Necessary and insufficient conditions for well-posed Cauchy problems

Local and global solutions of well-posed integrated Cauchy problems

In this paper we study the local well-posed integrated Cauchy problem, v′(t) = Av(t) + tα Γ(α + 1) x, v(0) = 0, t ∈ [0, κ), with κ > 0, α ≥ 0, and x ∈ X where X is a Banach space and A a closed operator on X. We extend solutions increasing the regularity in α. The global case (κ = ∞) is also treated in detail. Growths of solutions are given in both cases.

Some Results on Necessary Conditions for Tow Quasidifferentiable Optimization Problems

L well posed Cauchy Problems and Symmetrizability of First Order Systems

Quasireversibility Methods for Non-well-posed Problems

The nal value problem, ut + Au = 0 ; 0 < t < T u(T) = f with positive self-adjoint unbounded A is known to be ill-posed. One approach to dealing with this has been the method of quasireversibility, where the operator is perturbed to obtain a well-posed problem which approximates the original problem. In this work, we will use a quasi-boundary-value method, where we perturb the nal condition to ...

Cauchy-like Preconditioners for Two-Dimensional Ill-Posed Problems

Ill-conditioned matrices with block Toeplitz, Toeplitz block (BTTB) structure arise from the discretization of certain ill-posed problems in signal and image processing. We use a preconditioned conjugate gradient algorithm to compute a regularized solution to this linear system given noisy data. Our preconditioner is a Cauchy-like block diagonal approximation to an orthogonal transformation of ...