system,
that is, a Cauchy problem, which has the following properties:
1) it has a global solution,
2) this solution tends to a limit as time tends to infinity,
3) the limit solves the original linear or non-linear problem. New
convergence and discretization theorems are obtained. Examples of the
applications of this approach are given. The method works for a wide range of
well-posed problems as well.Comment: 21p
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