An efficient null space inexact Newton method for hydraulic simulation of water distribution networks

Null space Newton algorithms are efficient in solving the nonlinear equations arising in hydraulic analysis of water distribution networks. In this article, we propose and evaluate an inexact Newton method that relies on partial updates of the network pipes' frictional headloss computations to solve the linear systems more efficiently and with numerical reliability. The update set parameters are studied to propose appropriate values. Different null space basis generation schemes are analysed to choose methods for sparse and well-conditioned null space bases resulting in a smaller update set. The Newton steps are computed in the null space by solving sparse, symmetric positive definite systems with sparse Cholesky factorizations. By using the constant structure of the null space system matrices, a single symbolic factorization in the Cholesky decomposition is used multiple times, reducing the computational cost of linear solves. The algorithms and analyses are validated using medium to large-scale water network models.Comment: 15 pages, 9 figures, Preprint extension of Abraham and Stoianov, 2015 (https://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0001089), September 2015. Includes extended exposition, additional case studies and new simulations and analysi

Stable low-rank matrix recovery via null space properties

The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically. We derive sufficient conditions on the minimal amount of measurements ensuring recovery via convex optimization. We establish our results via certain properties of the null space of the measurement map. In the setting where the measurements are realized as Frobenius inner products with independent standard Gaussian random matrices we show that $10 r (n_1 + n_2)$ measurements are enough to uniformly and stably recover an $n_1 \times n_2$ matrix of rank at most $r$. We then significantly generalize this result by only requiring independent mean-zero, variance one entries with four finite moments at the cost of replacing $10$ by some universal constant. We also study the case of recovering Hermitian rank-$r$ matrices from measurement matrices proportional to rank-one projectors. For $m \geq C r n$ rank-one projective measurements onto independent standard Gaussian vectors, we show that nuclear norm minimization uniformly and stably reconstructs Hermitian rank-$r$ matrices with high probability. Next, we partially de-randomize this by establishing an analogous statement for projectors onto independent elements of a complex projective 4-designs at the cost of a slightly higher sampling rate $m \geq C rn \log n$. Moreover, if the Hermitian matrix to be recovered is known to be positive semidefinite, then we show that the nuclear norm minimization approach may be replaced by minimizing the $\ell_2$-norm of the residual subject to the positive semidefinite constraint. Then no estimate of the noise level is required a priori. We discuss applications in quantum physics and the phase retrieval problem.Comment: 26 page