Although some preconditioners are available for solving dense linear systems, there are still many matrices for which preconditioners are lacking, in particular in cases where the size of the matrix N becomes very large. Examples of preconditioners include ILU preconditioners that sparsify the matrix based on some threshold, algebraic multigrid, and specialized preconditioners, e.g., Calderón and other analytical approximation methods when available. Despite these methods, there remains a great need to develop general purpose preconditioners whose cost scales well with the matrix size N . In this paper, we propose a preconditioner, with broad applicability and with cost O(N ), for dense matrices, when the matrix is given by a smooth kernel. Extending the method using the same framework to general H 2 -matrices (i.e., algebraic instead of defined in terms of an analytical kernel) is relatively straightforward, but this won't be discussed here. These preconditioners have a controlled accuracy (e.g., machine accuracy can be achieved if needed) and scale linearly with N . They are based on an approximate direct solve of the system. The linear scaling of the algorithm is achieved by means of two key ideas. First, the H 2 -structure of the dense matrix is exploited to obtain an extended sparse system of equations. Second, fill-ins arising when performing the elimination of the latter are compressed as low-rank matrices if they correspond to well-separated interactions. This ensures that the sparsity pattern of the extended sparse matrix is preserved throughout the elimination, hence resulting in a very efficient algorithm with O(N log(1/ε) 2 ) computational cost and O(N log 1/ε) memory requirement, for an error tolerance 0 < ε < 1. The solver is inexact, although the error can be controlled and made as small as needed. These solvers are related to ILU in the sense that the fill-in is controlled. However, in ILU, most of the fill-in (i.e., below a certain tolerance) is simply discarded, whereas here it is approximated using low-rank blocks, with a prescribed tolerance. Numerical examples are discussed to demonstrate the linear scaling of the method and to illustrate its effectiveness as a preconditioner. ). 1 2. The inverse fast multipole method: key ideas and algorithm.2.1. Extended sparsification. It is the aim to obtain a fast solver for the dense linear system Ax = b, where A is an H 2 -matrix, i.e., a matrix constructed based on strong admissibility criteria (well-separated clusters are assumed to be low-rank only), and with a nested basis for the low-rank representations.The first step in obtaining such a solver is to exploit the H 2 -structure of the matrix in order to convert the original dense system into an extended sparse sys-2 The term fill-in is used here to denote the non-zero matrix entries. 3 The term elimination refers to removing variables from a system of equations and updating the coefficients of the remaining variables appropriately.
Abstract. Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability, but need to be used with an appropriate preconditioner (e.g., ILU, AMG, Gauss-Seidel, etc.) for proper convergence. The choice of an effective preconditioner is highly problem dependent. We propose a novel fully algebraic sparse matrix solve algorithm, which has linear complexity with the problem size. Our scheme is based on the Gauss elimination. For a given matrix, we approximate the LU factorization with a tunable accuracy determined a priori. This method can be used as a stand-alone direct solver with linear complexity and tunable accuracy, or it can be used as a black-box preconditioner in conjunction with iterative methods such as GMRES. The proposed solver is based on the low-rank approximation of fill-ins generated during the elimination. Similar to H-matrices, fill-ins corresponding to blocks that are well-separated in the adjacency graph are represented via a hierarchical structure. The linear complexity of the algorithm is guaranteed if the blocks corresponding to well-separated clusters of variables are numerically low-rank.
The working principle of particle-based solar receivers is to utilize the absorptivity of a dispersed particle phase in an otherwise optically transparent carrier fluid. In comparison to their traditional counterparts, which use a solid surface for radiation absorption, particle-based receivers offer a number of opportunities for improved efficiency and heat transfer uniformity. The physical phenomena at the core of such receivers involve coupling between particle transport, fluid turbulence, and radiative heat transfer. Previous analyses of particle-based solar receivers ignored delicate aspects associated with this three-way coupling. Namely, these investigations considered the flow fields only in the mean sense and ignored turbulent fluctuations and the consequent particle preferential concentration. In the present work, we have performed three-dimensional direct numerical simulations of turbulent flows coupled with radiative heating and particle transport over a range of particle Stokes numbers. Our study demonstrates that the particle preferential concentration has strong implications on the heat transfer statistics. We demonstrate that “for a typical setting” the preferential concentration of particles reduces the effective heat transfer between particles and the gas by as much as 25%. Therefore, we conclude that a regime with Stokes number of order unity is the least preferred for heat transfer to the carrier fluid. We also provide a 1D model to capture the effect of particle spatial distribution in heat transfer.
We study the case of inertial particles heated by thermal radiation while settling by gravity through a turbulent transparent gas. We consider dilute and optically thin regimes in which each particle receives the same heat flux. Numerical simulations of forced homogeneous turbulence are performed taking into account the two-way coupling of both momentum and temperature between the dispersed and continuous phases. Particles much smaller than the smallest flow scales are considered and the point-particle approximation is adopted. The particle Stokes number (based on the Kolmogorov time scale) is of order unity, while the nominal settling velocity is up to an order of magnitude larger than the Kolmogorov velocity, marking a critical difference with previous two-way coupled simulations. It is found that non-heated particles enhance turbulence when their settling velocity is sufficiently high compared to the Kolmogorov velocity. Energy spectra show that the non-heated particle settling impacts both the very small and very large flow scales, while the intermediate scales are weakly affected. When heated, particles shed plumes of buoyant gas, further modifying the turbulence structure. At the considered radiation intensities, clustering is strong but the classic mechanism of preferential concentration is modified, while preferential sweeping is eliminated or even reversed. Particle heating also causes a significant reduction of the mean settling velocity, which is caused by rising buoyant plumes in the vicinity of particle clusters. The turbulent kinetic energy is affected non-monotonically as the radiation intensity is increased due to the competing effects of the downward gravitational force and the upward buoyancy force. The thermal radiation influences all scales of the turbulence. The effects of settling and buoyancy on the turbulence anisotropy are also discussed.
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