We prove Hölder-continuous dependence results for the difference between solutions of certain ill-posed and approximate well-posed problems in both Hilbert and Banach spaces. We use operator-theoretic methods, including C-semigroups, to treat the abstract Cauchy problem du dt = Au, u(0) = χ, 0 ≤ t < T, where the operator −A is the infinitesimal generator of a holomorphic semigroup.

A new method due to Fokas for explicitly solving boundary-value problems for linear partial differential equations is extended to equations with mixed partial derivatives. The Benjamin-Bona-Mahony equation is used as an example: we consider the Robin problem for this equation posed both on the half line and on the finite interval. For specific cases of the Robin boundary conditions the boundary...

The posedness of extending a band-limited image known only within a cone is considered. The problem is globally ill posed in the sense that the output-error energy cannot be bound over the entire unknown image region corresponding to a given input-data error energy. We demonstrate, however, that the problem is locally well posed in the sense that, within a finite area of the unknown image regio...

Submitted for the DFD07 Meeting of The American Physical Society Regularization techniques for backward–in–time evolutionary PDE problems JONATHAN GUSTAFSSON, BARTOSZ PROTAS, McMaster University — Backward–in–time evolutionary PDE problems have applications in the recently–proposed retrograde data assimilation. We consider the terminal value problem for the Kuramoto–Sivashinsky equation (KSE) i...

SUMMARY Due to the influence of variations in landform, geophysical data acquisition is usually sub-sampled. Reconstruction of the seismic wavefield from sub-sampled data is an ill-posed inverse problem. Compressive sensing can be used to recover the original geophysical data from the sub-sampled data. In this paper, we consider the wavefield reconstruction problem as a com-pressive sensing and...

Image Reconstruction in Electric Impedance Tomography (EIT) is an ill-posed problem. Ill-posedness affects the reconstruction in two ways: 1) the image is very sensitive to measurement noise, instrumentation and modeling errors 2) in order to reconstruct meaningful images regularization technique need to be used. This limits the ability to reconstruct sharp images. Traditionally image reconstru...

Ill-posed inverse problem is commonly existed in signal processing such as image reconstruction from projection, parameter estimation on electromagnetic field, and path optimization in IP network. Usually, the solution of an inverse problem is unstable, not unique or does not exit. Traditional approach to solve this problem is to estimate the solution by optimizing a regularized objective funct...

In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and timevarying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable ...

Inversion of MT data is an inherently nonunique and unstable problem due to the ill-posedness of the electromagnetic inverse problem. A variety of models may fit the data very well. To overcome this illposed nature of the inverse problem, we use Tikhonov’s regularization in which the ill-posed problem is replaced by a family of well-posed problems. We also analyze the behavior of the Tikhonov r...

We consider an inverse initial value problem of the biparabolic equation; this problem is ill-posed and the regularization methods are needed to stabilize the numerical computations. This paper firstly establishes a conditional stability of Holder type, then uses a modified regularization method to overcome its ill-posedness and gives the convergence estimate under an a-priori assumption for th...