We solve the 2nd order wave equation, which is hyperbolic and linear in nature, to determine the pressure distribution for one-dimensional seismic problem with smooth initial pressure and rate of pressure change. With Dirichlet and Neumann boundary conditions, the pressure distribution is solved for a total of 500 time steps, which is slighter more than a periodic cycle. Our focus is the dependence of the output, the average surface pressure, on the system parameters ?, which consist of the earthquake source xs and the occurring time T.
The reduced basis method, offline-online computational procedures and a posteriori
error estimation are developed. We have shown that the reduced basis pressure
distribution is an accurate approximation to the finite element pressure distribution.
The greedy algorithm, the procedure of selecting the basis vectors which span
the reduced basis space, works well although a period of slow convergence is
experienced: this is because the finite element pressure distribution along
the edges of the earthquake source-time space cannot be accurately represented
as a linear combination of the existing basis vectors; hence, the greedy algorithm
has to bring these ?unique? finite element pressure distribution into the reduced
basis space individually,
accounting for the slow convergence rate. Lastly, applying the online stage
instead of the finite element method does not result in a reduction of computational
cost: the dimension of the finite element space = 200 is comparable with the
dimension of the reduced basis space N = 175; the efficiency of the offline-online
computational procedure should be more evident when applied to the two-dimensional
seismic problem where the dimension of the finite element space is the square
of its present value and the dimension of the reduced basis space N is expected
to increase but still has the same order of magnitude.
The a posteriori error estimation shows that the maximum effectivity, the maximum ratio of the error bound over the norm of the reduced basis error, is of magnitude O(10^3) and increases rapidly when the tolerance is lower. However, this high value is due to the norm of the reduced basis error having a low value and not a cause for concern. Furthermore, the ratio of the maximum error bound over the maximum norm of the reduced basis error has a constant magnitude of only O(10^2).
Finally, the inverse problem works reasonably well, giving a ?possibility region?
of the set of system parameters where the actual system parameters may reside.
We observed that the time steps selected for comparison should correspond to
the distinct characteristic of the reduced basis output in order to have a small
?possibility region?.