A necessary condition 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.

A Necessary Condition for Weyl-Heisenberg Frames

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 ...

Solving Ill-Posed Cauchy Problems by a Krylov Subspace Method

We study the numerical solution of a Cauchy problem for a self-adjoint elliptic partial differential equation uzz − Lu = 0 in three space dimensions (x, y, z) , where the domain is cylindrical in z. Cauchy data are given on the lower boundary and the boundary values on the upper boundary is sought. The problem is severely illposed. The formal solution is written as a hyperbolic cosine function ...

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 ...