Optimal Sampling

Solve weighted-LSP approximation of a function f on an irregular domain D, from samples points under a optimal measures.

• Function: Opt_sam_v1.m

  This function sampling points in any iteration with respect the dimension and polynomial degree.

  Basic sctructure:

  • Set up: Define the parameters.
  • Generate K points under uniform measure: Generate the grid.
  • Pre-processing:
    • Under hyperbolic cross index this generate n_max and N_max (polynomial degree and number of bases).
    • Reorder index set: The first index set correspond to n_1 and the second index until n_2 and continues.
    • Generate B matrix: This matrix contain the Legendre polynomials evaluates on the grid.
    • Compute measure mu.
    • Evaluate f on the grid.
  • Compute W-LS approximation:
    • Choose M points.
    • Compute the mu related to iteration.
    • sampling M points under mu.
    • solve w-LS approximation.
    • save singular values of A and Error.

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• Function: Opt_sam_v3.m

  This function use a sequantial sampling the dimension and polynomial degree.

  Basic sctructure:

  • Set up: Define the parameters.
  • Generate K points under uniform measure: Generate the grid.
  • Pre-processing:
    • Under hyperbolic cross index this generate n_max and N_max (polynomial degree and number of bases).
    • Reorder index set: The first index set correspond to n_1 and the second index until n_2 and continues.
    • Generate B matrix: This matrix contain the Legendre polynomials evaluates on the grid.
    • Compute measure mu.
    • Evaluate f on the grid.
  • Compute W-LS approximation:
    • Choose M points and k points.
    • sampling k points under mu_j.
    • sampling k_ad (additional) points under mu_{j-1}.
    • Re-order index.
    • solve w-LS approximation.
    • save singular values of A and Error.

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• Functions: Newsam_Recover_Plot_r1.m and Firstsam_Recover_plot_r1.m

  This function plot the result obtained under the sequentaly sampling(Opt_sam_v3.m), and optimal sampling(Opt_sam_v1.m). Note: The _r1 it is label for the Annular domain radio 1. You can change the script save for your convenience.

  Basic sctructure:

  • Save errors: The files related from d = 2 to d = 10 are separate of d = 15, because they have different size.
  • Compute the median.
  • plot the figures.