Multi-fidelity Modelling

The construction offers support for multi-fidelity modelling. A special type of knot function is included which simply maps a level to the same integer, to then be use to specify the model. This has a weight $w=1.0$ and a corresponding polynomial equal to $1$ for all inputs. In practice, for interpolation there is no parameter to be evaluated, but this set up provides the correct multi-fidelity structure using the existing single fidelity construction.

using SparseGridsKit
FidelityPoints()(3), Fidelity()(3)

(([3], [1.0]), 1)

A simple multi-fidelity model considers a constant function $f(x)=y_1^2 + sin(y_2) + y_3$, subject to errors $10^{-\alpha}$. Functions are specified a $f(\vec{\alpha},\vec{y})$ where the vector $\vec{\alpha}$ controls the fidelity and $\vec{y}$ is simply the parameter as usual.

f(y) = y[1]^2 + sin(y[2]) + y[3]

nfid = 1
nparams = 3
f_fidelities(alpha,y) = f(y) + 10^(-alpha[1])

f_fidelities (generic function with 1 method)

The rule that must be used for a fidelity is Fidelity. This returns $1$ for all input levels as there is only one model per level. The knots are FidelityPoints, which returns the input value plus a weight zero. The knot function is treated as a special case in the sparse grid construction.

maxfidelity = 5
rule = [fill(Fidelity(),nfid)..., fill(Doubling(),nparams)...]
knots = [fill(FidelityPoints(),nfid)..., fill(CCPoints(),nparams)...]

4-element Vector{Points}:
 FidelityPoints(Integer[1, 9223372036854775807])
 CCPoints{Float64}([-1.0, 1.0])
 CCPoints{Float64}([-1.0, 1.0])
 CCPoints{Float64}([-1.0, 1.0])

The domain for FidelityPoints is by default from 1 to the maximum integer, resulting in the large number above.

We wrap the function $f$ using multifidelityfunctionwrapper. This allows the user to use the single fidelity sparse grid construction, with function calls separated as $z=[\vec{\alpha},\vec{y}]$. Note that knots is passed as a parameter: this is used to identify the splitting for the function.

f_wrapped = multifidelityfunctionwrapper(f_fidelities,knots)

MI = create_smolyak_miset(nfid+nparams,2)

sg = create_sparsegrid(MI; knots=knots, rule=rule)
@show get_grid_points(sg)
@show f_eval = f_wrapped.(get_grid_points(sg))

33-element Vector{Float64}:
  0.1
  1.1
 -0.9
  0.8071067811865474
 -0.6071067811865475
  0.9414709848078965
 -0.7414709848078965
  1.9414709848078966
  0.2585290151921035
 -0.05852901519210349
  ⋮
  0.5999999999999999
  0.010000000000000002
  1.01
 -0.99
  0.8514709848078965
 -0.8314709848078965
  1.01
  1.01
  0.001

Finally, interpolation and integration are possible on the sparse grid. Interpolation currently requires dummy values to give a complete parameter vector $[\vec{\alpha}, \vec{y}]$. In the future this may be simplified.

interpolate_on_sparsegrid(sg, f_eval, [[fill(1,nfid)..., fill(0.0,nparams)...]])

pcl = precompute_lagrange_integrals(5, knots, rule)

f_on_grid = f_wrapped.(get_grid_points(sg))

# Test integrate_on_sparsegrid
integral_result = integrate_on_sparsegrid(sg, f_on_grid)

0.33433333333333376

It is also possible to use the adaptive sparse grid construction. A costfunction can be supplied to give a representative cost for each fidelity. This acts on a vector $\vec{α}$ of the fidelities. We must supply knots and rules explictly so that the algorithm can distinguish between model and input dimensions.

knots = [fill(FidelityPoints(),nfid)..., fill(CCPoints(),nparams)...]
rule = [fill(Fidelity(),nfid)..., fill(Doubling(),nparams)...]
(sg, f_on_Z) = adaptive_sparsegrid(f_wrapped, nfid + nparams; maxpts = 100, proftol=1e-4, rule = rule, knots = knots, θ=1e-4, type=:deltaint, costfunction=α->10^prod(α))

(SparseGrid(4, [[1.0, 9.223372036854776e18], [-1.0, 1.0], [-1.0, 1.0], [-1.0, 1.0]], [1 1 … 3 4; 1 2 … 1 1; 1 1 … 1 1; 1 1 … 1 1], [-1, 1, 0, 0, 1], Real[1.0 0.0 0.0 0.0; 1.0 1.0 0.0 0.0; 1.0 -1.0 0.0 0.0; 4.0 0.0 0.0 0.0], [-0.33333333333333326, 0.16666666666666669, 0.16666666666666669, 1.0], Points[FidelityPoints(Integer[1, 9223372036854775807]), CCPoints{Float64}([-1.0, 1.0]), CCPoints{Float64}([-1.0, 1.0]), CCPoints{Float64}([-1.0, 1.0])], Level[Fidelity(), Doubling(), Doubling(), Doubling()], SparseGridsKit.sparse_grid_data{4}(SparseGridsKit.SparseTerm{4}[SparseGridsKit.SparseTerm{4}(([1, 1], [1, 1], [1, 1], [1, 1])), SparseGridsKit.SparseTerm{4}(([1, 1], [2, 1], [1, 1], [1, 1])), SparseGridsKit.SparseTerm{4}(([1, 1], [2, 2], [1, 1], [1, 1])), SparseGridsKit.SparseTerm{4}(([1, 1], [2, 3], [1, 1], [1, 1])), SparseGridsKit.SparseTerm{4}(([4, 1], [1, 1], [1, 1], [1, 1]))], [(1, 1, 1, 1), (1, 2, 1, 1), (1, 3, 1, 1), (4, 1, 1, 1)], [1, 2, 1, 3, 4], [-1, 1, 1, 1, 1], [Dict([2, 1] => 2, [1, 1] => 1, [3, 1] => 3, [4, 1] => 4), Dict([4, 2] => 6, [1, 1] => 1, [3, 5] => 3, [4, 5] => 1, [2, 1] => 2, [3, 1] => 2, [4, 8] => 9, [4, 1] => 2, [4, 6] => 8, [2, 3] => 3…), Dict([4, 2] => 6, [1, 1] => 1, [3, 5] => 3, [4, 5] => 1, [2, 1] => 2, [3, 1] => 2, [4, 8] => 9, [4, 1] => 2, [4, 6] => 8, [2, 3] => 3…), Dict([4, 2] => 6, [1, 1] => 1, [3, 5] => 3, [4, 5] => 1, [2, 1] => 2, [3, 1] => 2, [4, 8] => 9, [4, 1] => 2, [4, 6] => 8, [2, 3] => 3…)], [[[1.0], [2.0], [3.0], [4.0]], [[0.0], [1.0, 0.0, -1.0], [1.0, 0.7071067811865475, 0.0, -0.7071067811865475, -1.0], [1.0, 0.9238795325112867, 0.7071067811865475, 0.38268343236508984, 0.0, -0.38268343236508984, -0.7071067811865475, -0.9238795325112867, -1.0]], [[0.0], [1.0, 0.0, -1.0], [1.0, 0.7071067811865475, 0.0, -0.7071067811865475, -1.0], [1.0, 0.9238795325112867, 0.7071067811865475, 0.38268343236508984, 0.0, -0.38268343236508984, -0.7071067811865475, -0.9238795325112867, -1.0]], [[0.0], [1.0, 0.0, -1.0], [1.0, 0.7071067811865475, 0.0, -0.7071067811865475, -1.0], [1.0, 0.9238795325112867, 0.7071067811865475, 0.38268343236508984, 0.0, -0.38268343236508984, -0.7071067811865475, -0.9238795325112867, -1.0]]], [[1.0, 2.0, 3.0, 4.0], [0.0, 1.0, -1.0, 0.7071067811865475, -0.7071067811865475, 0.9238795325112867, 0.38268343236508984, -0.38268343236508984, -0.9238795325112867], [0.0, 1.0, -1.0, 0.7071067811865475, -0.7071067811865475, 0.9238795325112867, 0.38268343236508984, -0.38268343236508984, -0.9238795325112867], [0.0, 1.0, -1.0, 0.7071067811865475, -0.7071067811865475, 0.9238795325112867, 0.38268343236508984, -0.38268343236508984, -0.9238795325112867]], [[[1.0], [1.0], [1.0], [1.0]], [[1.0], [0.16666666666666669, 0.6666666666666667, 0.16666666666666669], [0.033333333333333354, 0.26666666666666666, 0.4, 0.26666666666666666, 0.033333333333333354], [0.007936507936507936, 0.07310932460800906, 0.13968253968253969, 0.1808589293602449, 0.19682539682539682, 0.1808589293602449, 0.13968253968253969, 0.07310932460800906, 0.007936507936507936]], [[1.0], [0.16666666666666669, 0.6666666666666667, 0.16666666666666669], [0.033333333333333354, 0.26666666666666666, 0.4, 0.26666666666666666, 0.033333333333333354], [0.007936507936507936, 0.07310932460800906, 0.13968253968253969, 0.1808589293602449, 0.19682539682539682, 0.1808589293602449, 0.13968253968253969, 0.07310932460800906, 0.007936507936507936]], [[1.0], [0.16666666666666669, 0.6666666666666667, 0.16666666666666669], [0.033333333333333354, 0.26666666666666666, 0.4, 0.26666666666666666, 0.033333333333333354], [0.007936507936507936, 0.07310932460800906, 0.13968253968253969, 0.1808589293602449, 0.19682539682539682, 0.1808589293602449, 0.13968253968253969, 0.07310932460800906, 0.007936507936507936]]])), [0.1, 1.1, 1.1, 0.0001])

Function Reference

(f:Fidelity)(l)

(p::FidelityPoints)(n)

FidelityPoints()

fidelitymap(y, nfid, nparam)

fidelitypoints(n)

multifidelityfunctionwrapper(f, nfid, nparam)