Efficient and Adaptive Orthogonal Finite Element Representation of the Geopotential

Abstract

We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10⁻⁹ m s⁻², globally) are required near the Earth’s surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with efficiency optimized using radial adaptation.

Appendices

Appendix: A: Chebyshev Polynomials

Chebyshev polynomials are a set of orthogonal polynomials developed by the Russian mathematician Pafnuty Lvovich Chebyshev in 1857 [6, 12]. There are two kinds of Chebyshev polynomials. The k ^th Chebyshev polynomials of the first kind usually are denoted by T _k and the k ^th Chebyshev polynomials of the second kind usually are denoted by U _k . In this paper, we refer to Chebyshev polynomials of the first kind as Chebyshev polynomials. The Chebyshev polynomials can be computed through the recurrence relation as

$$ T_{0}(x)=1,\;\; T_{1}(x)=x,\;\; T_{k+1}(x)=2xT_{k}(x)-T_{k-1}(x), $$

(62)

or the Chebyshev polynomial of degree k can be defined by the identity T _k (x)=cos(kcos⁻¹(x)):x 𝜖[−1,1].

The continuous orthogonality conditions for Chebyshev polynomials are

$$ {\int}_{-1}^{1}w(x)T_{n}(x)T_{m}(x)dx=\left\{ \begin{array}{c} 0:n\neq m\\ \pi:n=m=0\\ \pi/2:n=m\neq0 \end{array}\right.\;\text{and}\; w(x)=(1-x^{2})^{-\frac{1}{2}}. $$

(63)

The discrete orthogonality conditions for the Chebyshev polynomials using the CGL nodes are

$$ \overset{M}{\underset{k=0}{\sum}}w_{k}T_{n}(x_{k})T_{m}(x_{k})\,=\,\!\left\{ \begin{array}{c} \!0:n\neq m\\ \!\!M\!\!:n\,=\,m\,=\,0\\ \!\!M/2\!:\!n=\!m\!\neq0 \end{array}\right.\;\mathrm{\!\!\!\!\!and}\; \!w_{0}\,=\,w_{M}\,=\,\frac{1}{2},\!\; \!w_{k}\,=\,1;\: k\,=\,1,2,...,M\,-\,1\!. $$

(64)

The (N+1) CGL (or “cosine”) nodes for the N ^th order Chebyshev polynomials are calculated from

$$ x_{k}=-\text{cos}\left( \frac{k\pi}{M}\right);\; k=0, 1, 2,...,M. $$

(65)

Indefinite integration of the Chebyshev polynomials has the property $\int T_{k}(x)dx=\frac {1}{2}\left (\frac {T_{k+1}}{k+1}-\frac {T_{k-1}}{k-1}\right )$.

The first derivative of the Chebyshev polynomials satisfies

$$\begin{array}{@{}rcl@{}} \frac{dT_{k}(x)}{dx} = kU_{k-1}(x) &=& \frac{k}{1-x^{2}} [-xT_{k}(x)+T_{k-1}(x)] \\ &=& \frac{2k}{1-T_{2}(x)}[-xT_{k}(x)+T_{k-1}(x)]. \end{array} $$

(66)

Thus integrals and the derivatives are expressed as recursions contiguous degree Chebyshev polynomials. The first six Chebyshev polynomials are shown in Fig. 21.

Fig. 21

Chebyshev Polynomials of the first kind

Appendix: B: Kronecker Factorization and Least Square

Approximation

Proof of the important property regarding Kronecker factorization in Least squares that if a matrix Φ of rank n with Φ ∈ R ^{m × n}; m ≥ n can be Kronecker factorized as

$$ {\Phi}={\Phi}_{x}\otimes{\Phi}_{y}, $$

(67)

then the classical normal equations

$$ a=\left\{ \left( {\Phi}^{T}{\Phi}\right){\Phi}^{T}\right\} \mathbf{\textbf{f}} $$

(68)

can, amazingly, be rewritten as

$$ a=\left\{ \left( {{\Phi}_{x}^{T}}{\Phi}_{x}\right)^{-1}{{\Phi}_{x}^{T}}\right\} \otimes\left\{ \left( {{\Phi}_{y}^{T}}{\Phi}_{y}\right)^{-1}{{\Phi}_{y}^{T}}\right\} \mathbf{\textbf{f}}. $$

(69)

That is, the large “least square operator” {(Φ^TΦ)⁻¹Φ^T} can be rewritten as simply the Kronecker product of two small matrices:

$$ \left\{ \left( {\Phi}^{T}{\Phi}\right)^{-1}{\Phi}^{T}\right\} =\left\{ \left( {{\Phi}_{x}^{T}}{\Phi}_{x}\right)^{-1}{{\Phi}_{x}^{T}}\right\} \otimes\left\{ \left( {{\Phi}_{y}^{T}}{\Phi}_{y}\right)^{-1}{{\Phi}_{y}^{T}}\right\} . $$

(70)

The matrices $\left ({{\Phi }_{x}^{T}}{\Phi }_{x}\right ),\left ({{\Phi }_{y}^{T}}{\Phi }_{y}\right )$ must obviously be non-singular. To prove this identity, we need the following three properties of Kronecker matrix operations for square and nonsingular matrices A and B

$$ \left( A\otimes B\right)^{T}=A^{T}\otimes B^{T}, $$

(71)

$$ \left( A_{1}\otimes A_{2}\right)\left( B_{1}\otimes B_{2}\right)=\left( A_{1}B_{1}\right)\otimes\left( A_{2}B_{2}\right), $$

(72)

$$ \left( A\otimes B\right)^{-1}=A^{-1}\otimes B^{-1}. $$

(73)

The property of Eq. 70 is proven as follows: Using the assumed factorization of Eq. 67, {(Φ^TΦ)⁻¹Φ^T} can be written as

$$ \left\{ \left( {\Phi}^{T}{\Phi}\right)^{-1}{\Phi}^{^{T}}\right\} =\left( \left( {\Phi}_{x}\otimes{\Phi}_{y}\right)^{T}\left( {\Phi}_{x}\otimes{\Phi}_{y}\right)\right)^{-1}\left( {\Phi}_{x}\otimes{\Phi}_{y}\right)^{T}. $$

(74)

Then using Eqs. 71, 74 re-arranges to

$$ \left\{ \left( {\Phi}^{T}{\Phi}\right)^{-1}{\Phi}^{^{T}}\right\} =\left( \left( {{\Phi}_{x}^{T}}{\otimes{\Phi}_{y}^{T}}\right)\left( {\Phi}_{x}\otimes{\Phi}_{y}\right)\right)^{-1}\left( {{\Phi}_{x}^{T}}{\otimes{\Phi}_{y}^{T}}\right), $$

(75)

and using Eq. 72, Eq. 75 becomes condition

$$ \left\{ \left( {\Phi}^{T}{\Phi}\right)^{-1}{\Phi}^{^{T}}\right\} =\left( \left( {{\Phi}_{x}^{T}}\otimes{\Phi}_{x}\right)\left( {{\Phi}_{y}^{T}}\otimes{\Phi}_{y}\right)\right)^{-1}\left( {{\Phi}_{x}^{T}}{\otimes{\Phi}_{y}^{T}}\right). $$

(76)

Using Eq. 73, Eq. 76 is

$$ \left\{ \left( {\Phi}^{T}{\Phi}\right)^{-1}{\Phi}^{^{T}}\right\} =\left( \left( {{\Phi}_{x}^{T}}\otimes{\Phi}_{x}\right)^{-1}\left( {{\Phi}_{y}^{T}}\otimes{\Phi}_{y}\right)^{-1}\right)\left( {{\Phi}_{x}^{T}}{\otimes{\Phi}_{y}^{T}}\right), $$

(77)

and finally, using the property of Eq. 72, Eq.77 becomes Eq. 70, Q.E.D.

This property extends to high dimensioned Kronecker factorizations, i.e., if

$$ {\Phi}={\Phi}_{x}\otimes{\Phi}_{y}\otimes{\Phi}_{z}, $$

(78)

then the large least square operator is written as the Kronecker product of three small least square operators as

$$ \left\{ \left( {\Phi}^{T}{\Phi}\right)^{-1}{\Phi}^{T}\right\} =\left\{ \left( {{\Phi}_{x}^{T}}{\Phi}_{x}\right)^{-1}{{\Phi}_{x}^{T}}\right\} \otimes\left\{ \left( {{\Phi}_{y}^{T}}{\Phi}_{y}\right)^{-1}{{\Phi}_{y}^{T}}\right\} \otimes\left\{ \left( {{\Phi}_{z}^{T}}{\Phi}_{z}\right)^{-1}{{\Phi}_{z}^{T}}\right\} . $$

(79)

These results are easily extended to include the weighted least square case, as well. For the special case that the basis functions in 1, 2, and 3 dimensions satisfy orthogonality conditions such that the off-diagonal elements of $\left ({{\Phi }_{x}^{T}}{\Phi }_{x}\right ),\left ({{\Phi }_{y}^{T}}{\Phi }_{y}\right ),\left ({{\Phi }_{z}^{T}}{\Phi }_{z}\right )$ vanish, then likewise the larger matrix (Φ^TΦ) is diagonal and we see that Eqs. 78 and 79 also provide very convenient means for generalizing one dimensional orthogonal approximation operators to higher dimensions. Care must always be taken to understand and properly choose the multidimensional nodal sample patterns and weight matrices, to ensure orthogonality of the basis functions with respect to both the weight matrices and nodal locations.