Computational Mathematics and Scientific Computing Seminar

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In this seminar the problem of numerical approximation of evolution PDEs is considered.
A peculiarity of the considered equations is their dependence on a few parameters, which
is associated with the need to calculate multiple solutions as the parameters vary, possibly
real-time.
 
The proposed method approximates a complex contour integral in order to invert numerically
the Laplace transform of the solution.
The considered contours are constructed in such a way as to approximate certain curves,
so-called pseudospectral, which characterize the space operator of the equation.
The proposed approach is particularly well-suited to parabolic equations, where the
contour can be suitably bounded.

For the purpose of real-time computations for several instances of the parameters, various
methodologies based on reduced bases or on model reduction methods have been proposed in the literature, which allow to solve small dimensional problems, maintaining a control of the
accuracy.

The use of reduction methods based on the Laplace transform on pseudospectral contours
is new and seems to have several advantages with respect to those previously considered
in the literature.

In fact, differently from time stepping integrators (like Runge-Kutta methods) the use of
the Laplace transform determines a significant reduction of the size of the reduced space.

When the operator is hyperbolic the application of the method becomes critical, as the
inversion of the Laplace transform on the continuous problem cannot be limited to a
bounded contour. However, many numerical approaches introduce an artificial diffusion, making the proposed method feasible.
On the other hand, when the operator is hyperbolic, reduced basis methods and model order reduction based on classical time stepping schemes fail to provide the desired performances due to a slow decay of the Kolmogorov n-width.

The communication is inspired by collaborations with M. Lopez Fernandez (Malaga) and
M. Manucci (GSSI and Stuttgart).