Iterative Optimization in Inverse Problems

Eigenvalue Bounds 143

9.6.5 The Simultaneous DROP Algorithm

Corollary 9.14 Let 0 <w

i

≤ 1,

α

i

= w

i

ν

−2

i

= w

i



J



j=1

|A

ij

|

2



−1

and β

j

= s

−1

j

,foreachi and j. Then the simultaneous algorithm with the

iterative step

x

k

j

= x

k−1

j

+ λ

k

I



i=1



w

i

A

ij

(b

i

− (Ax

k−1

)

i

)

s

j

ν

2

i



, (9.43)

converges to the minimizer of the function

I



i=1











w

i

(b

i

− (Ax)

i

)

ν

i











2

for which the function

J



j=1

s

j

|x

j

− x

0

j

|

2

is minimized.

For w

i

= 1, this is the CARP1 algorithm of [135] (see also [88, 89]). The

simultaneous DROP algorithm of [84] requires only that the weights w

i

be

positive, but dividing each w

i

by their maximum, max

i

{w

i

}, while multiply-

ing each λ

k

by the same maximum, gives weights in the in