The finite basis representation as the primary space in multidimensional pseudospectral schemes

1994 ◽  
Vol 101 (12) ◽  
pp. 10526-10532 ◽  
Author(s):  
Didier Lemoine
Keyword(s):  
Finite Basis ◽  
Finite Basis Representation ◽  
Primary Space

scholarly journals Nonparametric regression with adaptive truncation via a convex hierarchical penalty

Biometrika ◽  
2018 ◽  
Vol 106 (1) ◽  
pp. 87-107
Author(s):  
Asad Haris ◽  
Ali Shojaie ◽  
Noah Simon
Keyword(s):  
Nonparametric Regression ◽  
Empirical Studies ◽  
Finite Basis ◽  
Additive Models ◽  
Computationally Efficient ◽  
Penalized Estimation ◽  
Minimax Rates ◽  
Minimax Rate ◽  
Finite Basis Representation ◽  
Different Levels

SUMMARY We consider the problem of nonparametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well suited to high-dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties. In the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity. In contrast, a smoothing spline-type penalty on the component functions is adaptive but does not provide a parsimonious representation. Our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way. We demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class. We further establish minimax rates for a large class of sparse additive models. We also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size.

scholarly journals USING LEBEDEV GRIDS, SINE SPHERICAL HARMONICS, AND MONOMER CONTRACTED BASIS FUNCTIONS TO CALCULATE BENDING ENERGY LEVELS OF HF TRIMER

2003 ◽  
Vol 02 (04) ◽  
pp. 599-608 ◽  
Author(s):  
XIAO-GANG WANG ◽  
TUCKER CARRINGTON
Keyword(s):  
Spherical Harmonics ◽  
Energy Levels ◽  
Finite Basis ◽  
Chem Phys ◽  
Previous Method ◽  
Basis Functions ◽  
Bending Energy ◽  
Quadrature Scheme ◽  
Finite Basis Representation ◽  
Quadrature Grid

We calculate energy levels of a six-dimensional bending Hamiltonian for HF trimer using a finite basis representation (FBR) in conjunction with the Lanczos eigensolver. We improve on our previous method [J. Chem. Phys.115, 9781 (2001)] using three techniques: (1) Lebedev's quadrature scheme is used to reduce the size of quadrature grid by a factor of 3.4. (2) Since the barrier separating the two equivalent versions of HF trimer is high and wide, it is a good approximation to confine the bending motion to one well by using sine spherical harmonics basis functions (this reduces the size of the basis by a factor of 8). (3) The sine spherical harmonic basis is contracted for each monomer to generate a very efficient basis. It is shown that the best approach is to combine all the three techniques.

FINITELY BASED WORDS

2000 ◽  
Vol 10 (04) ◽  
pp. 457-480 ◽  
Author(s):  
OLGA SAPIR
Keyword(s):  
Sufficient Conditions ◽  
Basis Property ◽  
Free Monoid ◽  
Finite Basis ◽  
Finitely Based ◽  
Finite Basis Property ◽  
Letter Alphabet ◽  
Finite Language ◽  
Rees Quotient ◽  
The Ideal

Let W be a finite language and let Wc be the closure of W under taking subwords. Let S(W) denote the Rees quotient of a free monoid over the ideal consisting of all words that are not in Wc. We call W finitely based if the monoid S(W) is finitely based. Although these semigroups have easy structure they behave "generically" with respect to the finite basis property [6]. In this paper, we describe all finitely based words in a two-letter alphabet. We also find some necessary and some sufficient conditions for a set of words to be finitely based.

A nonassociative algebra having no finite basis for its laws

1978 ◽  
Vol 18 (6) ◽  
pp. 1007-1008 ◽  
Author(s):  
Yu. N. Mal'tsev ◽  
V. A. Parfenov
Keyword(s):  
Finite Basis ◽  
Nonassociative Algebra
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