M.Sc Thesis | |

M.Sc Student | Beliak Leonid |
---|---|

Subject | Adaptive Calculation of Variable Coefficients Elliptic Differential Equations via Wavelets |

Department | Department of Computer Science |

Supervisor | PROF. Moshe Israeli (Deceased) |

We propose a wavelet-based solver for strictly elliptic linear homogeneous PDE’s with non-constant coefficients of the form:

*ΔU -
b(x)U = f(x).*

We consider efficient discretization and solution of differential equations with non-oscillatory behavior that has possibly localized regions with irregular structures. Similar to other discretization schemes for the solution of PDEs, in wavelet schemes we have to solve a system of linear equations in the wavelet coordinate basis. Preconditioned Conjugate gradient method is one of the most efficient methods to solve such a system. Using this method only a constant number of steps is needed to obtain a solution with a prescribed accuracy.

We adaptively combine a
sparse multiplication algorithm with an existing diagonally preconditioned
conjugate gradient (CG) method based on wavelets. We use sparse data structures
to take advantage of the* O(Ns)* complexity of the algorithm, where Ns is
the number of significant coefficients (above a certain threshold) required for
a given accuracy. In this work we show that the usage of sparse multiplication
in wavelet space rather than in the original physical space can speed up by a
factor of 20 the performance of a regular solver. Therefore, we present an
algorithm and numerical results for adaptive multiplication scheme that can
solve fast mentioned above equation. We explored how the accuracy of
wavelet-based multiplication is affected by different input parameters of the
algorithm. We extended and integrated sparse multiplication into a PDE solver.
We also studied the relation between the performance of the solver and the
parameters of the wavelet based sparse multiplication.