 Landweber iteration

For a bounded linear operator $A$ between Hilbert spaces, Landweber iteration applies gradient descent to the squared residual:For nonzero $A$, the standard uniform step range is $0<\tau <2/\Vert A{\Vert }^2$. Starting at zero givesThe iteration introduces no component in $\ker A$. On data admitting a least-squares solution, it converges to the minimum-norm least-squares solution in this step range.

A singular value $\sigma$ has error multiplier $1-\tau {\sigma }^2$. Small singular values converge slowly, so a finite iteration admits well-determined components before aggressively fitting poorly determined ones. Continuing indefinitely on noisy data can amplify precisely those late components. Early stopping of Landweber iteration therefore acts as regularization; residual decrease alone is not a guarantee of reconstruction accuracy. The Landweber spectral filter makes this progression explicit.

Landweber iteration is an iterative method used to solve certain types of inverse problems, particularly those that can be framed as ill-posed problems. It is commonly applied in the context of linear inverse problems where one seeks to recover an unknown vector \( x \) from a given measurement \( b \) that is related to \( x \) through a linear operator \( A \): \[ Ax = b.