Low-rank approximate inverse for preconditioning tensor-structured linear systems

In this paper, we propose an algorithm for the construction of low-rank approximations of the inverse of an operator given in low-rank tensor format. The construction relies on an updated greedy algorithm for the minimization of a suitable distance to the inverse operator. It provides a sequence of approximations that are defined as the projections of the inverse operator in an increasing sequence of linear subspaces of operators. These subspaces are obtained by the tensorization of bases of operators that are constructed from successive rank-one corrections. In order to handle high-order tensors, approximate projections are computed in low-rank Hierarchical Tucker subsets of the successive subspaces of operators. Some desired properties such as symmetry or sparsity can be imposed on the approximate inverse operator during the correction step, where an optimal rank-one correction is searched as the tensor product of operators with the desired properties. Numerical examples illustrate the ability of this algorithm to provide efficient preconditioners for linear systems in tensor format that improve the convergence of iterative solvers and also the quality of the resulting low-rank approximations of the solution

Quantum annealing for systems of polynomial equations

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of $10^{-8}$.Comment: 11 pages, 4 figures. Added example for a system of quadratic equations. Supporting code is available at https://github.com/cchang5/quantum_poly_solver . This is a post-peer-review, pre-copyedit version of an article published in Scientific Reports. The final authenticated version is available online at: https://www.nature.com/articles/s41598-019-46729-

Spacetime emergence via holographic RG flow from incompressible Navier-Stokes at the horizon

We show that holographic RG flow can be defined precisely such that it corresponds to emergence of spacetime. We consider the case of pure Einstein's gravity with a negative cosmological constant in the dual hydrodynamic regime. The holographic RG flow is a system of first order differential equations for radial evolution of the energy-momentum tensor and the variables which parametrize it's phenomenological form on hypersurfaces in a foliation. The RG flow can be constructed without explicit knowledge of the bulk metric provided the hypersurface foliation is of a special kind. The bulk metric can be reconstructed once the RG flow equations are solved. We show that the full spacetime can be determined from the RG flow by requiring that the horizon fluid is a fixed point in a certain scaling limit leading to the non-relativistic incompressible Navier-Stokes dynamics. This restricts the near-horizon forms of all transport coefficients, which are thus determined independently of their asymptotic values and the RG flow can be solved uniquely. We are therefore able to recover the known boundary values of almost all transport coefficients at the first and second orders in the derivative expansion. We conjecture that the complete characterisation of the general holographic RG flow, including the choice of counterterms, might be determined from the hydrodynamic regime.Comment: 61 pages, 2 figures, 5 tables; matches with JHEP versio

Halo Shapes From Weak Lensing: The Impact of Galaxy--Halo Misalignment

We analyse the impact of galaxy--halo misalignment on the ability of weak lensing studies to constrain the shape of dark matter haloes, using a combination of the Millennium dark matter N-body simulation and different semi-analytic galaxy formation models, as well as simpler Monte Carlo tests. Since the distribution of galaxy--halo alignments is not known in detail, we test various alignment models, together with different methods of determining the halo shape. In addition to alignment, we examine the interplay of halo mass and shape, and galaxy colour and morphology with the resulting stacked projected halo shape. We find that only in the case where significant numbers of galaxy and halo minor axes are parallel does the stacked, projected halo axis ratio fall below 0.95. When using broader misalignment distributions, such as those found in recent simulations of galaxy formation, the halo ellipticity signal is washed out and would be extremely difficult to measure observationally. It is important to note that the spread in stacked halo axis ratio due to theoretical unknowns (differences between semi-analytic models, and between alignment models) are much bigger than any statistical uncertainty: It is naive to assume that, simply because LCDM predicts aspherical haloes, the stacked projected shape will be elliptical. In fact, there is no robust LCDM prediction yet for this procedure, and the interpretation of any such elliptical halo signal from lensing in terms of physical halo properties will be extremely difficult.Comment: 22 pages, 19 figures, accepted for publication in MNRAS. Minor changes for clarification and correcting typeo