JADE: Minimization of energy functionals with the spectral approximation

Purpose

• The package "Jade" minimizes energy functionals with the space-time spectral approximation (Gauss quadratures based on Legendre Lobatto points). The underline optimization tool is the L-BFGS package.

Specifications

• Name: Jade.
• Author: Feng Chen.
• Finishing date: 01/08/2009.
• Language: Fortran 90, Fortran 77.
• Required libraries: BLAS, L-BFGS.

Simple Example

• Energy functional: \begin{equation} \min_{u(x) \in C[-1,1]} \mathcal E [u(x)] = \int_{-1}^1 [\frac{|u'|^2}{2} + \frac{(u^2-1)^2}{4\epsilon^2}] \text{d} x, \end{equation} subjected to \begin{equation} u(-1) = 1, \quad u(1) = -1. \end{equation}
• Parameter: \begin{equation} \epsilon=0.03. \end{equation}
• Initial guess: \begin{equation} u_N(x_0) = 1, \quad u_N(x_N) = -1, \quad u_N(x_i)=0, \quad 1\leqslant i \leqslant N-1. \end{equation}

Quick Start

• Compiling and running:

  cd ./Jade
  make library
  ifort tst_Jade.f90 -llibrary -lblas
  ./a.out
  

• Output:

   
  *************************************************
    N=   31   NUMBER OF CORRECTIONS= 7
         INITIAL VALUES
   F=  2.51833D+03   GNORM=  2.22146D+03
  *************************************************
  
     I   NFN    FUNC        GNORM       STEPLENGTH
  
     1    2     1.58380D+03   9.04231D+02   4.50155D-04
     2    3     1.41239D+03   4.71418D+02   1.00000D+00
     3    4     1.28633D+03   2.94799D+02   1.00000D+00
                 ...
    58   64     1.27280D+02   1.15318D-04   1.00000D+00
    59   65     1.27280D+02   1.69359D-04   1.00000D+00
    60   66     1.27280D+02   2.82277D-05   1.00000D+00
  
   THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS.
   IFLAG = 3
   IFLAG =            3 Without error LBFGS done 
  

• CPU: Intel(R) Xeon(R) CPU X5550 @2.67GHz.
• OS: CentOS release 6.4 (Final).
• Compiler: ifort 12.1.5.

References

• Jinhae Park, Feng Chen, and Jie Shen. Modeling and simulation of switchings in ferroelectric liquid crystals, Discrete and Continuous Dynamical Systems (Series A), Volume 26, Issue 4, 1419-1440, (2010).
• Jorge Nocedal. Software for large-scale unconstrained optimization: L-BFGS in Fortran 77.

Code Highlights

!<
!! Evaluate the objective functional and its derivative.
!! These are required in the L-BFGS algorithm.
!! Global: epsinv = 1/(epsilon*epsilon)
!! Input:  n - number of Legendre collocation points
!!         nn = n + 1
!!         ua - function values
!! Output: f - functional value
!!         g - partial derivatives of f w.r.t. ua
!<
SUBROUTINE OBJ_GRAD_VALUE(n,nn,ua,f,G)
 INTEGER, INTENT(IN)          ::    N,NN
 DOUBLE PRECISION,INTENT(IN)  ::    ua(0:NN)
 DOUBLE PRECISION,INTENT(OUT) ::    F,G(1:N)
 DOUBLE PRECISION ::  du(0:NN),uf(0:NN)

 ! gradient of u
 du = MATMUL(D,ua)
 
 ! objective functional
 DO i=0,NN
    ! integrant values
    uf(i)=4.d0*du(i)*du(i)+epsinv*(ua(i)*ua(i)-1.d0)**2
 ENDDO
 ! quadrature
 f=DOT_PRODUCT(weight,uf)

 ! derivative of the functional
 DO j=1,NN-1
    G(j)=0.D0
    DO i=0,NN
    G(j)=G(j)+weight(i)*du(i)*D(i,j)
    ENDDO
    G(j)=8.D0*G(j)+4.D0*epsinv*weight(j)*(ua(j)**3-ua(j))
 ENDDO

 RETURN
END SUBROUTINE OBJ_GRAD_VALUE