# Reduced basis method for 2nd order wave equation : application to one-dimensional seismic problem

Alternative Title:
Reduced basis method for second order wave equation : application to 1D seismic problem

Author:
Tan Yong Kwang, Alex

Abstract:
In this thesis, we solve the 2nd order wave equation, which is hyperbolic and linear in nature, to determine the pressure distribution for a one-dimensional seismic problem with smooth initial pressure and rate of pressure change with time. 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 on the dependence of the output, the average surface pressure as it varies with time, on the system parameters ,u, which consist of the earthquake source x8 and the occurring time T. The reduced basis method, the offline-online computational procedures and the associated 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 reasonably 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 are fairly "unique" and cannot be accurately represented as a linear combination of the existing basis vectors;(cont.) 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 Af = 200 is comparable with the dimension of the reduced basis space N = 175; however, when the two-dimensional model problem is run, the dimension of the finite element space is A = 3.98 x .04 while the dimension of the reduced basis space is N = 267 and the online stage is around 62.2 times faster then the finite element method. The proposition for the a posteriori error estimation developed shows that the maximum effectivity. the maximum ratio of the error bound over the norm of the reduced basis error, is of magnitude O(103) 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 hence 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 0(102).(cont.) Lastly, the maximum output effectivity is significantly larger than the maximum effectivity of the pressure distribution due to a conservative bound for the dual contribution. The offline-online computational procedures work well in determining the reduced basis pressure distribution. However, during the a posteriori error estimation, heavy canceling of the various offline stage matrices results in small values for the square of the dual norm of the residuals which decreases as the tolerance is lowered. When the tolerance is of magnitude 0(10-6), the square of the dual norm of the residuals is of magnitude 0(10-14) which is very close to machine precision. Hence, precision error sets in and the offline-online computational procedures break down. 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 note that at least 9 time steps should be selected for observation to ensure that the rising and dropping region of the output is detected. Lastly, the greater the measured field error, the larger the "possibility region" we obtain.

Description:
Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006.; MIT Institute Archives copy: pages 93 and 94 bound in reverse order.; Includes bibliographical references (p. 93-95).