I propose two Galerkin methods based on the transfer-of-approximation property for static and dynamic acoustic boundary value problems in seismic applications. For problems with heterogeneous coefficients, the polynomial finite element spaces are no longer optimal unless special meshing techniques are employed. The transfer-of-approximation property provides a general framework to construct the optimal approximation subspace on regular grids. The transfer-of-approximation finite element method is theoretically attractive for that it works for both scalar and vectorial elliptic problems. However the numerical cost is prohibitive. To compute each transfer-of-approximation finite element basis, a problem as hard as the original one has to be solved. Furthermore due to the difficulty of basis localization, the resulting stiffness and mass matrices are dense. The 2D harmonic coordinate finite element method (HCFEM) achieves optimal second-order convergence for static and dynamic acoustic boundary value problems with variable coefficients at the cost of solving two auxiliary elliptic boundary value problems. Unlike the conventional FEM, no special domain partitions, adapted to discontinuity surfaces in coe cients, are required in HCFEM to obtain the optimal convergence rate. The resulting sti ness and mass matrices are constructed in a systematic procedure, and have the same sparsity pattern as those in the standard finite element method. Mass-lumping in HCFEM maintains the optimal order of convergence, due to the smoothness property of acoustic solutions in harmonic coordinates, and overcomes the numerical obstacle of inverting the mass matrix every time update, results in an efficient, explicit time step.

The least mean square (LMS) algorithm is widely expected to operate near the corresponding Wiener filter solution. An exception to this popular perception occurs when the algorithm is used to adapt a transversal equalizer in the presence of additive narrowband interference. The steady-state LMS equalizer behavior does not correspond to that of the fixed Wiener equalizer: the mean of its weights is different from the Wiener weights, and its mean squared error (MSE) performance may be significantly better than the Wiener performance. The contributions of this study serve to better understand this so-called non-Wiener phenomenon of the LMS and normalized LMS adaptive transversal equalizers. The first contribution is the analysis of the mean of the LMS weights in steady state, assuming a large interference-to-signal ratio (ISR). The analysis is based on the Butterweck expansion of the weight update equation. The equalization problem is transformed to an equivalent interference estimation problem to make the analysis of the Butterweck expansion tractable. The analytical results are valid for all step-sizes. Simulation results are included to support the analytical results and show that the analytical results predict the simulation results very well, over a wide range of ISR. The second contribution is the new MSE estimator based on the expression for the mean of the LMS equalizer weight vector. The new estimator shows vast improvement over the Reuter-Zeidler MSE estimator. For the development of the new MSE estimator, the transfer function approximation of the LMS algorithm is generalized for the steady-state analysis of the LMS algorithm. This generalization also revealed the cause of the breakdown of the MSE estimators when the interference is not strong, as the assumption that the variation of the weight vector around its mean is small relative to the mean of the weight vector itself. Both the expression for the mean of the weight vector and for the MSE estimator are analyzed for the LMS algorithm at first. The results are then extended to the normalized LMS algorithm by the simple means of adaptation step-size redefinition. / Ph. D.