Accurate and efficient full-degree multidimensional polynomial interpolation
5 Stars
Updated Last
8 Months Ago
Started In
April 2018


Build Status

Note: this package is under active development and is subject to change.

Interpolation, accuracy, and the curse of dimensionality

Interpolation is the process of building a model from data available at discrete locations. Full-degree multidimensional polynomial interpolation differs from most other interpolation schemes in that it is not, in general, continuous over space. Instead, the goal is to find a local polynomial fit of full degree and use this for detailed investigations of local structure---in essence, prioritizing accuracy of local interpolation over any global property like continuity or smoothness.

An important application full-degree interpolation is derivative-free optimization (DFO), where many algorithms work by constructing a quadratic fit to the local function values and then minimizing the quadratic over some trust region---since the goal is to find the location of the minimum (or minima, when there are several), global continuity of the interpolation scheme is not an urgent priority. For use in DFO (and presumably most other interpolation applications), one therefore requires a rule to select a "complete" subset of evaluation positions to determine the model coefficients. If different point subsets are selected in different regions of space, the polynomial will change discontinuously as one moves around the space; but for purposes of optimization, this is an acceptable tradeoff that enables one to quickly find the exact minimum via quasi-Newton methods.

Unfortunately, when n is large the computational burden of building the model can be high. For example, the number of parameters in a full-degree quadratic model in n dimensions is (n+1)*(n+2)/2. This implies that data from (at least) an equal number of evaluation points must be used; moreover, solving for the model via linear regression is in the general case an O(n^6) problem. This becomes prohibitive for n in the hundreds and extremely costly even for much more modest problems. Powell described a method for updating a quadratic model with a new data point; this method has a cost in the range O(n^2)--O(n^4) per update depending on the amount of history maintained, and for local DFO this represents a significant improvement. However, this method is less attractive in the context of global optimization, where multiple regions in space may be pursued simultaneously but for which one may not wish to maintain storage of O(n^2) parameters for many different local models.

Reducing the burden with CSp-trees

This repository implements a new (?) data structure, the coordinate splitting tree of degree p or CSp-trees for short. As shown here, this data structure supports efficient and accurate full-degree multidimensional polynomial interpolation. For example, a CS1-tree reduces the burden of fitting a quadratic model to ~O(n^4), and a CS2-tree reduces it to O(n^2). Moreover, the resulting coefficients are determined with significantly higher precision than by traditional approaches. CSp trees naturally implement adaptive mesh refinement (AMR), as adding new evaluation points to the interpolation scheme corresponds to adding new nodes to the tree. This package is written in the Julia programming language.

CSp-trees have a close relationship with k-d trees, and indeed CS1 trees can be implemented essentially as a k-d tree with only one evaluation point (and substantial restrictions on its position) per box. Specifically, a k-d tree "splits" space along one dimension at a time; a CS1 tree selects a new evaluation point that differs from its "parent" position in only one coordinate, thus creating a new box that splits the parent along this coordinate. A general CSp tree splits p dimensions simultaneously, creating a set of 2^p-1 new boxes each associated with a unique position and new function evaluation.

For now, to learn how to use this pacakge just type ?CoordinateSplittingPTrees at the Julia prompt, and then read the help associated with each function. For convenience, that overall summary help-text is reproduced below.

Usage summary

Note that more detail is available in the help for each function.

The world bounds are specified by creating a World object (see ?World).

The tree is initialized with root = Box{p}(world), and new leaves are added with Box(parent, dims, xs, metas) (see ?Box).

You can interact with boxes using the following API (again, this is currently under development and may be unimplemented, outdated, or in flux):

Polynomial fits

  • polynomial_full: construct a full polynomial of the specified degree
  • polynomial_minimal: construct a minimal polynomial of the specified degree

Tree API

  • position: retrieve the position of a Box

  • meta: retrieve the metadata associated with a Box

  • value: retrieve just the function value associated with a Box

  • boxbounds: retrieve the edges of a Box

  • isroot: determine whether a given Box is the root-node

  • isleaf: determine whether a given Box is a leaf-node

  • getroot: get the root node associated with a tree

  • getleaf: get the leaf node associated with a (possibly-parent) box

  • find_leaf_at: find the leaf node containing a particular position

  • addpoint!: add a new n-dimensional evaluation point

  • leaves: returns an iterator for visiting all leaf-nodes


  • print_tree: display the tree structure (from AbstractTrees.jl)
  • splitprint: a more compact display of the tree structure
  • splitprint_colored: similar to splitprint but with highlights