This package implements Smolyak's method for approximating multivariate continuous functions. Two different types of interpolation schemes are allowed: Chebyshev polynomials and piecewise linear.
To install this package you need to type in the REPL
using Pkg Pkg.add("SmolyakApprox")
Then the package can be used by typing
The nodes are computed using Chebyshev-Gauss-Lobatto, with the approximation grid and the multi-index computed by
grid, multi_ind = smolyak_grid(chebyshev_gauss_lobatto,d,mu,domain)
d is the dimension of the function,
mu is the layer or approximation order, and domain is a 2d-array (2xd) containing the upper and lower bound on each variable. If domain is not provided, then it is assumed that the variables reside on the [-1,1] interval. If
mu is an integer, then an isotropic grid is computed whereas if
mu is a 1d array of integers with length
d, then an anisotropic grid is computed. Because the chebyshev_gauss_lobatto points are the same as the Chebyshev extrema points you can use
chebyshev_extrema in place of
With the grid and multi-index in hand, we can compute the weights, or coefficients in the approximation, according to
weights = smolyak_weights(y,grid,multi_ind,domain)
y is a 1d-array containing the evaluations at each grid point of the function being approximated. Computation of the weights can be made more efficient by computing the inverse interpolation matrix (this generally needs to be done only once, outside any loops)
inv_interp_mat = smolyak_inverse_interpolation_matrix(grid,multi_ind,domain)
with the weights now computed through
weights = smolyak_weights(y,inv_interp_mat)
Lastly, we can evaluate the Smolyak approximation of the function at any point in the domain by
y_hat = smolyak_evaluate(weights,point,multi_ind,domain)
point (a 1d-array) is the point in the domain where the approximation is to be evaluated.
For piecewise linear approximation equidistant nodes are used where the number of nodes is determined according to the Clenshaw-Curtis grid structure: 2^(mu-1)+1
grid, multi_ind = smolyak_grid(clenshaw_curtis_equidistant,d,mu,domain)
Then the weights are computed using
weights = smolyak_pl_weights(y,grid,multi_ind,domain)
and the approximation computed via
y_hat = smolyak_pl_evaluate(weights,point,grid,multi_ind,domain)
mu can be either an integer or a 1d array of integers depending on whether an isotropic or an anisotropic approximation is desired, and the argument
domain is unnecessary where the grid resides on [-1,1]^d.
There are multi-threaded functions to compute the polynomial weights and the interpolation matrix. These multi-threaded functions are accessed by adding
_threaded to the end of the funtion, as per
weights = smolyak_weights_threaded(y,inv_interp_mat)
The key structure to be aware of is the SApproxPlan, which contains the key information needed to approximate a function.
d = 3 mu = 3 domain = [2.0 2.0 2.0; -2.0 -2.0 -2.0] grid, mi = smolyak_grid(chebyshev_extrema,d,mu,domain) plan = SApproxPlan(:chebyshev_extrema,grid,mi,domain)
plan = smolyak_plan(chebyshev_extrema,d,mu,domain)
Once the approximation plan has been constructed it can be used to create functions to interpolate and to compute gradients and hessians.
f = smolyak_interp(y,plan) g = smolyak_gradient(y,plan) h = smolyak_hessian(y,plan) point = [1.0, 1.0, 1.0] f(point) g(point) h(point)
There are threaded versions of
My primary references when writing this package were:
Judd, K., Maliar, L., Maliar, S., and R. Valero, (2014), "Smolyak Method for Solving Dynamic Economic Models: Lagrange Interpolation, Anisotropic Grid and Adaptive Domain," Journal of Economic Dynamics and Control, 44, pp.92--123.
Klimke, A., and B. Wohlmuth, (2005), "Algorithm 846: spinterp: Piecewise Multilinear Hierarchical Grid Interpolation in MATLAB," ACM Transactions on Mathematical Software, 31, 4, pp.561--579.