## Approximation of the scattering amplitude and linear systems

Gene H. Golub, Martin Stoll, and Andy Wathen

### Abstract

The simultaneous solution of $Ax=b$ and $A^{T}y=g$, where $A$ is a non-singular matrix, is required in a number of situations. Darmofal and Lu have proposed a method based on the Quasi-Minimal Residual algorithm (QMR). We will introduce a technique for the same purpose based on the LSQR method and show how its performance can be improved when using the generalized LSQR method. We further show how preconditioners can be introduced to enhance the speed of convergence and discuss different preconditioners that can be used. The scattering amplitude $g^{T}x$, a widely used quantity in signal processing for example, has a close connection to the above problem since $x$ represents the solution of the forward problem and $g$ is the right-hand side of the adjoint system. We show how this quantity can be efficiently approximated using Gauss quadrature and introduce a block-Lanczos process that approximates the scattering amplitude, and which can also be used with preconditioning.

Full Text (PDF) [337 KB]

### AMS subject classifications

65F10, 65N22, 65F50, 76D07.

### Links to the cited ETNA articles

 [43] Vol. 13 (2002), pp. 56-80 Zdeněk Strakoš and Petr Tichý: On error estimation in the conjugate gradient method and why it works in finite precision computations

 Vol. 43 (2014-2015), pp. 70-89 Paraskevi Fika, Marilena Mitrouli, and Paraskevi Roupa: Estimates for the bilinear form $x^T A^{-1} y$ with applications to linear algebra problems

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