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A Multilevel Method for Ill-Posed Problems

A Multilevel Method for Ill-Posed Problems. Malena I. Español Tufts University. Outline. Problem Regularization Multilevel method Implementation Future work. Discrete Ill-Posed problem. Need for regularization. Regularization methods. LSQR: Regularization method. relative error.

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A Multilevel Method for Ill-Posed Problems

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  1. A Multilevel Method for Ill-Posed Problems Malena I. Español Tufts University January 2006 - The Mathworks

  2. Outline • Problem • Regularization • Multilevel method • Implementation • Future work January 2006 - The Mathworks

  3. Discrete Ill-Posed problem January 2006 - The Mathworks

  4. Need for regularization January 2006 - The Mathworks

  5. Regularization methods January 2006 - The Mathworks

  6. LSQR: Regularization method relative error iterations January 2006 - The Mathworks

  7. Multilevel: Introduction January 2006 - The Mathworks

  8. Downsampling (“going down”) January 2006 - The Mathworks

  9. Interpolation (“going up”) January 2006 - The Mathworks

  10. Basic multilevel algorithm downsampling A and b interpolating x Solve small system January 2006 - The Mathworks

  11. Implementation function x=nm(A,b,iter) n=size(A,1); while length(b)>31 b=fwei(b); end x=lsqr(@pap,b,1e-8,iter,[],[],[],A); while length(x)<n x=interpol(x); end downsampling b interpolating Solve small system January 2006 - The Mathworks

  12. Implementation function z=fwei(y) n=length(y); m=floor(n/2); z=zeros(m,1); z(1:m)=.5*y(2:2:n-1) + .25*(y(1:2:n-2)+y(3:2:n)); downsampling b January 2006 - The Mathworks

  13. Implementation function y = pap(x,A,transp_flag) n=size(A,1); m=length(x); if (nargin > 2) & strcmp(transp_flag,'transp') while length(x)>n x=interpol(x); end y=A'*x; while length(y)>m y=fwei(y); end else while length(x)>n x=interpol(x); end y=A*x; while length(y)<m y=fwei(y); end end x=lsqr(@pap,b,1e-8,iter,[],[],[],A); downsampling b Solve small system using LSQR January 2006 - The Mathworks

  14. Implementation function z=interpol(x) n=length(x); m=2*n+1; z=zeros(m,1); z(1)=x(1)/2; z(m)=x(n)/2; z(2:2:m-1)=x(1:n); z(3:2:m-2)=.5*(x(1:n-1) +x(2:n)); downsampling b interpolating Solve small system using LSQR January 2006 - The Mathworks

  15. Results January 2006 - The Mathworks

  16. Future Work • Fine tune parameters • Find where it can be used as a preconditioner • Extend to 2D, 3D January 2006 - The Mathworks

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