Levenberg-Marquardt for nonlinear elliptical equation

1. 给出初值 \(x_i \in \mathbb{R}^n; k=1,\eta \in(0,1)\)
2. 若 \(\|J_k^{T} F_k\|=0\), 停;
   求解 \(\displaystyle (J_k^TJ_k+\lambda_k I)d=-J_k^T F_k,\lambda_k=\|F_k\|\), 得到搜索方向 \(d_k\)；
3. 若\(d_k\) 满足 \(\displaystyle \|F(x_k+d_k) \| \leq \eta \|F_k\|\), 则 \(x_{k+1}=x_k+d_k\);
   否则： \(x_{k+1}=x_k+\alpha_k d_k\), 这里的 $\alpha_k $ 由Armijio 线搜索得到；
   注：这里的Armijio 线搜索如下：
   \(\alpha_k=\xi^t,\xi \in(0,1)\), \(t\in \mathbb{Z}^+\) 是满足如下不等式的最小非负整数：
   \[ {\| F(x_k+\xi^t d_k) \|}^2 \leqslant {\|F(x_k) \|}^2+\beta_1 \xi^t F^T_k J_k d_k \]
4. \(k=k+1\), 转2.
   

function T69
%  全局二次收敛的 Levenberg-Marquardt 
% yuewen_chen@qq.com

n=300;
r=linspace(-pi*5,pi*5,n)';
h=r(2)-r(1);
rb=[r(1)-h;r;r(end)+h];
f=-sin(r)+sin(r).^2;
x0=sin(rb);
u=cos(r).*sin(r);  % initial guess




% ************ D2********************
cn1=-1*ones(n-1,1);
cp1=1*ones(n-1,1);
c=-2*ones(n,1);
D2=sparse((diag(c,0)+diag(cp1,1)+diag(-cn1,-1))/(h^2));
%******************************************
M=D2;
eta=0.9;


for k=1:200
    lam=norm(F(u));
    L=g(u)'*g(u)+lam*speye(n,n);
    R=-g(u)*F(u);
    d=L\R;
    if norm(F(u+d))<=norm(eta*F(u))
        u=u+d;
    else
        alp=Armijo(u,d);
        u=u+alp*d;
    end
    plot(r,u,'b.',r,sin(r),'r')
    title(['res=',num2str(nF(u)), ',  k=',num2str(k),' , \alpha=',num2str(norm(d))])
    drawnow
end


function y=F(u)
        y=M*u+u.^2-f;
        y(1)=y(1)+x0(1)/h^2;
        y(end)=y(end)+x0(end)/h^2;
end

   function y=g(u)
         J=M+diag(2*u,0);
         y=J;
   end

    function y=nF(u)
         y=norm(F(u));
    end
    function alp=Armijo(u,d)
        et=1/5;
        bet1=0.5;
        t=0;
         while t<=500
             
             if (nF(u+et^t*d)^2<=nF(u)^2+bet1*et^t*F(u)'*g(u)*d)
                  alp=et^t;
                  break
             else
                 t=t+1;
             end
             alp=et^t;
         end
    end
end

Levenberg-Marquardt for nonlinear elliptical equation

原文：https://www.cnblogs.com/yuewen-chen/p/11524458.html