Preconditioning and Solver Optimization Ideas for Radiative Transfer

2005 ◽  
Author(s):  
David B. Carrington ◽  
Vincent A. Mousseau
Keyword(s):  
Radiative Transfer ◽  
Search Algorithm ◽  
Linear Equations ◽  
Computational Domain ◽  
System Of Linear Equations ◽  
Participating Media ◽  
Preconditioning Techniques ◽  
Radiation Diffusion ◽  
Diagonal Scaling ◽  
Optimization Search

In this paper, radiative transfer and time-dependent transport of radiation energy in participating media are modeled using a first-order spherical harmonics method (P1) and radiation diffusion. Partial differential equations for P1 and radiation diffusion are discretized by a variational form of the equations using support operators. Choices made in the discretization result in a symmetric positive definite (SPD) system of linear equations. Modeling multidimensional domains with complex geometries requires a very large system of linear equations with 10s of millions of elements. The computational domain is decomposed into a large number of subdomains that are solved on separate processors resulting in a massively parallel application. The linear system of equations is solved with a preconditioned conjugate gradient method. Various preconditioning techniques are compared in this study. Simple preconditioning techniques include: diagonal scaling, Symmetric Successive Over Relaxation (SSOR), and block Jacobi with SSOR as the block solver. Also, a two-grid multigrid-V-cycle method with aggressive coarsening is explored for use in the problems presented. Results show that depending on the test problem, simple preconditioners are effective, but the more complicated preconditioners such as an algebraic multigrid or the geometric multigrid are most efficient, particularly for larger problems and longer simulations. Optimal preconditioning varies depending on the problem and on how the physical processes evolve in time. For the insitu preconditioning techniques—SSOR and block Jacobi—a fuzzy controller can determine the optimal reconditioning process. Discussions of the current knowledge-based controller, an optimization search algorithm, are presented. Discussions of how this search algorithm can be incorporated into the development of data-driven controller incorporating clustering and subsequent construction of the fuzzy model from partitions are also discussed.

Finite-Volume Solution of the P3 Equations of Radiative Transfer and Coupling to Reacting Flow Calculations

2009 ◽  
Author(s):  
Mahesh Ravishankar ◽  
Sandip Mazumder ◽  
Ankan Kumar
Keyword(s):  
Radiative Transfer ◽  
Finite Volume ◽  
Unstructured Mesh ◽  
Laminar Flame ◽  
Cell Group ◽  
Reacting Flow ◽  
Computational Domain ◽  
Algebraic Equations ◽  
Participating Media ◽  
Cpu Time

The method of spherical harmonics (or PN) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P3 equations of radiative transfer in two-dimensional geometry—a set of four coupled second-order partial differential equations. The FV procedure, presented here, is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the Binary Spatial Partitioning algorithm, followed by fully implicit coupled solution within each cell group using a pre-conditioned Generalized Minimum Residual (GMRES) solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for a benchmark problem. For completeness, results using the P1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P3 results increase as the optical thickness is decreased. The P3 approximation is found to be more accurate than the P1 approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the 4 P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the 10 equations that are solved for the reacting flow calculations.

Modeling Radiative Transfer

2017 ◽  
Author(s):  
Kelly Chance ◽  
Randall V. Martin
Keyword(s):  
Radiative Transfer ◽  
Linear Equations ◽  
Single Scattering ◽  
Usual Method ◽  
Integrodifferential Equations ◽  
Practical Calculation ◽  
System Of Linear Equations ◽  
Basic Concepts ◽  
Radiative Transfer Modeling ◽  
Upwelling And Downwelling

Basic concepts and definitions of radiative transfer modeling are introduced. The applicability of single scattering to aerosol retrievals is demonstrated. A two-stream formulation of radiative transfer is introduced. The two streams, upwelling and downwelling radiation, are selected to angularly represent average atmospheric properties: Relatively simple, two-stream calculations form the basis for much practical calculation, particularly of hemispherical fluxes in stratified atmospheres. Following this development, the most usual method for replacing the general integrodifferential equations obtained when setting up a scattering scenario by a system of linear equations is demonstrated.

Finite-Volume Formulation and Solution of the P3 Equations of Radiative Transfer on Unstructured Meshes

2009 ◽  
Vol 132 (2) ◽  
Author(s):  
Mahesh Ravishankar ◽  
Sandip Mazumder ◽  
Ankan Kumar
Keyword(s):  
Radiative Transfer ◽  
Finite Volume ◽  
Unstructured Mesh ◽  
Laminar Flame ◽  
Cell Group ◽  
Benchmark Problems ◽  
Reacting Flow ◽  
Computational Domain ◽  
Participating Media ◽  
Cpu Time

Abstract The method of spherical harmonics (or PN) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P3 equations of radiative transfer in two-dimensional geometry—a set of four coupled, second-order partial differential equations. The FV procedure presented here is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the binary spatial partitioning algorithm, followed by fully implicit coupled solution within each cell group using a preconditioned generalized minimum residual solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for two benchmark problems. For completeness, results using the P1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P3 results increase as the optical thickness is decreased. The P3 approximation is found to be more accurate than the P1 approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the four P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the case without radiation.

scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

scholarly journals Numerical approximation for the solution of linear sixth order boundary value problems by cubic B-spline

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
A. Khalid ◽  
M. N. Naeem ◽  
P. Agarwal ◽  
A. Ghaffar ◽  
Z. Ullah ◽  
...  
Keyword(s):  
Numerical Approximation ◽  
Analytic Solution ◽  
Linear Equations ◽  
Computational Cost ◽  
Spline Method ◽  
System Of Linear Equations ◽  
Approximation Solution ◽  
B Spline ◽  
Novel Technique ◽  
Derivatives Of

AbstractIn the current paper, authors proposed a computational model based on the cubic B-spline method to solve linear 6th order BVPs arising in astrophysics. The prescribed method transforms the boundary problem to a system of linear equations. The algorithm we are going to develop in this paper is not only simply the approximation solution of the 6th order BVPs using cubic B-spline, but it also describes the estimated derivatives of 1st order to 6th order of the analytic solution at the same time. This novel technique has lesser computational cost than numerous other techniques and is second order convergent. To show the efficiency of the proposed method, four numerical examples have been tested. The results are described using error tables and graphs and are compared with the results existing in the literature.

scholarly journals Hybrid Bernstein Block-Pulse Functions Method for Second Kind Integral Equations with Convergence Analysis

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Mohsen Alipour ◽  
Dumitru Baleanu ◽  
Fereshteh Babaei
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Convergence Analysis ◽  
Bernstein Polynomials ◽  
Linear Equations ◽  
The Other ◽  
New Combination ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Block Pulse Functions

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

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