Scilab Function
Last update : 3/11/2006

me_nep2d - 2D Moreau envelope for convex functions, NEP algorithm

Calling Sequence

M = me_nep2d(Xr,Xc,f,Sr,Sc)

Parameters

Description

Compute numerically the discrete Moreau envelope of a set of spatial points (Xr(i1),Xc(i2),f(i1,i2)) at slopes (Sr(j1),Sc(j2)), i.e.

						     2		         2
	M(j1,j2) =  min [ f(i1,i2) + (Sr(j1) - Xr(i1) + (Sc(j2) - Xc(i2))  ].
		   i1,i2
It reduces computation to one dimension, and computes the Legendre conjugate with the Non-Expansive Prox algorithm thereby resulting in a theta(n*m + m1*m2) linear-time algorithm.

Note that the algorithm requires the underlying function f to have a nonexpansive proximal mapping to return the correct result (see me_nep ).

Examples

	    function f=f(lambda,x),f=lambda * x.^2,endfunction
	    function g=g(lambda1,lambda2,x,y),g=f(lambda1,x)+f(lambda2,y),endfunction
	    lambda1=1;lambda2=2;
	    x1=(-10:10)';x2=(-5:5)';
	    [X, Y]=ndgrid(x1,x2);F=g(lambda1,lambda2,X,Y);
	    s1=(-4:4)';s2=(-5:6)';
	    Xr=x1;Xc=x2;Sr=s1;Sc=s2;
	    desired=me_nep2d(x1,x2,F,s1,s2);
	    //1d computation for separable function
	    Ms1=me_direct(x1,f(lambda1,x1),s1);
	    Ms2=me_direct(x2,f(lambda2,x2),s2);
	    t1 = Ms1 * ones(1,size(Ms2,1));
	    t2 = ones(size(Ms1,1),1) * Ms2';
	    correct=t1+t2;
	    b = and(correct == desired);
  

See Also

me_brute2d ,   me_direct2d ,   me_llt2d ,   me_pe2d ,   me_nep ,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

See me_nep

Used Function

Computation is reduced to one dimension, which is then handled by me_nep .