Approximate Approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function on a uniform grid to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe the application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.Comment: 29 pages, 17 figure

Computation of volume potentials over bounded domains via approximate approximations

We obtain cubature formulas of volume potentials over bounded domains combining the basis functions introduced in the theory of approximate approximations with their integration over the tangential-halfspace. Then the computation is reduced to the quadrature of one dimensional integrals over the halfline. We conclude the paper providing numerical tests which show that these formulas give very accurate approximations and confirm the predicted order of convergence.Comment: 18 page

On the computation of high-dimensional potentials of advection-diffusion operators

We study a fast method for computing potentials of advection-diffusion operators $-D +2bcdot abla+c$ with $binC^n$ and $cin C$ over rectangular boxes in R^n. By combining high-order cubature formulas with modern methods of structured tensor product approximations we derive an approximation of the potentials which is accurate and provides approximation formulas of high-order. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. The action of volume potentials on the basis functions allows one-dimensional integral representations with separable integrands i.e. a product of functions depending only on one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Since only one-dimensional operations are used, the resulting method is effective also in high-dimensional case

Approximate approximations from scattered data

AbstractThe aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators

Activation of G protein-coupled receptors entails cysteine modulation of agonist binding

The increase of the affinity of agonists with an increase in pH and experiments using thiol-specific reagents indicate that G protein-coupled receptors contain an ionizable cysteine residue at the ligand binding site. Since treatment of receptors with reducing agents produces functional activation and potentiates agonist stimulation, it is likely that this free sulfhydryl modulates receptor activation. We have derived a two-state acid-base model for cysteine modulation of ligand binding which leads to a description of ligand efficacy. We have shown that pH-dependent binding of agonists is closely correlated with measurements of ligand efficacy at the 5-HT2A receptor. In general, efficacy is determined by the preference of a ligand for the base of the receptor. Efficacy may also be described in thermodynamic terms as the coupling free energy involving a ligand and the acid and base states of the receptor. Molecular modeling of the third transmembrane domain of the 5-HT2A receptor, which contains a conserved cysteine, shows that efficacy is determined by the difference between the electrostatic interaction energies of a ligand with the acid and base forms of the receptor model. The difference in interaction energy between the two forms of cysteine makes the largest contribution to this electrostatic interaction energy difference. Therefore, the cysteine makes the largest contribution to ligand efficacy. Using this approach, we can distinquish between the efficacies of agonists with varying molecular structures and account for the differences between the properties of agonists and antagonists

Fast computation of elastic and hydrodynamic potentials using approximate approximations

We propose fast cubature formulas for the elastic and hydrodynamic potentials based on the approximate approximation of the densities with Gaussian and related functions. For densities with separated representation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures. We obtain high order approximations up to a small saturation error, which is negligible in computations. Results of numerical experiments which show approximation order O(h2M) , M= 1 , 2 , 3 , 4 , are provided

Approximation of Uncoupled Quasi-Static Thermoelasticity Solutions Based on Gaussians

A fast approximation method to three dimensional equations in quasi-static uncoupled thermoelasticity is proposed. We approximate the density via Gaussian approximating functions introduced in the method approximate approximations. In this way the action of the integral operators on such functions is presented in a simple analytical form. If the density has separated representation, the problem is reduced to the computation of one-dimensional integrals which admit efficient cubature procedures. The comparison of the numerical and exact solution shows that these formulas are accurate and provide the predicted approximation rate 2 , 4 , 6 and 8