CFD Utilities Files

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!
   program usinterp
!
!  Purpose:
!
!     Read a list of data points with any number of dimensions and any number of
!     function values at each data point as a column dataset with any number of
!     header lines that are ignored.  Also read a similar table of coordinates
!     at which to calculate estimates of those functions.  This serves as a
!     driver for the optimal interpolation module originally made available by
!     Alexander Barth (University of Liege, Belgium) and extended by David
!     Saunders at NASA Ames Research Center.  Interpolating/smoothing of an
!     unstructured dataset is intended.
!
!  Method:
!
!     The optimal interpolation scheme is designed to minimize interpolation
!     errors under certain statistical assumptions about the function samples.
!     Experience with smoothly varying test datasets shows remarkably accurate
!     results that may be artificially good.  This driver is part of setting up
!     and running further test cases, including working with noisy datasets.
!
!  Unstructured Input Dataset Format:
!
!     Header records (zero or more lines that are ignored)
!     x1 x2 ... xn  f1 f2 .. fm  ! First data point: n dimensions, m functions
!     x1 x2 ... xn  f1 f2 .. fm  ! Further data points, any order
!      :  : ...  :   :  :     :  ! n >= 1, m >= 1
!      :  : ...  :   :  :     :
!     x1 x2 ... xn  f1 f2 .. fm
!
!  Target Coordinates Dataset Format:
!
!     Header records (one or more lines, ignored)
!     x1 x2 ... xn
!     x1 x2 ... xn
!      :  : ...  :  ! Any number of target points, in some order or not
!      :  : ...  :
!     x1 x2 ... xn
!
!  Input Control File Format (on Standard Input):
!
!     USINTERP Control File
!     nd                    ! Number of coordinate dimensions; nd >= 1
!     nf                    ! Number of functions at each data point
!     unstructured.dat      ! Input dataset
!     target.dat            ! Target coordinates dataset
!     usinterp.dat          ! Output results
!     (3es16.8, 4es15.7)    ! Run-time output format
!     m                     ! Number of nearest neighbors to interpolate within
!     normalize             ! T|F; T => normalize the input coordinates
!     verbose               ! T|F; T => diagnostic output is written to fort.50
!     cf1 cf2 ...           ! nd correlation functions: each is g|l|G|L
!     cl1 cl2 ...           ! nd correlation lengths in (0, 1]
!     ovar1 ovar2 ...       ! nf observation variances >= 0.
!     adapt                 ! -1.|0.|a < 1.; -1. => simple inverse dist. weights
!
!  Output Dataset Format:
!
!     Same as the target coordinates, with interpolated function(s) appended
!     as in the input data format, no header lines, and interpolated function
!     variances as nf additional functions:
!
!     x1 x2 ... xn  f1 f2 .. fm v1 v2 .. vm
!     x1 x2 ... xn  f1 f2 .. fm v1 v2 .. vm
!      :  : ...  :  ! Target coords., interpolated fns., interpolated variances
!      :  : ...  :
!     x1 x2 ... xn  f1 f2 .. fm v1 v2 .. vm
!
!  Notes:
!
!     1. Best choices for the multiple controls tend to require trial and error
!        because of inadequate documentation.
!     2. The normalization option converts the working coordinates to [0, 1]
!        in case of disparate raw units; unnormalized coordinates are output.
!     3. Forcing some nonzero contribution from the furthest of m neighbors to
!        a given target point may or may not be a good idea.  Increasing the
!        correlation length(s) should have a similar effect.
!     4. See the available test_optiminterp.f90 for testing on a dataset that
!        is constructed from a trigonometric function, with an option to impose
!        Gaussian noise on the data point function values.
!     5. The observation variance inputs are assumed to be a measure of the
!        uncertainty in the function data.  Originally, they could differ at
!        each data point for a single function.  Handling of multiple functions
!        led to one variance per function type instead.  Experience suggests
!        that specifying zeros for the variances is preferable, but this is
!        not well understood.
!     6. The adaptation option is not working as intended because the trig.
!        squared test case suggests that large correlation lengths produce the
!        best results.
!
!  History:
!
!     01/31/2022  D.A.Saunders  Initial implementation to enable more testing.
!     02/11/2022    "     "     [After a severed internet cable hiatus:]
!                               trig-squared test case for Optimal_Interpolation
!                               reproduced here from tabular datasets.
!     02/27/2022    "     "     Perform a second set of interpolations at the
!                               data point coordinates and show the sum of
!                               squared differences from the data functions.
!
!  Author: David Saunders, AMA, Inc. at NASA Ames Research Center, CA
!
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