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

January The Mathworks Outline Problem Regularization Multilevel method Implementation Future work

January The Mathworks Discrete Ill-Posed problem

January The Mathworks Need for regularization

January The Mathworks Regularization methods

January The Mathworks LSQR: Regularization method iterations relative error

January The Mathworks Multilevel: Introduction

January The Mathworks Downsampling (“going down”)

January The Mathworks Interpolation (“going up”)

January The Mathworks Basic multilevel algorithm Solve small system downsampling A and b interpolating x

January 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 while length(x)<n x=interpol(x); end

January 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));

January The Mathworks Implementation downsampling b Solve small system using LSQR 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

January 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));

January The Mathworks Results

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