Steven Bürger: About an autoconvolution problem arising in ultrashort laser pulse characterization

Author(s):

Steven Bürger

Title:

Steven Bürger: About an autoconvolution problem arising in ultrashort laser pulse characterization

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 16, 2014

Mathematics Subject Classification:

    47J06 []
    78A60 []
    65J20 []

Abstract:

We are investigating a kernel-based autoconvolution problem, which has its origin in the physics of ultra short laser pulses. The task in this problem is to reconstruct a complex-valued function $x$ on a finite interval from measurements of its absolute value and a kernel-based autoconvolution of the form \[[F(x)](s)=\int k(s,t)x(s-t)x(t)\de t.\] This problem has not been studied in the literature. One reason might be that one has more information than in the classical autoconvolution case, where only the right hand side is available. Nevertheless we show that ill posedness phenomena may occur. We also propose an algorithm to solve the problem numerically and demonstrate its performance with artificial data. Since the algorithm fails to produce good results with real data and we suspect that the data for $|F(x)|$ are not dependable we also consider the whole problem with only $\arg(F(x))$ given instead of $F(x)$.

Keywords:

Autoconvolution equation, inverse problems, local well-posedness and ill-posedness

Language:

English

Publication time:

10/2014