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\begin{document}

\title{Slow Exponential Sparse Grids - (MISSPELLED DOCUMENT) }

\author{John Burkardt \\
Department of Scientific Computing \\
Florida State University \\ \\
http://people.sc.fsu.edu/$\sim$jburkardt/latex/hunspell/bogus.tex\\
http://people.sc.fsu.edu/$\sim$jburkardt/latex/hunspell/bogus.pdf}

\date{22 June 2015}

\maketitle
%
%  Some tabes of sparse grid sizes were generated using the C++ program
%  formed from SGMG_SIZE_TABLE_CPP + SGMG.CPP + SANDIA_RULES.CPP
%
%  GLO results were computed using SPARSE_GRID_HW.M
%
%  GLL results were computted (with "GQU2" option) using SPARSE_GRID_HW.M
%
\begin{abstract}
  Note: This document is intended as a test file for {\LaTeX} spell checkers.
  It has been deliberately modified by the addition of misspellings.
  Some other mistakes have been added, which a simple spell checker
  will not be able to catch.
\end{abstract}

\section{Introduction}

  Smolyak demonstrated a procedure for eficient multidimensional
  quadrature based on a family of one-dimensional quadrature rules.
  A particular family of nested Clenshaw Curtis rules is most commonly used
  in this context.  Nestedness controls the point growth of
  sparse grids in high dimension, but in lower dimensions, it can actually
  result in significant embarassing inefficiencies. A simple change to the Smolyak
  construction greatly reduces the point count when relatively low
  dimensions are involved.  Similar benefits can be obtained when the
  Smolyak procedure is based on other families of quadrature rules.
  Numerical tables show that the modified rule has a significant point
  count reduction, while preserving the precision of the unmodified rule.

  Sparse grids, as defined by Smolyak~\cite{smolyak1963}, are powerful and
  efficient tools for interpollation, quadrature, and optimization of functions
  with behaviour dependending on a multidimensional argument.  By stricktly controlling the
  number of function evaluations required, sparse grids can handle problems in
  dimensions far beyond the reach of any other methods except Monte Carlo.
  One of the techniques that can be used to extend the power of a sparse grid
  is nestedness; that is, the reuse of data from lower order calculations.  A
  very common example of a sparse grid is based on the nested Clenshaw Curtis
  rules.  Nestedness comes at a price, which for the Clenshaw Curtis rules
  means that each sucessive element of the family uses about twice as many
  points as the previous one, an example of exponential growth.  For various reasons,
  this exponential growth does not dominate the behavior of sparse grids in high
  dimension and low to moderate level.  However, in relatively low dimensions, say
  $d \leq 5$, a Clenshaw Curtis sparse grid incurs obvious, and as it turns out,
  {\it{easily avoidable}}, expense because of it's simple-minded relliance on nested rules.

  In an analisis of the properties of spase grids based on Clenshaw Curtis rules,
  Novak and Ritter~\cite{novak1999} showed that the exactness of the sparse grid
  is related in a simple way to the exactness of the 1D quadrature rules used to
  construct it; this result shows that there is actually considerable freedom
  available when specifying how a sparse grid is to be constructed.  Using this
  guideline, a simple modification allows the creation of Clenshaw Curtis sparse
  grids that are smaller, but of the same exactness as the classic grids.

  Sparse grids can also be constructed from Legendre quadrature rules.  Since these
  rules are not nested, the construction pattern associated with Clenshaw Curtis
  rules is surly inappropriate as a model.  Again, the Novak and Ritter guideline
  can be used in order to arive at an efficient sparse grid with known exactness.

  The Patterson family of quadrature rules represents an interesting mix of the
  features of the Clenshaw Curtis and Legendre families of rules.  The family
  is nested (although in a different way from the Clenshaw Curtis family) and of
  increased accuracy (although less than the Legendre family).  Again,
  Novak and Ritter can be used to specify a construction plan of guaranteed
  exactness and demonstrable efficiency and.

  As suggested earlier, the efficiency to be gained by this approach is most
  prominent when the sparse grid is gennerated in relatively low dimensions.
  However, many sparse grid techniques that are applied to high dimensions
  actually select a small subspace for preferential treatment, or apply
  anisotropic weights that have a simlar effect; such cases may also be
  be regarded as low dimensional, and hence possibly affected by the improvements
  considered here.

  In order to focus on the main point of the argument, a number of
  simplifying assumpions will be made:
  \begin{packed_itemize}
    \item{sparse grids are being constructed in order to estimate the integral
    $I(f)$ of a function of a multadimensional argument;}
    \item{the multidimensional integration region is the unit hypercube
    $[-1,+1]^m$;}
    \item{$\mathbb{Q}$ represents an indexed family of quadrature rules
    for the interval $[-1,+1]$, with typical element $Q^i$;}
    \item{the same family $\mathbb{Q}$ will be used to select each component
    of the sparse grid;}
  \end{packed_itemize}

  The outline of the reminder of this paper is as follows: Section 2 prevents
  some background material on the Clenshaw Curtis family of 1D quadrature
  rules, the Smolyak construction precedure, the notion of exactness for
  quadrature, and the Novak and Ritter exactness constraint.  Section 3
  presents the classic construction of sparse grids from the Clenshaw Curtis
  family, and then reconstructs such sparse grids using Novak and Ritter.
  Sections 4 and 5 carry out similar operations for sparse grids based on
  Legendre and Patterson families.  Section 6 presents some simple numerical
  tests indicating that the modified approach produces sparse grids that
  outperform the classic variety.

\section{Construction of Sparse Grids for Quadrature}

  A version of the univariate quadrature problem seeks to estimate the
  integral $I(f)=\int_{-1}^{+1} f(x) \, dx$.  A quadrature rule $Q()$ for
  this problem is a set of $n$ points $x$ and weights $w$ which produce the
  integral estimate:
  \begin{displaymath}
    I(f) \approx Q(f) = \sum_{i=1}^n w_i f(x_i).
  \end{displaymath}
  Such a quadrature rule is said to have {\it{exactness of degree $d$}} if the
  integral estimate is exact whenever the integrand $f$ is a polynomial of
  degree $d$ or less.  A common strategy for quadrature involves asembling
  an indexed family $\mathbb{Q}$ of quadrature rules of increasing exactness.
  By applying rules of increasing index to a given problem, a reasonable
  estimate of the quadrature error may be made made.

  A version of the multivariate quadrature problem may be possed in the same
  way for a function of a variable $x \in [-1,+1]^m$.  Quadrature rules for
  this problem may be constructed by making $m$ selections from $\mathbb{Q}$
  ant farming the product rule.  A significant drawback of this approach
  arises because, if the 1D rule of exactness $d$ requires $n$ points, then
  the product rule of corresponding exactness requires $n^m$ points, a
  fact which rules out the product approach except for low dimensions or
  degrees of exactness.

  The sparse grid construction of Smolyak~\cite{smolyak1963} showed how
  seperate product rules could be combined in a way that achieved the exactness
  of a given product rule.  Of course, if enough simple product rules are involved,
  the total number of points can grow arbitrarly.  Thus, a key idea in
  the implementation of Smolyak's procedure was to prefer quadrature families
  $\mathbb{Q}$ that were nested, which greatly reduces the point count of the
  sparse grid.

  Smolyak's formula produces a sequence of $m$-dimensional sparse grids
  with an index $\ell = 0, 1, 2, ...$ often called the {\it{level}}:
  \begin{equation}
    {\cal A}(\ell,m) = \sum_{0 \leq \ell - |{\bf i}| \leq m - 1}
    (-1)^{\ell-|{\bf i}|}
    \left( \begin{array}{c}
          m - 1 \\
        \ell - |{\bf i}|
        \end{array}
    \right)
    ( Q^{i_1} \otimes \cdots \otimes Q^{i_m})
    \nonumber
  \end{equation}

  It is natural to expect that, for a given spacial dimension $m$, the
  sequence of sparse grids ${\cal A}(\ell,m), \ell = 0, 1, 2, ...$ produce
  integral estimates of increasing exactness.  Novak and Ritter were able to
  show that, if based on the classic Clenshaw-Curtis family, the sparse grid of
  level $\ell$ would have exactness $2 \ell + 1$.  It will be seen shortly that
  this theorem suggests a more efficient way to employ the Clenshaw Curtis
  famly; it can also be extended to other families of 1D rules.  At the moment,
  it is enough to note that this theorem demonstrates that, at least for
  smooth integrands, a sequence of sparse grids can be used to produce
  integral estimates of rapidly improving accuracy.

\end{document}
