Example Parameter Estimation/Tweaking Problem:

Parameter Estimation/Tweaking Problems
solved with
Calculus Programming
_________

This section shows how to solve equations of the following form:.

Uxx = f(x, U, Ux; a, b, c, ...)
where the parameters (a, b, c, etc.) are tweaked/estimated

This is a general form of a partial differential equation. To tweak parameters a, b, c, etc. through Calculus-level programming we replace the following code for the "call xAxis" statement in these examples:


         a = 1:   b = 1:    c = 1		! Initial Estimations
         find a, b, c, ...   In xAxis   ooo   to optimize xyz

     where 'optimize' may be the term 'maximize', 'minimize', 'match',
          or 'extreme'; 'match' implies your objective function 
          converges to zero.

plus, add an objective function in your math model.
ooo

xyz = some function of parameters a, b, c, etc.

This 'find' statement will vary your parameters until 'xyz' is optimal. Thus meeting your objective requirements.

Enjoy learning Calculus-level Programming!


Example Parameter Estimation/Tweaking Problem Source Code:



A Parameter Estimation/Tweaking Problem:



      global all
      problem nonLinPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: non-linear PDE (1D) Parameter
C --- Estimation Problem solved.
C ------------------------------------------------------------------------
C
C User parameters ...
!       rho = ...
        e0 = 8.854187817e-12 ! F/m or A2 s4 kg-1m−3 permittivity of free space
!        ipoints = 10          ! grid pts. over x-axis
C
C x-parameter initial settings: x ==> i
!        xFinal =  1:    xPrint = xFinal/ipoints:       ip = ipoints
        pi= 4*atan(1)
C
        a = 1:	b = 1:    c = 1		! Initial Estimations
        find a, b, c, ... ;   in xAxis;    by ?solver?;   to ?optimize? xyz
      end
      model xAxis
C ... Integrate over x-axis
C
        x= 0:    xPrt = xPrint:      dx = xPrt / 10
        Initiate janus;  for PDE;
     ~       equations Uxx/Ux, Ux/U;  of x;  step dx;  to xPrt
!        U = 1.23		! @ x = 0 ... an Initial Value Problem
        do while (x .lt. xFinal)
          Integrate PDE;  by janus
          if( x .ge. xPrt) then
            print 79, x, U, Ux, Uxx
            xPrt = xPrt + xPrint
          end if
        end do
 79     format( 1h , f8.4, 2x, 10(g14.5, 2x))
      end
      model PDE                         ! Partial Differential Equation
        Uxx = -rho/e0 * (1.23 + sin( Ux * pi) - .543) ooo	! your Diff Eq goes here
        xyz = function of parameters to be optimized ...
! e.g.   xyz = U + c - b / (1 + Uxx * c**2)
      end

Example Parameter Estimation/Tweaking Problem Output:



selected output goes here ...
HTML code for linking to this page:


<a href="http://fortranCalculus.info/math-problems/parameter-estimation.html"><img style="margin-left: 5%; width:100px" src="http://fortranCalculus.info/image/fc-compiler-icon.png"/> <strong>Example Parameter Estimation/Tweaking Problem</strong> </a>; Simulation to Optimization, Tweak Parameters for Optimal Solution.

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