[Dottorcomp] Seminari di Matematica Applicata. Martedì 17 ottobre. Dave Hewett.

[Stefano Lisini]

Seminari di Matematica Applicata, Dipartimento di Matematica "F. Casorati"
e Istituto del CNR IMATI "E. Magenes" di Pavia.

Martedì 17 ottobre 2023, alle ore 15.00 precise, presso l'aula Beltrami del
Dipartimento di Matematica,

Dave Hewett (University College London)

terrà un seminario dal titolo:

Numerical evaluation of oscillatory integrals via automated steepest
descent contour deformation.

Il seminario verrà anche trasmesso in diretta su zoom.

Link Zoom:
https://us02web.zoom.us/j/87518137128?pwd=eHg2eG9QZUdydFJEU1NESWN5a1lPQT09

Abstract. Oscillatory integrals arise in many areas of mathematics, science
and technology, including Fourier analysis, special function theory, signal
processing, quantum mechanics, and acoustic and electromagnetic wave
propagation. By an oscillatory integral we mean an integral over some
subset of R^n in which the integrand undergoes a large number of
oscillations. Typically, the integrand takes the form of a non-oscillatory
prefactor multiplied by an oscillating exponential of the form exp(ikg(x)),
for some “phase function” g(x) governing the oscillations. Such integrals
can be prohibitively expensive to evaluate numerically using a conventional
interpolatory quadrature method (e.g. Gauss quadrature, trapezoidal rule
etc), because one needs a very large number of quadrature points to
properly capture the oscillations in the integrand. Thankfully, a number of
alternative quadrature approaches have been developed to bypass this
problem. Our attention in this talk will be on one-dimensional oscillatory
integrals, and the numerical steepest descent (NSD) method. The basic idea
of NSD is to deform the integration contour into the complex plane onto a
contour on which the integrand is non-oscillatory and exponentially
decaying, so that conventional quadrature rules can be applied. However,
unless the phase function governing the oscillation is particularly simple,
the application of NSD requires a significant amount of a priori analysis
and expert user input, to determine the appropriate contour deformation,
and to deal with the non-uniformity in the accuracy of standard quadrature
techniques associated with the coalescence of stationary points (zeros of
the derivative of the phase function) with each other, or with the
endpoints of the original integration contour. In this talk we present a
novel algorithm for the numerical evaluation of oscillatory integrals with
general polynomial phase functions, which automates the contour deformation
process and avoids the difficulties typically encountered with coalescing
stationary points and endpoints. The inputs to the algorithm are simply the
phase and amplitude functions, the endpoints and orientation of the
original integration contour, and a small number of numerical parameters.
As an application, we use our algorithm to evaluate cuspoid canonical
integrals from scattering theory. A Matlab implementation of the algorithm
is available online and is called PathFinder.

This is joint work with Andrew Gibbs (UCL) and Daan Huybrechs (KU Leuven).
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Pagina web dei Seminari di Matematica Applicata
https://matematica.unipv.it/ricerca/cicli-di-seminari/seminari-di-matematica-applicata/
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