This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large. As a consequence a few strategies are developed whose goal is to achieve this feature. A number of "templates" for enabling implementations of these factorizations are presented. Numerical experiments show that the resulting ILUs offer a good compromise between robustness and efficiency.

The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real space by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are successfully suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H-transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The new approach works not only with analytical bands but also with full-band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN BJT. For the first time the convergence of the spherical harmonics expansion is shown for a device and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved memory efficiently by the numerically robust package ILUPACK.

An algebraic multilevel (ML) preconditioner is presented for the Helmholtz equation in heterogeneous media. It is based on a multilevel incomplete LDL T factorization and preserves the inherent (complex) symmetry of the Helmholtz equation. The ML preconditioner incorporates two key components for efficiency and numerical stability: symmetric maximum weight matchings and an inverse-based pivoting strategy. The former increases the block-diagonal dominance of the system, whereas the latter controls L −1 for numerical stability. When applied recursively, their combined effect yields an algebraic coarsening strategy, similar to algebraic multigrid methods, even for highly indefinite matrices. The ML preconditioner is combined with a Krylov subspace method and applied as a "black-box" solver to a series of challenging two-and three-dimensional test problems, mainly from geophysical seismic imaging. The numerical results demonstrate the robustness and efficiency of the ML preconditioner, even at higher frequency regimes.

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