Two Method Solving Self-adjoint Operator Equation By Introducing Complex Operator

In real life, ill-posed problem is widely used. For example in scanning image formation, nature survey and so on. The usual methods dealing with ill-posed problem are Tikhonov method , Landweber method and so on. In this paper, in order to deal with a special kind of ill-posed problem-self-adjoint ill-posed problem. We introduce two new method, there are complex parameter method and iterative complex parameter method. The study of ill-posed problem include three basic questions: (1)solution existence (2)solution uniqueness (3)solution stability. In this paper, we prove the regularization and the orders of convergence of the two method. Finally, we verify the result by numerical experiment.Firstly, we discuss the equation:Kx=y (0.7)and the disturbed equation:Kx=y^δ(0.8)Here, X is a Hilbert Space with infinite dimensions, K : X→X be self-adjoint, linear, compact operator. y^δeeeee X and subject to:‖y^δ-y‖≤δ,δ>0 (0.9)In the second chapter of this paper, we introduce complex parameter to the equation, and the main results are as follows: Theorem 1 Let (μ_j,x_j) be the singular system of self-adjoint compact operatorK : X→X. Because K is self-adjoint, we can conclude thatμ_j is real and the operator yield fromiαx^α,δ+Kx^α,δ=y^δ(0.10)is a regularized operator.Theorem 2 Let x = Kz∈K(X), and‖z‖≤E, chooseα= (?), then yield the following estimation:x is the exact solution of Kx=y, x^{α(δ),δ} is the solution of (0.10).Theorem 3 Letα> 0, x^αis the solution of (0.10). Then x^αis continuouslyrely on y andα,α→‖x^α‖is monotonously unincreased. and lim x^α= 0:α→‖Kx^α-y‖is monotonously undecreased and lim_{α→0}Kx^α=y,if Ky≠0, then the above two mappings are both strictly monotonous.Theorem 4 K : X→X is a linear, compact, one to one. self-adjoint operator,let Kx=y and x,y∈X subject to‖y -y^δ‖≤δ<‖y‖. if x^{α(δ),δ} subject to‖Kx^{α(δ),δ}- y^δ‖=δ,δ∈(0,δ₀), then we get :(1) ifδ→0, then x^{α(δ),δ}→x, so the stopping rule is responsible.(2) if x = Kz∈K(X),‖z‖≤E, then‖x^{α(δ),δ} - x‖≤2(?)Theorem 5 K : X→X is a linear, compact, one to one, self-adjoint operator, and X is an infinite dimension space, x∈X, continuous functionα(δ) : [0,∞)→[0,∞),α(0) = 0, if lim_{δ→0}‖x^{α(δ),δ}- x‖δ^1/2= 0, here y^δ∈X, and subject to‖y^δ- Kx‖≤δ, and x^{α(δ),δ} is the solution of (0.10), then we can get x=0.In the third chapter of this paper, we establish the following iterative method with complex parameter: In this chapter, we mainly discuss the case that chooseαas regularization parameter.From (0.11) we getThe main results are as follows:Theorem 6 Let K: X→X be a self- adjoint, linear, compact operator, R_α^m is defined as above, then R_α^mdefine a regularization strategy with regularizationparameterα, and‖R_α^m‖≤m/α, every strategyα(δ)→0(δ→0) withδ/α(δ)→0(δ→0) is admissible.Theorem 7 Let K : X→X be a self-adjoint, linear, compact operator R_α^m is defined as above. For m≥r, if we chooseα(δ) = (mδ/CrE)^1/r+1, then we yield the error estimate:‖x_α,δ^m-x‖≤C₁^mδ^r/r+1E^1/r+1,here C₁^m= m^r/r+1C^1/r+1(r^1/r+1+ r^-r/r+1), so the orders of convergence is optimal.In the forth chapter of this paper, we give numerical experiments of the two method, and they verify the results of the two method.