A rescaled method for RBF approximation

In the recent paper [8], a new method to compute stable kernel-based interpolants has been presented. This \textit{rescaled interpolation} method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties

A rescaled method for RBF approximation

A new method to compute stable kernel-based interpolants has been presented by the second and third authors. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allow us to consider its error and stability properties. First, we prove that the method is an instance of the Shepard\u2019s method, when certain weight functions are used. In particular, the method can reproduce constant functions. Second, it is possible to define a modified set of cardinal functions strictly related to the ones of the not-rescaled kernel. Through these functions, we define a Lebesgue function for the rescaled interpolation process, and study its maximum - the Lebesgue constant - in different settings. Also, a preliminary theoretical result on the estimation of the interpolation error is presented. As an application, we couple our method with a partition of unity algorithm. This setting seems to be the most promising, and we illustrate its behavior with some experiments

Neurohumoral stimulation in type-2-diabetes as an emerging disease concept

Neurohumoral stimulation comprising both autonomic-nervous-system dysfunction and activation of hormonal systems including the renin-angiotensin-aldosterone system (RAAS) was found to be associated with Type-2-diabetes (T2D). Therapeutic strategies such as RAAS interference proved to be beneficial in both T2D treatment and prevention. In addition to an activated RAAS, hyperleptinemia in obesity, hyperinsulinemia in conditions of peripheral insulin resistance and overall oxidative stress in T2D represent known activators of the sympathetic component of the autonomic nervous system. Here, we hypothesize that sympathetic activation may cause peripheral insulin resistance defined as partial blocking of insulin effects on glucose uptake. Resulting hyperinsulinemia or hyperglycemia-related oxidative stress may further aggravate sympatho-excitation. This notion leads to a secondary hypothesis: sympathetic activation worsens from obesity towards insulin resistance, and further towards T2D. In this review, existing evidence relating to neurohumoral stimulation in T2D and consequences thereof, such as oxidative stress and inflammation, are discussed. The aim of this review is to provide a rationale for therapies, which are able to intercept neuroendocrine pathways in T2D and precursor states such as obesity

A Meshfree Solver for the MEG Forward Problem

Noninvasive estimation of brain activity via magnetoencephalography (MEG) involves an inverse problem whose solution requires an accurate and fast forward solver. To this end, we propose the Method of Fundamental Solutions (MFS) as a meshfree alternative to the Boundary Element Method (BEM). The solution of the MEG forward problem is obtained, via the Method of Particular Solutions (MPS), by numerically solving a boundary value problem for the electric scalar potential, derived from the quasi-stationary approximation of Maxwell’s equations. The magnetic field is then computed by the Biot-Savart law. Numerical experiments have been carried out in a realistic single-shell head geometry. The proposed solver is compared with a state-of-the-art BEM solver. A good agreement and a reduced computational load show the attractiveness of the meshfree approach