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Banana-shaped minima region in objective function

One of the challenges of inverse problems is that the line search often must go through minima regions that have challenging shapes. This tutorial illustrates a "banana"-shaped minimum region
imb=  mk_common_model('a2c2',16);
img= mk_image(imb,1); %show_fem(img,[0,0,1]);
vv = fwd_solve(img); vv = vv.meas;
p1 = [29,28]; p2 = [19,20];

sp = logspace(-2.5,2.5,40);
for i = length(sp):-1:1
   for j = length(sp):-1:1
      img.elem_data(p1) = sp(i);
      img.elem_data(p2) = sp(j);
      v2 = fwd_solve(img); 
      M(i,j) = norm(vv-v2.meas);
   end
end
img.elem_data(p1) = 10^(+.1);
img.elem_data(p2) = 10^(-.1);
subplot(121);show_fem(img,[0,0,1]);
clim = [min(M(:)),max(M(:))]*(eye(2)-[1,0;1,0]/20);
subplot(122);imagesc(log10(sp),log10(sp),M,clim); axis image

print_convert minima_region_01a.jpg


Figure: Left: forward model with two regions (blue and red). Right: objective function as a function of the conductivity in each region

Last Modified: $Date: 2017-02-28 13:12:08 -0500 (Tue, 28 Feb 2017) $ by $Author: aadler $