Fast Projected Newton-like Method for Precision Matrix Estimation with Nonnegative Partial Correlations

We study the problem of estimating precision matrices in multivariate Gaussian distributions where all partial correlations are nonnegative, also known as multivariate totally positive of order two ($\mathrm{MTP}_2$). Such models have received significant attention in recent years, primarily due to interesting properties, e.g., the maximum likelihood estimator exists with as few as two observations regardless of the underlying dimension. We formulate this problem as a weighted $\ell_1$-norm regularized Gaussian maximum likelihood estimation under $\mathrm{MTP}_2$ constraints. On this direction, we propose a novel projected Newton-like algorithm that incorporates a well-designed approximate Newton direction, which results in our algorithm having the same orders of computation and memory costs as those of first-order methods. We prove that the proposed projected Newton-like algorithm converges to the minimizer of the problem. We further show, both theoretically and experimentally, that the minimizer of our formulation using the weighted $\ell_1$-norm is able to recover the support of the underlying precision matrix correctly without requiring the incoherence condition present in $\ell_1$-norm based methods. Experiments involving synthetic and real-world data demonstrate that our proposed algorithm is significantly more efficient, from a computational time perspective, than the state-of-the-art methods. Finally, we apply our method in financial time-series data, which are well-known for displaying positive dependencies, where we observe a significant performance in terms of modularity value on the learned financial networks.Comment: 43 pages; notation updated for section

Development of an implicit framework for the two-fluid model on unstructured grids

The two-fluid model is an efficient method for simulating multiphase flows, based on an averaged description of the phases as interpenetrating and interacting continua. It is particularly attractive for the simulation of dispersed gas-solid flows in which the large number of particles in practical devices can impose an insurmountable computational burden for particle tracking methods, given currently available computing resources. Whilst the two-fluid model is more efficient than particle tracking methods, it results in large, strongly coupled and highly non-linear systems of equations, placing a premium on efficient solution algorithms. Additionally, the constitutive models used to describe the solid phase introduce stiff source terms, requiring a robust solution algorithm to handle them. In this thesis a fully-coupled algorithm is developed for the two-fluid model, based on a Newton linearisation of the underlying equation system, resulting in an algorithm treating all inter-equation couplings implicitly. For comparison, a semi-coupled algorithm, based on a Picard linearisation of the two-fluid model is also implemented, yielding a smaller implicitly coupled pressure-velocity system and a segregated system for the transport of phase concentrations. Motivating this work is the highly non-linear nature of the two-fluid model and the stiff source terms arising in the models of the dispersed phase, these are treated explicitly in the semi-coupled algorithm and may impose stability limits on the algorithm. By treating these terms implicitly, it is expected that the fully-coupled solution algorithm will be more robust. The algorithms are compared by application to test cases ranging from academic problems to problems representative of industrial applications of the two-fluid model. These comparisons show that with increasing problem complexity, the robustness of the fully-coupled algorithm leads to an overall more efficient solution than the semi-coupled algorithm.Open Acces

A butterfly‐based direct solver using hierarchical LU factorization for Poggio‐Miller‐Chang‐Harrington‐Wu‐Tsai equations

A butterfly‐based hierarchical LU factorization scheme for solving the PMCHWT equations for analyzing scattering from homogenous dielectric objects is presented. The proposed solver judiciously re‐orders the discretized integral operator and butterfly‐compresses blocks in the operator and its LU factors. The observed memory and CPU complexities scale as O(N log2 N) and O(N1.5 log N), respectively. The proposed solver is applied to the analyses of scattering several large‐scale dielectric objects.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143676/1/mop31166.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/143676/2/mop31166_am.pd

Algebraic and parametric solvers for the power flow problem: towards real-time and accuracy-guaranteed simulation of electric systems

The final publication is available at Springer via http://dx.doi.org/10.1007/s11831-017-9223-6The power flow model performs the analysis of electric distribution and transmission systems. With this statement at hand, in this work we present a summary of those solvers for the power flow equations, in both algebraic and parametric version. The application of the Alternating Search Direction method to the power flow problem is also detailed. This results in a family of iterative solvers that combined with Proper Generalized Decomposition technique allows to solve the parametric version of the equations. Once the solution is computed using this strategy, analyzing the network state or solving optimization problems, with inclusion of generation in real-time, becomes a straightforward procedure since the parametric solution is available. Complementing this approach, an error strategy is implemented at each step of the iterative solver. Thus, error indicators are used as an stopping criteria controlling the accuracy of the approximation during the construction process. The application of these methods to the model IEEE 57-bus network is taken as a numerical illustration.Peer ReviewedPostprint (author's final draft