Functions
AMA_MltvGrdApprox.c File Reference
#include <AMA.h>

Functions

long int AMA_MltvGrdApprox (AMA_OPTIONS *options, long int nind, long int *ng, double **x, double *z, double *epsilon, long int *degree, AMA_SPLINE **spline)
 Approximation of Multivariate Gridded Data More...
 

Function Documentation

long int AMA_MltvGrdApprox ( AMA_OPTIONS options,
long int  nind,
long int *  ng,
double **  x,
double *  z,
double *  epsilon,
long int *  degree,
AMA_SPLINE **  spline 
)

Approximation of Multivariate Gridded Data

This function employs cnspla to compute a spline approximation of independent variable data ${\bf X}\in{\rm R}^n$ and dependent variable data ${\bf Z}\in{\rm R}^1$. The independent variable data is given as $x_{k,\ell_k}$, for $\ell_k=1,\ldots,N^g_k$ and $1\le k\le n$, and it defines the rectilinear grid

\[ (x_{1,1},x_{1,2},\cdots,x_{1,N^g_1-1},x_{1,N^g_1})\times\cdots\times(x_{n,1},x_{n,2},\cdots,x_{n,N^g_n-1},x_{n,N^g_n}). \]

The dependent variable data lies on the grid and is given as $z_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$; that is, there are $N=\prod_{k=1}^nN^g_k$ dependent variable values. Similarly, the approximation tolerances are given as $\epsilon_{\ell_1,\cdots,\ell_n}$ for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$.

For a given set of data this function employs cnspla to compute the spline

\[ s({\bf X}) = \sum_{j_n=1}^{m_n}\cdots\sum_{j_1=1}^{m_1}\alpha_{j_1,\ldots,j_n}B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda_n}) \]

that minimizes

\[ F^{(2)}(s({\bf X})) = \sum_{k=1}^n\left[\int_R\left(\partial^2 s({\bf X})\over\partial^2 x_k\right)^2dR \right] \]

subject to the approximation constraints

\[ z_{\ell_1,\cdots,\ell_n} - \epsilon_{\ell_1,\cdots,\ell_n} \le s(x_{1,\ell_1},\cdots,x_{n,\ell_n}) \le z_{\ell_1,\cdots,\ell_n} + \epsilon_{\ell_1,\cdots,\ell_n}, \]

for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$.

In the above definition of $s({\bf X})$ the $\alpha_{j_1,\ldots,j_n}$, for $j_k=1,\ldots,m_k$ and $1\le k\le n$, are the coefficients of the tensor product B-splines

\[ \Phi_{j_1,\ldots,j_n}({\bf X}) = B_{d_1,j_1}(x_1|{\bf\Lambda}_1)\cdots B_{d_n,j_n}(x_n|{\bf\Lambda}_n) \]

where $B_{d_k,j_k}(x_k|{\bf\Lambda}_k)$ are the $m_k$ univariate B-splines of degree $1 \le d_k\le 5$ defined by the knot vector ${\bf\Lambda}_k\in{\rm R}^{M_k}$. The knot vectors ${\bf\Lambda}_k$ are

\[ {\bf\Lambda}_k = ( \lambda_{k,1},\cdots,\lambda_{k,d_k+1}, \lambda_{k,d_k+2}, \cdots, \lambda_{k,m_k}, \lambda_{k,m_k+1},\cdots,\lambda_{k,m_k+d_k+1} )^T \]

and they depend on the independent variable data and the degree $d_k$. The knot vector ${\bf\Lambda}_k$ is defined by AMA_LamdaInterp() based on the points $(x_{k,1},x_{k,2},\cdots,x_{k,N^g_k-1},x_{k,N^g_k})$.

This function employs an efficient algorithm for computing spline approximations of multivariate gridded data. The algorithm consists of computing the splines

\[ \bar s_k({\bf X}) = \sum_{q_n=1}^{\bar m_n}\cdots\sum_{q_1=1}^{\bar m_1}\bar\alpha^k_{q_1,\ldots,q_n}B_{\bar d_1,q_1}(x_1|{\bf\bar\Lambda}_1)\cdots B_{\bar d_n,q_n}(x_n|{\bf\bar\Lambda_n}) \]

for $1\le k\le n$. The coefficients $\bar\alpha^k_{q_1,\ldots,q_n}$ of $\bar s_k({\bf X})$ are the coefficients of the tensor product B-splines

\[ \Phi_{q_1,\ldots,q_n}({\bf X}) = B_{\bar d_1,q_1}(x_1|{\bf\bar\Lambda}_1)\cdots B_{\bar d_n,q_n}(x_n|{\bf\bar\Lambda}_n) \]

where $B_{\bar d_k,q_k}(x_k|{\bf\bar\Lambda}_k)$ are the $\bar m_k$ univariate B-splines of degree $\bar d_k$ defined by the knot vector ${\bf\bar\Lambda}_k\in{\rm R}^{\bar m_k + \bar d_k + 1}$. The degree $\bar d_{k_o}$ is

\[ \bar d_{k_o} = \cases { d_{k_o}, & for $1\le k_o\le k$, \cr 1 , & for $k < k_o\le n $; \cr } \]

the number of coefficients $\bar m_{k_o}$ is

\[ \bar m_{k_o} = \cases { m_{k_o} , & for $1\le k_o\le k$, \cr n^g_{k_o}, & for $k < k_o\le n $; \cr } \]

and the knot vector ${\bf\bar\Lambda}_{k_o}$ is

\[ {\bf\bar\Lambda}_{k_o} = \cases { {\bf\Lambda}_{k_o} , & for $1\le k_o\le k$, \cr ( x_{k_o,1}, x_{k_o,1}, x_{k_o,2}, \cdots, x_{k_o,n^g_{k_o}-1}, x_{k_o,n^g_{k_o}}, x_{k_o,n^g_{k_o}})^T & for $k < k_o\le n $. \cr } \]

The spline $s({\bf X})$ is based on the spline $\bar s_n({\bf X})$ and is defined by setting its coefficents $\alpha_{j_1,\ldots,j_n}=\bar\alpha^n_{j_1,\ldots,j_n}$ for $j_k=1,\ldots,m_k$ and $1\le k\le n$. The spline $\bar s_n({\bf X})$ is defined through multiple invocations of AMA_UnvApprox() in the following manner.

First, the spline $\bar s_1({\bf X})$ is defined by invoking AMA_UnvApprox() $\prod_{k=2}^nN^g_{k}$ times to compute the univariate spline

\[ \bar s_{q_2,\cdots,q_n}(x_1) = \sum_{q_1=1}^{m_1}\bar\alpha^1_{q_1,\ldots,q_n}B_{d_1,q_1}(x_1|{\bf\Lambda}_1) \]

that minimizes

\[ F^{(2)}(\bar s_{q_2,\cdots,q_n}(x_1)) = \int_{x_{1,1}}^{x_{1,N^g_1}}\left( \bar s^{(2)}_{q_2,\cdots,q_n}(x_1) \right)^2 dx_1 \]

subject to the approximation constraints

\[ z_{\ell_1,q_2,\cdots,q_n} - \epsilon_{\ell_1,q_2,\cdots,q_n} \le \bar s_{q_2,\cdots,q_n}(x_{1,\ell_1}) \le z_{\ell_1,q_2,\cdots,q_n} + \epsilon_{\ell_1,q_2,\cdots,q_n} \]

for $\ell_1=1,\ldots,N^g_1$. Computing the univariate splines $\bar s_{q_2,\cdots,q_n}(x_1)$, for $q_k=1,\ldots,N^g_k$ and $2\le k\le n$, defines the coefficients $\bar\alpha^1_{q_1,\ldots,q_n}$ of $\bar s_1({\bf X})$.

After computing $\bar s_1({\bf X})$ the splines $\bar s_k({\bf X})$, for $2\le k\le n$, are defined by invoking AMA_UnvApprox() $\prod_{k_o=1}^{k-1}m_{k_o}\prod_{k_o=k+1}^nn^g_{k_o}$ times to compute the univariate spline

\[ \bar s_{q_1,\cdots,q_{k-1},q_{k+1},\cdots,q_n}(x_k) = \sum_{q_k=1}^{m_k}\bar\alpha^k_{q_1,\ldots,q_n}B_{d_k,q_k}(x_k|{\bf\Lambda}_k) \]

that minimizes

\[ F^{(2)}(\bar s_{q_1,\cdots,q_{k-1},q_{k+1},\cdots,q_n}(x_k)) = \int_{x_{k,1}}^{x_{k,n^g_k}}\left( s^{(2)}_{q_1,\cdots,q_{k-1},q_{k+1},\cdots,q_n}(x_k) \right)^2 dx_k \]

subject to the interpolation constraints

\[ \bar s_{q_1,\cdots,q_{k-1},q_{k+1},\cdots,q_n}(x_{k,\ell_k}) = \bar\alpha^{k-1}_{q_1,\cdots,q_{k-1},\ell_k,q_{k+1},\cdots,q_n} \]

for $\ell_k=1,\ldots,N^g_k$. Computing the univariate splines $\bar s_{q_1,\cdots,q_{k-1},q_{k+1},\cdots,q_n}(x_k)$, for $q_{k_o}=1,\ldots,m_{k_o}$ when $1\le k_o\le k-1$ and for $q_{k_o}=1,\ldots,N^g_{k_o}$ when $k+1 \le k_o \le n$, defines the coefficients $\bar\alpha^k_{q_1,\ldots,q_n}$ of $\bar s_k({\bf X})$.

In total this algorithm consists of $\sum_{k=1}^n\left(\prod_{k_o=1}^{k-1}m_{k_o}\prod_{k_o=k+1}^nN^g_{k_o}\right)$ invocations of AMA_UnvApprox() to compute a multivariate spline which satisfies the approximation constraints.

This function does the following:

  • Checks the input parameters for validity, see AMA_MltvGrdInputCheck().
  • Defines the spline $s({\bf X})$ based on the knot vectors ${\bf\Lambda}_k$, for $1\le k\le n$.
  • Defines the splines $\bar s_k({\bf X})$, for $1\le k\le n$, through multiple invocations of AMA_UnvApprox().
  • Stores the approximation $s({\bf X})=\bar s_n({\bf X})$ into spline.
Note
By default the spline coefficients are initialized to zero but a different initial value for the spline coefficients can be set with AMA_OptionsSetCoefficients().
Because this function employs the univariate spline approximation function AMA_UnvApprox(), it does not minimizes the cross partial terms of the penalty term utilized by AMA_MltvApprox().
Since the approximation constraints are only imposed during the computation of $\bar s_1({\bf X})$, the approximation depends on the order of the independent variables.

Parameter Note: In the parameters description given below the limits on $k$ are $1\le k\le n$, k = $k-1$ and N $=\prod_{k=1}^nN^g_k$.

Parameters
options[in] Pointer to AMA_OPTIONS. Must be initialized with AMA_Options() prior to calling AMA_MltvGrdApprox().
nind[in] The number of independent variables $n$. Must satisfy $2\le$ nind $\le$ AMA_MXNIND.
ng[in] Array of size nind containing the number of points $N^g_k$ where ng[k] $= N^g_k$. Must satisfy ng[k] $\ge 2$.
x[in] Array of size nind containing arrays of size ng[k] where x[k] contains the independent variable data $x_{k,\ell_k}$, for $\ell_k=1,\ldots,N^g_k$. The values of x[k] must be in strictly ascending order.
z[in] Array of size N containing the dependent variable data $z_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$.
epsilon[in] Array of size N containing the approximation toleraneces $\epsilon_{\ell_1,\cdots,\ell_n}$, for $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. Must satisfy $\epsilon_{\ell_1,\cdots,\ell_n}\ge 0.0$ for all $\ell_1=1,\ldots,N^g_1,\cdots,\ell_n=1,\ldots,N^g_n$. If epsilon = NULL, then this function uses $\epsilon_{\ell_1,\cdots,\ell_n}=0.0$.
degree[in] Array of size nind containing the degree $d_k$ where degree[k] $= d_k$. Must satisfy $0\le$ degree[k] $\le 5$.
spline[out] Pointer to AMA_SPLINE pointer containing the spline approximation. Must satisfy spline $\ne$ NULL.
Returns
Success/Error Code.

User Callable Function - Documented 103015