i am dealing with gradieent descent based minimization problem , my matrix is heavily ill conditioned , leading to false recovery in forward inverse problem

clc;
clear all;
close all;
%% greens function between source to target
soux=[3,3,4,4,5,5];               % source x coordinate
souy=[3,4,3,4,3,4];               % source y coordinate
w2=zeros(6,6);
for k=1:6
    hg=1;
    for i=3:5                  %target x coordinate
        for j=3:4                  %target y coordinate
            p=soux(k);
            q=souy(k);
            temp1=(i-p)*(i-p)+(j-q)*(j-q)+(0.2-0)^2 ;
            dt1=sqrt(temp1);
            %kg=9.9260
            kg=0.3897;
            grf2(hg,:)=(exp(-(kg*dt1)))/(2.0673*dt1);
            hg=hg+1;
        end
    end
    
    w2(:,k)=grf2;
end
G=w2

%%  greens function between target to detector

a=3
b=4
c=5
dtx=[a,a,b,b,c,c];
dty=[a,b,a,b,a,b];

% dtx=[3,3,4,4,5,5];    % detector x coordinate
% dty=[3,4,3,4,3,4];    % detector y coordinate

w1=zeros(6,6);
% greens function between targets to detectors
t=1;
for i=3:5 %target x coordinate
    for j=3:4 %target y coordinate
        for k=1:6
            g=dtx(k);
            m=dty(k);
            temp=(i-g)*(i-g)+(j-m)*(j-m)+(0.4-0.2)^2;% euclidean distance
            dt=sqrt(temp);
            kk=0.3897;% this kk is in centimeters not in millimeters and 20.422 is in centimeters
            grf1(k,:)=(exp(-(kk*dt)))/(2.0673*dt);% greens function computation from target to detectors
        end
        w1(:,t)=grf1;% output of the sensing matrix
        %      w2=normc(w1);
        t=t+1;
        %    
    end
    %    
end
H=w1;
%%
I = eye(6,6);

f   =[1     1     0     0     0     0;
    1     1     0     0     0     0;
    0     0     1     1     0     0;
    0     0     1     1     0     0;
    0     0     0     0     0     0;
    0     0     0     0     0     0]
%%
A = I - G*diag(f);


% here we are trying to minimize u and tring to find values for u
% A = I - G*diag(f)
%u_in = ones(6,6);

u_ini = zeros(6,6)

u_in=ones(6,6)

% u_in=[1.995 0.997 0.955 0.996 0.988 0.992;
%     0.978 1.985 0.956 0.968 0.9823 0.934;
%     0.987 0.999 1.991 0.991 0.956 0.925;
%     0.965 0.989 0.963 1.989 0.993 0.973;
%     0.968 0.934 0.982 0.996 1.945 0.924;
%     0.988 0.989 0.954 0.9786 0.969 1.991]

% imagesc(u_in)
% colormap(gray)

%% gradient and objective function

funval =@(u)  0.5 * ((norm(A*u - u_in))^2);               % computing value of objective function
gradu = @(u) [A'*(A*u - u_in)]                            % computing the gradient for the trikhonov case

%u_ini = ones(6,6);
u_size =size(u_ini);
% A_size=size(A);

iterations =200;

stepsize = 0.001;


u_next = u_ini;
u_rec = [u_ini]   ;                         % to record the u iterative value

ff = zeros(iterations,1)
for  i = 1 : iterations
    u_next = u_next - stepsize * gradu(u_next);
    recordedguess_u = [u_rec , u_next ];
    ff(i,1) =funval(u_next);
    ii(i,1)=i;
end

% rec = reshape(u_next, u_size);
u_tilda=u_next;
funval(u_tilda)

figure()
plot(ii,ff)

%display(u_tilda);
%C = u -u_next

%% calculating filed at detector
Y = (H*diag(u_tilda)).*f;
%Y = abs(Y)
figure()
imagesc(Y)
colormap(gray)



pp=H*diag(u_tilda)
pp_inv = pinv(pp)
fff = pp_inv.* Y

figure()
imagesc(f)
colormap(gray)

figure()
imagesc(fff)
colormap(gray)

[Matt J]

It's not clear what your question is. If your matrix is ill-conditioned, bad results from steepest descent are to be expected.

[asim asrar]

thank you matt for the response , can you suggest how such problems can be irradicated.

Answer 1

[Matt J]

Try Newton's method instead of steepest descent. If you have the Optimization Toolbox, fminunc() basically has Newton-like method's already impleneted for you.