We show that point-neuron models with a Heaviside firing rate function can be ill posed. More specifically, the initial-condition-to-solution map might become discontinuous in finite time. Consequently, if finite precision arithmetic is used, then it is virtually impossible to guarantee the accurate numerical solution of such models. If a smooth firing rate function is employed, then standard O...

Several thermomechanical models have been proposed from a heuristic point of view. A mathematical analysis should help to clarify the applicability of these models, among those recent thermal or viscoelastic models. Single-phase-lag and dual-phase-lag heat conduction models can be interpreted as formal expansions of delay equations. The delay equations are shown to be ill-posed, as are the form...

Ill posed quadratic optimization frequently occurs in control and inverse problems and are not covered by the Lax-Milgram-Riesz theory. Typically small changes in the input data can produce very large oscillations on the output. We investigate the conditions under which the minimum value of the cost function is finite and we explore the ‘hidden connection’ between the optimization problem and t...

A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed i.e., with a unique solution regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters. We will see that choosing appropriate hypotheses on the parameters, we will obt...

The regularized least squares methods for the solution of ill-posed inverse problems are summarized, and appropriate references are stated. Additionally, an adap-tive multi-scale algorithm is proposed to solve highly ill-posed inverse problems with fewer degrees of freedom and comparable performance. The algorithm controls the level of detail in the reconstruction by distributing the ¯ne scale ...

In this paper we solve large scale ill-posed problems, particularly the image restoration problem in atmospheric imaging sciences , by a trust region-cg algorithm. Image restoration involves the removal or minimization of degradation (blur, clutter, noise, etc.) in an image using a priori knowledge about the degradation phenomena. Our basic technique is the so-called trust region method, while ...

In this paper we solve large scale ill-posed problems, particularly the image restoration problem in atmospheric imaging sciences, by a trust region-CG algorithm. Image restoration involves the removal or minimization of degradation (blur, clutter, noise, etc.) in an image using a priori knowledge about the degradation phenomena. Our basic technique is the so-called trust region method, while t...

Cone beam X-ray luminescence tomography can realize fast X-ray luminescence tomography imaging with relatively low scanning time compared with narrow beam X-ray luminescence tomography. However, cone beam X-ray luminescence tomography suffers from an ill-posed reconstruction problem. First, the feasibility of experiments with different penetration and multispectra in small animal has been teste...

The \L{curve" is a plot (in ordinary or doubly{logarithmic scale) of the norm of (Tikhonov{) regularized solutions of an ill{posed problem versus the norm of the residuals. We show that the popular criterion of choosing the parameter corresponding to the point with maximal curvature of the L{curve does not yield a convergent regularization strategy to solve the ill{posed problem. Nevertheless, ...

For joint estimation of state variables and unknown parameters, adaptive observers usually assume some persistent excitation (PE) condition. In practice, the PE condition may not be satisfied, because underlying recursive problem is ill-posed. To remedy lack condition, inspired by ridge regression, this paper proposes a regularized observer with enhanced parameter adaptation gain. Like in typic...