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

IL METODO DELLE SOLUZIONI FONDAMENTALI PER LA SOLUZIONE DEL PROBLEMA DIRETTO M/EEG

The research already started on the mesh-free solution of the M / EEG direct problem has led to the development of a solver based on the method of fundamental solutions (MFS, method of fundamental solutions) able to manage the physical-geometric complexity of realistic models of the head more efficiently than traditional

An Improved Solver for the M/EEG Forward Problem

Noninvasive investigation of the brain activity via electroencephalography (EEG) and magnetoencephalography (MEG) involves a typical inverse problem whose solution process requires an accurate and fast forward solver. We propose the Method of Fundamental Solutions (MFS) as a truly meshfree alternative to the Boundary Element Method (BEM) for solving the M/EEG forward problem. The solution of the forward problem is obtained, via the Method of Particular Solutions (MPS), by numerically solving a set of coupled boundary value problems for the 3D Laplace equation. Numerical accuracy and computational load are investigated for spherical geometries and comparisons with a state-of-the-art BEM solver shows that the proposed method is competitive

Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations (SPDEs). Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the SPDE centered at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.Comment: Updated Version in International Journal of Computer Mathematics, Closed to Ye's Doctoral Thesi