ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

15. Numerical Analysis and Scientific Computing

1991 MS Classification: 45, 65 Solodky, Sergei G., Institute of Mathematics, Kiev, Ukraine Optimization of projection methods for solving linear ill-posed problems We investigate the efficiency of projection methods for solving ill-posed first kind Fredholm equations

Ax(t) := ∫_{0}^{1} a(t, τ)x(τ)dτ = f(t)  (1)

with normal solution from Range(|A|^{ν}), 0 < ν ≤ 1. Suppose that instead of (1) we have a perturbed equation A_{h}x = f_{δ}, where A, A_{h} ∈ L(L_{2}, L_{2}), f, f_{δ} ∈ L_{2}, ∥A − A_{h}∥ ≤ h, ∥f − f_{δ}∥ ≤ δ. By card(ν, δ, h, H^{r}) we mean the minimal amount of discrete information in the form of Galerkin functionals guaranteeng optimal order of accuracy O(δ^{ν/(ν+1)}) on the class of equations

(1) with operators A ∈ H^{r} such that ∑_{i,j=0}^{r} ∫_{0}^{1} ∫_{0}^{1} |\frac{∂^{i+j}a(t,τ)}{∂t^{i}∂τ^{j}} |^{2} dtdτ < ∞. We obtain that for sufficiently small δ and h = O(δ) it holds

c_{1}δ^{−1/r} ≤ card(ν, δ, h, H^{r}) ≤ c_{2}δ^{−1/r} log \frac{1}{δ}.

The indicated optimal order of card(ν, δ, h, H^{r}) is attained by a projection method connected with some hyperbolic cross.