Scilab Function
Last update : 3/11/2006

me_brute2d - [For comparison only] 2D Moreau envelope, brute force computation

Calling Sequence

M = me_brute2d(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 performs a brute force computation for a worst-case time complexity for a grid of size N=n^2 of O(N^2)=O(n^4), when n=length(Xr)=length(Xc)=length(Sr)=length(Sc). This function is available for comparison purposes only, use more efficient linear-time algorithms for efficient computation.

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_brute2d(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_direct2d ,   me_llt2d ,   me_nep2d ,   me_pe2d ,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

Y. Lucet, 2006, Fast Moreau Envelope Computation I: Numerical Algorithms, Numerical Algorithms, 43 (2006), 235-249