PHYSOR 2012 Advances in Reactor Physics Linking Research, Industry, and Education Knoxville, Tennessee, USA, April 15-20, 2012, on CD-ROM, American Nuclear Society, LaGrange Park, IL (2012)
PSEUDOSPECTRAL CHEBYSHEV REPRESENTATION OF FEW-GROUP CROSS SECTIONS ON SPARSE GRIDS Pavel M. Bokov∗ and Danni¨ell Botes South African Nuclear Energy Corporation (Necsa) Building 1900, PO Box 582, Pretoria 0001, South Africa
[email protected] Vyacheslav G. Zimin Department “Automatics”, National Research Nuclear University “MEPhI”, Moscow, Kashirskoe shosse, 31, 115409, Russia
[email protected]
ABSTRACT This paper presents a pseudospectral method for representing few-group homogenised cross sections, based on hierarchical polynomial interpolation. The interpolation is performed on a multi-dimensional sparse grid built from Chebyshev nodes. The representation is assembled directly from the samples using basis functions that are constructed as tensor products of the classical one-dimensional Lagrangian interpolation functions. The advantage of this representation is that it combines the accuracy of Chebyshev interpolation with the efficiency of sparse grid methods. As an initial test, this interpolation method was used to construct a representation for the two-group macroscopic cross sections of a VVER pin cell. Key Words: sparse grids, homogenised neutron cross sections, Lagrange interpolation
1. INTRODUCTION There are several steps in the typical deterministic reactor calculational path and it is important that each step should be performed as accurately as possible. The particular step that will be the focus of this paper is the representation of few-group, homogenised neutron cross sections as they are passed from the cell or assembly code to the full core simulator. These homogenised neutron cross sections may depend on a number of physical parameters, such as burnup, fuel and moderator temperature, etc. Several approaches have been applied to the construction of a representation of these dependencies and a brief overview follows below. The conceptually simplest way to create a multi-dimensional cross section representation is to take a one-dimensional representation method (either an interpolation or an approximation) and apply a tensor product to extend it into multiple dimensions. This method quickly runs into the problem that the number of samples required to achieve a given accuracy grows exponentially with the number of dimensions, a phenomenon known as the curse of dimensionality [1]. As an example, consider a one-dimensional interpolation performed on 5 samples. To extend that interpolation to 7 dimensions in the standard tensor product manner would require 57 = 78125 samples of the function being interpolated. The approach based on tensor products has, however, been used for cross section representation, as shown in [2]. ∗ Corresponding
author
P. M. Bokov, D. Botes and V. G. Zimin
An alternative is to sample the function being approximated on a non-tensor product mesh, and then construct the approximation by either regression [3,4] or quasi-regression [5–7]. The curse of dimensionality may still appear in this case, but in a different form: if the dependencies are unknown, one can try to build the model by using, for example, a linear combination of tensor products of one-dimensional functions. If no special measures are used, the number of such combinations grows exponentially with the number of dimensions. These special measures may include an iterative construction of the model with criteria for rejection/retention of trial functions based on e.g. statistical [3] or high dimensional model representation [4–7] principles. Another way to mitigate the curse of dimensionality is to use techniques that are specially designed to deal with it, such as Monte Carlo or sparse grid methods. Monte Carlo methods do not suffer from the curse of dimensionality and is known to have a steady, but slow rate of convergence, which does not depend on the smoothness of the underlying function. Dependencies of few-group neutron cross sections are known to be smooth, and there would be an advantage in using a method that takes this property into account in order to improve the rate of convergence. Such a method for the construction of an interpolation or a quadrature was proposed by Smolyak [8]. It involves the combination of tensor products of hierarchical one-dimensional functionals (interpolation or quadrature), so as to optimise the convergence of the error for certain classes of functions. The multidimensional discretisation mesh that results from this process is called a sparse grid. A combinatorial formulation of sparse grid methods was later proposed in [9]. Sparse grid methods have been applied to represent few-group neutron cross section dependencies in the past with some success [5,7]. This was done by constructing a global polynomial approximation through quasi-regression. It was found that the global approximation missed localised behaviour (such as the xenon build-up), unless polynomials of a very high order were used. Hierarchical multi-linear interpolation [10] of the cross section dependencies on sparse grids followed the efforts at approximation, with the expectation of better capturing the small-scale phenomena of the dependencies. The multi-linear interpolation showed promising results, but came with some important drawbacks. Firstly, it suffers from unacceptable discontinuities in the first derivatives of the interpolation functions. These discontinuities are a concern, given the necessity of calculating quantities such as reactivity coefficients. Secondly, linear interpolation has a relatively slow rate of convergence for the interpolation error, which can be improved on for smooth functions by using a high order polynomial interpolation. A method for hierarchical polynomial interpolation on sparse grids, which is intended to overcome the above drawbacks is presented and tested in this paper. This methodology for higher order multidimensional polynomial interpolation will be based on the one-dimensional Lagrange interpolation method. The Lagrange interpolation method has long been considered to be mainly of academic interest with little practical value, and accepted wisdom claimed that higher order polynomials should be avoided due to Runge’s phenomenon, a problem that plagues most polynomial interpolation methods. This perception has recently changed due to the work of Berrut and Trefethen [11], who discussed an efficient algorithm for Lagrange interpolation and have pointed out that Runge’s phenomenon is mitigated if the interpolation is performed on the roots or extrema of Chebyshev polynomials. Interpolation nodes chosen in this way are referred to as Chebyshev nodes and the representation method is referred as pseudo-spectral [12]. 2012 Advances in Reactor Physics Linking Research, Industry, and Education (PHYSOR 2012) Knoxville, Tennessee, USA April 15-20, 2012
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Pseudospectral Chebyshev Representation of Few-Group Cross Sections on Sparse Grids
Following these arguments, a description of polynomial interpolation on a Clenshaw-Curtis sparse grid, i.e. a sparse grid built from Chebyshev nodes, is provided. This allows for our method to combine the well-known efficiency of Chebyshev interpolation with the low calculation and storage requirements of sparse grid methods, which makes the method scalable in dimensionality. In addition, due to the direct, hierarchical manner in which our interpolation is constructed, it is possible to assess the importance of each term individually, which allows us to reject unimportant terms and thereby compress the representation without having to regenerate it. The same mechanism allows us to estimate the accuracy of the representation. The interpolation method was used to construct a representation for the two-group macroscopic cross sections of a VVER pin cell and the results obtained will be presented and discussed. 2. SPARSE GRID INTERPOLATION 2.1. Problem Statement Consider the problem of performing polynomial interpolation on a real-valued, multivariate function defined on a finite d-dimensional rectangular domain. This rectangular problem domain can be easily mapped to the canonical domain [−1, 1]d and back by using a linear transformation. For this reason, the problem of interpolating a real valued function f : [−1, 1]d → R is discussed in the rest of this paper. 2.2. One-Dimensional Lagrange Interpolation on Chebyshev Nodes Let us start our description of the interpolation method with a discussion of the one-dimensional case. This discussion allows us to introduce all the necessary notations, conventions and concepts, that will be used later on for the multidimensional interpolation on sparse grids. 2.2.1. Interpolation nodes Let us consider the problem of interpolating a univariate function, f : [−1, 1] → R, based on a finite set of samples. The quality of the interpolation depends on the choice of interpolation nodes x j . Many options exist and the so-called Chebyshev nodes, used in our work, are in some sense optimal, as they allow one to minimise the maximum interpolation error and to mitigate the Runge phenomenon [11–13]. Let us denote X and refer to it as the interpolation grid the set {x j }Nj=0 ⊂ [−1, 1] of interpolation nodes with x j chosen as the Chebyshev points of the second kind jπ , (1) x j = − cos N also known as Chebyshev-Gauss-Lobatto points. Here j = 0, . . . , N for some given non-negative integer N ∈ N0 and, by convention, x0 = 0 for j = N = 0. Parameter N defines, therefore, the degree of the interpolation polynomial constructed using N + 1 interpolation points. 2012 Advances in Reactor Physics Linking Research, Industry, and Education (PHYSOR 2012) Knoxville, Tennessee, USA April 15-20, 2012
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N` Let us now consider a sequence {X` }∞ `=0 of grids X` = {x j } j=0 , where N` is defined as:
( 0, if N` = 2` , if
` = 0, ` ≥ 1.
(2)
The sequence index ` ∈ N0 , introduced in Eq. (2), will be referred to as the level of the corresponding one-dimensional grid. Let us introduce the delta set D` as a subset of X` , which contains the nodes that have their first appearance at some grid level `: D` = X` \ X`−1 . The level of a node x j ∈ X` is said to be l if this node has its first appearance at grid level l: level x j = l ⇔ x j ∈ Dl ⊆ X` ,
(3)
(4)
where 0 ≤ l ≤ `. Sets X` defined by Eqs. (1–2) are nested, i.e. X`−1 ⊂ X` for all ` ≥ 1. Grid X` S can, therefore, be presented as the union of non-overlapping delta sets X` = 0≤m≤` Dm . In other words, X` is a collection of nodes with the levels l less than or equal to `. These nested sets X` of Chebyshev nodes can be built in practice by generalising the dyadic representation approach discussed in [10]. To this effect, consider a sequence of integer numbers ν ∈ Z, ν ≥ −1. To each ν there corresponds a Chebyshev node with the abscissa: xν = − cos (πuν ) .
(5)
Here parameter uν is defined as follows: u−1 = 0.5, u0 = 0, u1 = 1 (for ν = −1, 0, 1), and lν
uν =
∑ b j 2− j
(for ν ≥ 2),
(6)
j=1
where b j ∈ {0, 1} is the bit value in the binary representation of the integer number ν. Here the level lν of a node xν is defined as the position of the last (counting from the rightmost position) non-zero bit in the binary representation of integer ν and, by convention, l−1 = 0 and l0 = l1 = 1. As an example, consider ν = 6. Its binary representation is 6 = 1102 which leads to: l6 = 3, u6 = 0.0112 = 0 · 2−1 + 1 · 2−2 + 1 · 2−3 = 3/8 and x6 = − cos(3π/8). It should be emphasised that the abscissa of a node, its level and, as will be discussed later, the corresponding basis function are uniquely identified by a single index ν. The interpolation grid X` can be constructed by incrementing ν, starting from ν = −1. The abscissa xν and level lν of the node are calculated using the binary operations defined above and treating the first three terms (i.e. ν = −1, 0, 1) as exceptions. The process is continued until the first node with a level lν = ` + 1 is achieved. The set of interpolation nodes and the delta set can, therefore, be defined as X` = {xν : lν ≤ `} and D` = {xν : lν = `} ,
(7)
respectively. 2012 Advances in Reactor Physics Linking Research, Industry, and Education (PHYSOR 2012) Knoxville, Tennessee, USA April 15-20, 2012
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Pseudospectral Chebyshev Representation of Few-Group Cross Sections on Sparse Grids
2.2.2. Lagrange interpolating polynomial Consider a set X` of interpolation nodes for some fixed `. An interpolating polynomial of degree N` can be constructed using the classical Lagrange interpolation formula: PN` (x) =
f (xν )Lν (x) ,
∑
(8)
xν ∈X`
where Lν (x) are the Lagrange Basis Polynomials, corresponding to the node xν ∈ X` , defined as Lν (x) =
(x − xµ ) , (x − xµ ) ∈X \x ν
∏
xµ
`
(9)
ν
for ν ≥ 0 and, in order to avoid ambiguity, let us define L−1 (x) = 1 for ` = 0. By their construction, the Lagrange Basis Polynomials have the following property: if xν ∈ X` and xµ ∈ X` then Lν (xµ ) = δν µ ,
(10)
which will be crucial for our further derivations. A function that satisfies this requirement is referred to as the cardinal function, though some authors use: “cardinal basis”, “Lagrange basis”, or “the fundamental polynomials of interpolation” [12]. As one may observe, the denominator in Eq. (9) depends on the distribution of the interpolation nodes and does not depend on x. The so-called barycentric weights [11] ων =
∏ (xν − xµ )−1,
(11)
xµ ∈X` \xν
can, therefore, be pre-calculated (analytically or numerically) and the Lagrange Basis Polynomial can then be written in a simplified way: Lν (x) = ω ν
∏ (x − xµ ).
(12)
xµ ∈X` \xν
Note that it follows from the definition of the Lagrange interpolation that ω−1 = 1 for ` = 0. 2.2.3. Hierarchical interpolation Although a function in the form of a Lagrange polynomial, as given by Eq. (8), is perfectly capable of fulfilling the interpolation role, we will represent the interpolation function in the so-called hierarchical form. The hierarchical interpolation is an iterative procedure, which consists in applying the next level interpolation to the remainder of the current level interpolation. The resulting interpolation is a sum of contributions from different levels. The crucial advantage of the hierarchical approach is its flexibility in the process of both building the interpolation and its posterior optimization [16]. The hierarchical interpolation procedure will be described in full detail for the multidimensional case in the next section. This section is used to discuss and to illustrate some essential facts related to the hierarchical interpolation, as well as some implications of using Chebyshev nodes. 2012 Advances in Reactor Physics Linking Research, Industry, and Education (PHYSOR 2012) Knoxville, Tennessee, USA April 15-20, 2012
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Let fˆ` (x) ≡ PN` (x) be the interpolation given by Eq. (8) for some level ` and R` (x) ≡ f (x) − fˆ` (x) be the remainder of such an interpolation. Applying the next level interpolation to the remainder yields: (13) ∑ R`(xν )Lν (x) = ∑ R`(xν )Lν (x) + ∑ R`(xν )Lν (x). xν ∈X`+1
xν ∈X`
xν ∈D`+1
Due to the nestedness of the interpolation nodes, the property given by Eq. (10) is valid for arbitrary xν and xµ under an additional condition: lν ≥ lµ . As a result, the first sum on the right-hand side of Eq. (13) vanishes for all xν ∈ X` : R` (xν ) = f (xν ) − fˆ` (xν ) = f (xν ) −
∑
f (xµ )Lµ (xν ) = f (xν ) −
xµ ∈X`
∑
f (xµ )δµν = 0.
(14)
xµ ∈X`
Moreover, contributions of nodes xµ ∈ D`+1 to the second term in the right-hand side of Eq. (13) are independent:
∑ R`(xν )Lν (xµ ) = ∑ R`(xν )δµν = R`(xµ ) = f (xµ ) − fˆ`(xµ )
xν ∈D`+1
(15)
xν ∈D`+1
and characterised by quantities sµ = f (xµ ) − fˆ` (xµ ), which are called hierarchical surpluses. Finally, due to the nestedness of interpolation grids and the properties of the cardinal functions, the higher level terms do not affect the approximation constructed on previous levels. As a result, the contribution, corresponding to each node xν from the sequence {xν }∞ ν=−1 of interpolating nodes, appears only once in the hierarchical interpolation formula `
fˆ` (x) =
∑ ∑
sν Bν (x)
(16)
m=0 xν ∈Dm
according to the level lν of the node. The spacial dependence of the related hierarchical term is described by a unique hierarchical basis function Bν (x), which corresponds to the cardinal function (12) with ` = lν : Bν (x) = ω ν ∏ (x − xµ ), (17) xµ ∈Xlν \xν
weighted with the value of the hierarchical surplus sν . These surpluses can be calculated individually at each level during the process of iterating over levels. A side effect of the hierarchical approach to the approximation is a simplification of basis functions. Saltzer [13] has demonstrated that the barycentric weights can be calculated analytically for the distribution of interpolation nodes given by Eq. (1). Moreover, one may observe that the number of nodes, given by Eq. (2), is always odd. Also, starting from level ` = 2, all new points appear at odd positions (i.e. they have odd index j, used in Eq. (1)). Applying this information to the analytical formulas from [13] one deduces that the barycentric weights depend only on the level lν of the node: if lν = 0; 1, −1 2 , if lν = 1; ων = (18) 2lν −lν −1 −2 , if lν > 1. 2012 Advances in Reactor Physics Linking Research, Industry, and Education (PHYSOR 2012) Knoxville, Tennessee, USA April 15-20, 2012
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As a result, there is no need to calculate barycentric weights for each node: they can be pre-calculated level-wise in advance with very little calculational effort (one calculation per level). 2.3. Multivariate Interpolation The Lagrange interpolation approach allows a straightforward generalization to the multi-dimensional space via a tensor product of one dimensional interpolation formulas. If one uses an interpolation with the same scale resolution in each direction, this leads to Nd = (N` + 1)d points in the mesh. A simple analysis shows that this approach is not scalable: for instance, for level ` = 2, i.e. for an interpolation with a 4th order polynomial, and d = 5 it requires Nd = 55 = 3125 and for d = 7 it increases to Nd = 57 = 78125. Using the next level (i.e. ` = 3) resolution leads to Nd = 95 = 59049 for the five dimensional problem. This indicates that, for the application discussed here, a naive implementation of this approach does not provide sufficient resolution for the state parameter dependencies, or the number of samples achievable in practice. An alternative way to construct the interpolation, which is based on sparse grid method, will be pursued in this paper. This method is known to provide for smooth functions an accuracy comparable with the tensor product construction while requiring substantially fewer samples [9,14,16]. 2.3.1. Multidimensional nodes and basis functions Let ~ν = (ν1 , ν2 , . . . , νd ) be a d-dimensional vector of integers greater than or equal to −1. Each vector ~ν represents a unique multidimensional node ~x~ν = (xν1 , xν2 , . . . , xνd ) ⊂ [−1, 1]d ,
(19)
where xνi are defined by Eq. (5) for i = 1, 2, . . . , d. The word unique in this context means that ~x~ν =~x~µ if and only if ~ν = ~µ. Therefore, new multidimensional nodes can be constructed by incrementing the components νi of ~ν. Let us introduce the level of the d-dimensional node ~x~ν as a sum of the levels of its components: d
d
l~ν = level (~x~ν ) = ∑ level (xνi ) = ∑ lνi . i=1
(20)
i=1
As it follows from its definition above, l~ν ∈ N0 , i.e. the level of the node is a non-negative integer and l~ν = 0 if lνi = 0 for all i = 1, 2, . . . , d. A d-dimensional sparse grid of level ` can be introduced, therefore, as a collection of nodes ~x~ν with a level l~ν at most `: X`d = {~x~ν : l~ν ≤ `}. (21) When the coordinates of grid points are Chebyshev nodes, the sparse grid is usually referred to as the Clenshaw-Curtis grid because of the related quadrature rule [14]. The delta set of sparse grid nodes can, by analogy, be introduced as a set of nodes with the level equal to `: D`d = {~x~ν : l~ν = `}. 2012 Advances in Reactor Physics Linking Research, Industry, and Education (PHYSOR 2012) Knoxville, Tennessee, USA April 15-20, 2012
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d ⊂ X d holds for the sparse grid used in this paper and the sparse grid The nested property X`−1 ` S can be presented as a union of delta sets [14]: X`d = 0≤m≤` Dmd .
The multivariate basis function, corresponding to a node ~x~ν , is defined via the tensor product d
B~ν (~x) = Bν1 (x1 ) ⊗ · · · ⊗ Bνd (xd ) = ∏ Bνi (xi )
(23)
i=1
of one-dimensional cardinal functions Bνi (xi ), given by Eq. (17). Here the tensor product is replaced with the ordinary product because Bνi (xi ) are real valued functions. For the multivariate basis functions B~ν (~x) the following property holds: B~ν (~x~µ ) = δ~ν~µ
(24)
for a given level ` and for all nodes ~x~ν ∈ D`d and ~x~µ ∈ X`d , i.e. the basis functions vanish at all the other nodes from the same and previous levels. This property of basis functions allow the building of an interpolation hierarchically and point-by-point in a way that will be described in the next section. 2.3.2. Interpolation The interpolation function of level ` can be written as a linear combination of the multivariate basis functions (23) with known coefficients: fˆ` (~x) =
∑ s~ν B~ν (~x).
(25)
~x~ν ∈X`d
This interpolation can also be written in an equivalent hierarchical form as a weighted contribution of the basis functions belonging to different levels: `
fˆ` (~x) =
∑ ∑
s~ν B~ν (~x).
(26)
m=0 ~x~ν ∈Dmd
The hierarchical surpluses s~ν can be calculated recursively for m = 0, . . . , ` and for all interpolation nodes ~x~ν from the set ~x~ν ∈ Dmd , using the formula s~ν = f (~x~ν ) − fˆm−1 (~x~ν )
(27)
with the starting condition for the recursion: fˆm
