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

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

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

4. January 2006 - The Mathworks Need for regularization

5. January 2006 - The Mathworks Regularization methods

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

7. January 2006 - The Mathworks Multilevel: Introduction

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

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

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

11. January 2006 - The Mathworks Implementation Solve small system downsampling b interpolating 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

12. January 2006 - The Mathworks Implementation downsampling b 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));

13. January 2006 - The Mathworks Implementation downsampling b Solve small system using LSQR x=lsqr(@pap,b,1e-8,iter,[],[],[],A); 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

14. January 2006 - The Mathworks Implementation downsampling b Solve small system using LSQR interpolating 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));

15. January 2006 - The Mathworks Results

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