Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / A new method, the Dynamical Systems Method (DSM), justified
recently, is applied to solving ill-conditioned linear algebraic
system (ICLAS). The DSM gives a new approach to solving a wide class
of ill-posed problems. In Chapter 1 a new iterative scheme for
solving ICLAS is proposed. This iterative scheme is based on the DSM
solution. An a posteriori stopping rules for the proposed method is
justified. We also gives an a posteriori stopping rule for a
modified iterative scheme developed in A.G.Ramm, JMAA,330
(2007),1338-1346, and proves convergence of the solution obtained by
the iterative scheme. In Chapter 2 we give a convergence analysis of
the following iterative scheme:
u[subscript]n[superscript]delta=q u[subscript](n-1)[superscript]delta+(1-q)T[subscript](a[subscript]n)[superscript](-1) K[superscript]*f[subscript]delta, u[subscript]0[superscript]delta=0,
where T:=K[superscript]* K, T[subscript]a :=T+aI, q in the interval (0,1),\quad
a[subscript]n := alpha[subscript]0 q[superscript]n, alpha_0>0, with finite-dimensional
approximations of T and K[superscript]* for solving stably Fredholm integral
equations of the first kind with noisy data. In Chapter 3 a new
method for inverting the Laplace transform from the real axis is
formulated. This method is based on a quadrature formula. We assume
that the unknown function f(t) is continuous with (known) compact
support. An adaptive iterative method and an adaptive stopping rule,
which yield the convergence of the approximate solution to f(t),
are proposed in this chapter.
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/8109 |
| Date | January 1900 |
| Creators | Indratno, Sapto Wahyu |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
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