Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

Два примера, связанные со свойствами дискретных мер

Предложены два примера, основанные на свойствах дискретных мер. В первой части статьи доказывается, что для произвольной единичной меры $\mu$, $\operatorname{supp}{\mu}=[-1,1]$, логарифмический потенциал которой непрерывный на $[-1,1]$, существовует (дискретная) мера $\sigma=\sigma(\mu)$, $\operatorname{supp}{\sigma}=[-1,1]$, такая, что для соответствующих ортогональных полиномов $P_n(x;\sigma)=x^n+\dotsb$ справедливо соотношение: $$ \frac1n \chi(P_n( \cdot ;\sigma))\xrightarrow{*}\mu,\qquad n\to\infty, $$ где $\chi( \cdot )$ - мера, считающая нули полинома. Доказательство существования меры $\sigma$ основано на свойствах обобщенных точек Лея (weighted Leja points). Во второй части приводится пример компакта и последовательности дискретных мер с носителями на этом компакте, обладающей следующим свойством. Эта последовательность мер сходится в $*$-слабой топологии к равновесной мере компакта, но соответствующая последовательность логарифмических потенциалов не сходится по емкости к равновесному потенциалу ни в одной окрестности компакта. Библиография: 11 названий.

A Collocation-Based Multi-Configuration Time-Dependent Hartree Method Using Mode Combination and Improved Relaxation

<p>Although very useful, the original multi-configuration time-dependent Hartree (MCTDH) method has two weaknesses: (1) its cost scales exponentially with the number of atoms in the system; (2) the standard MCTDH implementation requires that the PES be in sum-of-product (SOP) form in order to reduce the cost of computing integrals in the MCTDH basis. One way to deal with (1) is to lump coordinates into groups. This is called mode combination (MC). One way to deal with (2) is to reformulate MCTDH using collocation so that there are no integrals. In this paper we combine MC and collocation to formulate a mode combination collocation multiconfiguration time-dependent Hartree method (MC-C-MCTDH). In practice, its cost does not scale exponentially with the number of atoms in the system and it can be easily used with any general potential energy surfaces (PES); the PES need not be a SOP and need not have a special form. No integrals and hence no quadratures are necessary. We demonstrate the accuracy and e ciency of the new method by computing vibrational energy eigenstates of the methyl radical, methane, and acetonitrile. To do this, we use MC-C-MCTDH with a variant of improved relaxation, derived by evaluating a residual at points rather than starting from a variational principle. Because the MC basis functions are multivariate, collocation points in multidimensional spaces are required. We use two types of collocation points: 1) DVR (discrete variable representation)-like points obtained from (approximate) simultaneous diagonalisation of matrices; and 2) Leja points, which are known to be good interpolation points, determined from a generalised recipe suitable for any (not necessarily polynomial-type) basis.<br></p>

A Collocation-Based Multi-Configuration Time-Dependent Hartree Method Using Mode Combination and Improved Relaxation

<p>Although very useful, the original multi-configuration time-dependent Hartree (MCTDH) method has two weaknesses: (1) its cost scales exponentially with the number of atoms in the system; (2) the standard MCTDH implementation requires that the PES be in sum-of-product (SOP) form in order to reduce the cost of computing integrals in the MCTDH basis. One way to deal with (1) is to lump coordinates into groups. This is called mode combination (MC). One way to deal with (2) is to reformulate MCTDH using collocation so that there are no integrals. In this paper we combine MC and collocation to formulate a mode combination collocation multiconfiguration time-dependent Hartree method (MC-C-MCTDH). In practice, its cost does not scale exponentially with the number of atoms in the system and it can be easily used with any general potential energy surfaces (PES); the PES need not be a SOP and need not have a special form. No integrals and hence no quadratures are necessary. We demonstrate the accuracy and e ciency of the new method by computing vibrational energy eigenstates of the methyl radical, methane, and acetonitrile. To do this, we use MC-C-MCTDH with a variant of improved relaxation, derived by evaluating a residual at points rather than starting from a variational principle. Because the MC basis functions are multivariate, collocation points in multidimensional spaces are required. We use two types of collocation points: 1) DVR (discrete variable representation)-like points obtained from (approximate) simultaneous diagonalisation of matrices; and 2) Leja points, which are known to be good interpolation points, determined from a generalised recipe suitable for any (not necessarily polynomial-type) basis.<br></p>