Table of Contents

Adaptive Refinement

ChebyshevSharp provides adaptive-refinement APIs for selecting piecewise knots and per-dimension node counts.

For smooth dense-grid interpolation, start with Error-Driven Construction. It describes the ChebyshevApproximation auto-node loop controlled by errorThreshold, maxN, and nullable entries in nNodes.

ChebyshevSpline.AutoKnots

AutoKnots builds a ChebyshevSpline after scanning each dimension for second-difference spikes with the other dimensions fixed at their midpoints. It is a heuristic for candidate knot placement, not a proof that every non-smooth point has been found.

double F(double[] p, object? _) => Math.Abs(p[0]);
var sp = ChebyshevSpline.AutoKnots(F, 1,
    new[] { new[] { -1.0, 1.0 } },
    new[] { 16 });
// Discovers a knot near x=0; the resulting Spline has 2 pieces.

Tuning parameters: thresholdFactor (default 5.0), maxKnotsPerDim (default 5), and nScanPoints (default 200). Set maxKnotsPerDim: 0 to disable auto-knot insertion and build a single-piece spline. Always inspect sp.Knots and validate held-out points after using the scan.

SobolIndices

Variance decomposition from spectral Chebyshev coefficients. No Monte Carlo, no extra evaluations beyond what's already in TensorValues. The reported variance uses the Chebyshev orthogonality weight on each normalized input dimension, not a sampled uniform input distribution.

double F(double[] p, object? _) => Math.Sin(p[0]) + p[1] * p[2];
var ap = new ChebyshevApproximation(
    function: F,
    numDimensions: 3,
    domain: new[]
    {
        new[] { -1.0, 1.0 },
        new[] { -1.0, 1.0 },
        new[] { -1.0, 1.0 }
    },
    nNodes: new[] { 16, 16, 16 });
ap.Build();
SobolResult s = ap.SobolIndices();
Console.WriteLine($"FirstOrder: [{string.Join(", ", s.FirstOrder)}]");
Console.WriteLine($"TotalOrder: [{string.Join(", ", s.TotalOrder)}]");
Console.WriteLine($"Variance: {s.Variance}");

s.Variance == 0 or a value at the documented numerical noise floor indicates a constant function; the indices are zero and meaningless. For ChebyshevSpline, indices are computed from the full piecewise expansion, combining within-piece Chebyshev variance, between-piece mean variance, and interactions between interval membership and local Chebyshev modes.

For a Chebyshev expansion

\[ f(x) = \sum_{\alpha} c_\alpha \prod_{j=1}^{d} T_{\alpha_j}(x_j), \]

the variance under the Chebyshev weight is the weighted energy of all non-constant coefficients:

\[ V = \sum_{\alpha \ne 0} c_\alpha^2 \prod_{j=1}^{d} \langle T_{\alpha_j}, T_{\alpha_j}\rangle, \qquad \langle T_0,T_0\rangle=\pi,\quad \langle T_k,T_k\rangle=\pi/2\ (k>0). \]

First-order energy for dimension \(j\) sums coefficients with \(\alpha_j>0\) and all other degrees zero. Total-order energy for dimension \(j\) sums every coefficient with \(\alpha_j>0\). ChebyshevTT.SobolIndices() computes the same quantities by contracting TT coefficient cores, avoiding dense coefficient materialization.

ChebyshevTT.WithAutoOrder + Reorder

TT compression rank depends on dim order; some functions admit much lower-rank TTs under a non-canonical permutation.

double F(double[] p) => Math.Sin(p[0] * p[2]) + Math.Cos(p[1]);
var tt = ChebyshevTT.WithAutoOrder(F, 3,
    new[] { new[] { -1.0, 1.0 }, new[] { -1.0, 1.0 }, new[] { -1.0, 1.0 } },
    new[] { 16, 16, 16 },
    nTrials: 5, method: "greedy_swap");
Console.WriteLine($"DimOrder: [{string.Join(", ", tt.DimOrder)}]");
// Eval/Slice/Extrude/etc. transparently remap user coordinates by tt.DimOrder.
double v = tt.Eval(new[] { 0.3, -0.4, 0.5 });

// Manual realignment to a different permutation:
var realigned = tt.Reorder(new[] { 1, 2, 0 }, maxRank: 16, tolerance: 1e-10);

Binary algebra (+, -) between TTs requires matching DimOrder; call Reorder on one operand first if they differ.

References

  • Sobol, I. M. (2001). "Global Sensitivity Indices for Nonlinear Mathematical Models and Their Monte Carlo Estimates."
  • Saltelli, A. et al. (2010). "Variance Based Sensitivity Analysis of Model Output."
  • Trefethen, L. N. (2013). Approximation Theory and Approximation Practice. SIAM.
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