Discrepancy Principle And Orders Of Convergence For Inverse Problems

A character of Inverse Problem is that it is an ill-posed problem usually. Operator formula:Method used usually of solving ill-posed problems is regularization strategy:a family of well-posed problem converge ill-posed problem. Tikhonov Variational regularization method for stable solution of (1) consists of solving the problem F(u) := ||Au-f_δ||²+ a||u||²=min,where a is regularization parameters > 0,α= const. An important method for choosing regularization parameter is discrepancy principle.Firstly.this paper study a discrepancy principle and orders of convergence based Dynamical Systems Method (DSM).Assume (l)is a solvable linear equation in a Hilbert space. ||A||<∞, and R(A) is not closed, so problem (1) is ill-posed. A solvable quation (1) is equivalent toBu = A^*f,B :=A^*A. Let (?)(t) be a monotone, decreasing function,(?)(t)>0,lim_t→0(?)(t)= 0,lim_{t→∞}sup_1/2≤s≤t|(?)|(?)^-2(t) = 0.A DSM (dynamical systems method) for solving (1), consists of solving the followingCauchy problem:where u₀ is arbitrary, and proving that, for any u₀.Cauchy problem has a unique solution for all t>0, there exists y:=u(∞) := lirn_{t→∞}u(t), Ay =f, and y is the unique minimal-norm solution to (1). These results are proved in [3]. If f_δis given. such that ||f-f_δ||≤δ, then u_δ(t) is denned as the solution to Cauchy problem with f replaced by fg. The stopping time is defined as a number t_δ为such that lim_{δ→0}||u_δ(t_δ)-y||=0,and lim_{δ→0}t_δ=∞.Let us assume f_δ⊥N(A^*). Then this discrepancy principle:theorem 1 If A is a bounded linear operator in a Hilbert space H, equation (1) is solvable,||f_δ||>δ,fδ⊥N(A^*) , and (?)(t) satisfies the assumptions stated above, then equationhas a unique solution t_δ, andholds, where y is the unique minimalnorm solution to (1).Furthermore,we estimate the convergence of solution for the discrepancy principle: theorem 2 If A is a bounded linear operator in a Hilbert space H, equation (1)is solvable, u_δ,∈(t_δl) from DSMobtained.thenLety=A^*z,||z||≤E , can choose t(t_δ),such thatThe second part of this paper study some DSM solving ill-posed problem and another discrepancy principle:(1)DSM of selfadjoint operatorThe DSM for solving equation (1) with a linear selfadjoint operator can be constructedas follows. Consider the problemwhere a=const > 0. Our first result is formulated as Theorem 3.theorem 3 If Ay = f and y⊥N , then Our second result shows that the method, based on Theorem 1, gives a stable solution of the equation Au = f. Assume that ||f_δ-f||≤δ, and let u_a,δ(t)be the solution towith fs in place of f.theorem 4 There exist t = t_δ, lim_{δ→0}t_δ=∞:and a=a(δ),lim_{δ→0}a(δ)=0 , such that u_δ:= u_a(δ),δ(t_δ) satisfiesWe will discuss the ways to choose a(δ) and t_δafter the proofs of these theorems are given.(2)Improved DSM of selfadjoint operatorConsider problem u=i(A + ia)u_α-if,u(0)=0;u =((du)/(dt)), with a=a(t).theorem 5 Assume that a(t) > 0 is a continuous function monotonically decayingto zero as t→∞andThe solution to this problem isthenFurtlicrmorc,Thoorem 5 yields a stable solution to equation (1).theorem 6 There exists a stopping time t_δ,lim_{δ→0}t_δ=∞, such thathold , where u_δ=u_δ(t_δ) solving||f_δ-f||≤δ.(3) Another discrepancy principle based DSM Suppose that the assumption f_δ⊥N(A^*) does not hold. Then one can use the discrepancy principle of the form:theorem 7 Ifwhere f = Ay,y⊥N ,that a(t) > 0 is a continuous function monotonically decaying to zero as t→∞andthenDSM method u = -u + T_a(t)^-1A^*f, u(0) = 0, yields a stable solution to problem (1).theorem 8 If t_δis chosen so thatholds,then the solution ofsatisfiestheorem 9 Assume that a(t)>0 is a monotonically decaying twice continuously differentiate function.The equationhas a solution t = t_δ,lim_{δ→0}t_δ=∞,such that holds,where u_δis the solution to||f_δll>Cδ.The third part of this paper study a new discrepancy principle and orders of convergence :theorem 10 Assume that A is a bounded linear operator in a Hilbert space H, that f = Ay,y⊥N,||f_δ-f||≤6,||f_δ||≥Cδ,C=const,C∈(1.2),and u_α,βis any element which satisfies the inequality:whereb=const > 0 , and C²>1+b. Then equationhas a solution for any fixedδ>0 , lim_{δ→0}a(δ)=0, andAnother version of theorem 10 :theorem 10 ' Assume thati)A is a bounded linear operator in a Hilbert spaceâ…¡,ii)Thc equation Au = f solvable , y is the unique minimalnorm solution ,iii) ||f_δ-f||≤δ,||f_δ||≥Cδ.C = const>1.thena)The equationhas a solution tor any fixedδ>0, where u_δ,(?) satisfied F(u_δ,(?))≤m + (C²-1-b)δ²,F(u) := ||A(u)-f_δ||₂+∈||u||₂,m=m(δ.∈) := inf_uF(u), b=canst>0,C²>1+b,b) If (?) = (?)(δ) solved ||Au_δ,(?)-f_δ||=Cδ,u_δ:=u_δ.(?)(δ),then lim_{δ→0}||u_δ-y||=0.Furthermore,we estimate the convergence of solution for the discrepancy principle: theorem 11 Assume thati)A is a bounded linear operator in a Hilbert space H,ii)The equation Au = f solvable , y is the unique minimalnorm solution ,iii) ||(?)_δ-(?)||≤δ,||(?)_δ||≥Cδ,C = const>1.thenc)If t = (?)(δ) solved ||Au_δ,(?)-f_δ||= Cδ,u_δ:= u_δ,(?)(δ),then ||u_δ-y||=O(δ^1/2).d)If ||u_δ-y||=o(δ^1/2), then R(A) has to be finite dimension, that is, orders of convergence O(δ^1/2)is optimal.