Decomposition techniques for the ocean wave identification problem. (English) Zbl 0588.76025
This paper deals with the solution of the ocean wave identification problem by means of decomposition techniques, A discrete formulation is assumed. An ocean test structure is considered, and wave elevation and velocities are assumed to be measured with a number of sensors. Within the frame of linear wave theory, a Fourier series model is chosen for the wave elevation and velocities. Then, the following poblem is posed (Problem P): Find the amplitudes of the various wave components of specified frequency and direction, so that the assumed model of wave elevation and velocities provides the best fit to the measured data. Here, the term best fit is employed in the least-square sense over a given time interval.
Problem P is numerically difficult because of its large size 2MN, where M is the number of frequencies and N is the number of directions. Therefore, both the CPU time and the memory requirements are considerable [e.g. the first two authors, J. C. Heideman and J. N. Sharma, ibid. 44, 453-484 (1984; Zbl 0543.76023)].
In order to offset the above difficulties, decomposition techniques are employed in order to replace the solution of problem P with the sequential solution of two groups of smaller subproblems. The first group (Problems F) involves S subproblems, having size 2M, where S is the number of sensors and M is the number of frequencies; these S subproblems are least-square problems in the frequency domain. The second group (Problems D) involves M subproblems, having size 2N, where M is the number of frequencies and N is the number of directions; these M subproblems are least-square problems in the direction domain.
In the resulting algorithm, called the discrete formulation decomposition algorithm (DFDA) [see: the authors, Rice Univ., Aero-Astronautics Report No.185 (1986)], the linear equations are solved with the help of the Householder transformation in both the frequency domain and the direction domain. By contrast, in the continuous formulation decomposition algorithm (CFDA) [see: the authors, ibid. No.185 (1985)], the linear equations are solved with Gaussian elimination in the frequency domain and with the help of the Householder transformation in the direction domain.
Mathematically speaking, there are three cases in which the solution of the decomposed problem and the solution of the original, undecomposed problem are identical: (a) the case where the number of sensors equals the number of directions; (b) the case where problem P is characterized by a vanishing value of the functional being minimized; and (c) the case where the wave component periods are harmonically related to the sampling time.
Problem P is numerically difficult because of its large size 2MN, where M is the number of frequencies and N is the number of directions. Therefore, both the CPU time and the memory requirements are considerable [e.g. the first two authors, J. C. Heideman and J. N. Sharma, ibid. 44, 453-484 (1984; Zbl 0543.76023)].
In order to offset the above difficulties, decomposition techniques are employed in order to replace the solution of problem P with the sequential solution of two groups of smaller subproblems. The first group (Problems F) involves S subproblems, having size 2M, where S is the number of sensors and M is the number of frequencies; these S subproblems are least-square problems in the frequency domain. The second group (Problems D) involves M subproblems, having size 2N, where M is the number of frequencies and N is the number of directions; these M subproblems are least-square problems in the direction domain.
In the resulting algorithm, called the discrete formulation decomposition algorithm (DFDA) [see: the authors, Rice Univ., Aero-Astronautics Report No.185 (1986)], the linear equations are solved with the help of the Householder transformation in both the frequency domain and the direction domain. By contrast, in the continuous formulation decomposition algorithm (CFDA) [see: the authors, ibid. No.185 (1985)], the linear equations are solved with Gaussian elimination in the frequency domain and with the help of the Householder transformation in the direction domain.
Mathematically speaking, there are three cases in which the solution of the decomposed problem and the solution of the original, undecomposed problem are identical: (a) the case where the number of sensors equals the number of directions; (b) the case where problem P is characterized by a vanishing value of the functional being minimized; and (c) the case where the wave component periods are harmonically related to the sampling time.
MSC:
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
Keywords:
ocean wave identification problem; decomposition techniques; discrete formulation; ocean test structure; linear wave theory; Fourier series model; model of wave elevationCitations:
Zbl 0543.76023Software:
LINPACKReferences:
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