An Efficient Implementation of the Thomas-Algorithm for Block Penta-diagonal Systems on Vector Computers

Benkert, Katharina; Fischer, Rudolf

doi:10.1007/978-3-540-72584-8_19

Cited by 14 publications

(7 citation statements)

References 8 publications

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“…The Thomas algorithm represents a standard method to solve blocktridiagonal systems [43]. In this paper, we propose an algorithm based on a generalization of the Thomas algorithm (see [3][4][5]) which provides us a block-pentadiagonal solver with optimal complexity. Although this procedure is standard (Gaussian elimination for banded matrices), for the sake of clearness in the exposition of our work we present the used algorithm.…”

Section: Endmentioning

confidence: 99%

Multigrid second-order accurate solution of parabolic control-constrained problems

González-Andrade

Borzì

2010

Comput Optim Appl

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A mesh-independent and second-order accurate multigrid strategy to solve control-constrained parabolic optimal control problems is presented. The resulting algorithms appear to be robust with respect to change of values of the control parameters and have the ability to accommodate constraints on the control also in the limit case of bang-bang control. Central to the development of these multigrid schemes is the design of iterative smoothers which can be formulated as local semismooth Newton methods. The design of distributed controls is considered to drive nonlinear parabolic models to follow optimally a given trajectory or attain a final configuration. In both cases, results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed schemes are able to solve parabolic optimality systems with textbook multigrid efficiency. Further results are presented to validate second-order accuracy and the possibility to track a trajectory over long time intervals by means of a receding-horizon approach.

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Section: Endmentioning

confidence: 99%

Multigrid second-order accurate solution of parabolic control-constrained problems

González-Andrade

Borzì

2010

Comput Optim Appl

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“…Due to the sparse nature of both the left and right side solver matrices, this equation can be stored in vector form extracting only the populated diagonal vectors a, b, c, d , e and f along with the temperature vectors T mÀ1 and T m . The penta-diagonal matrix can be solved efficiently using a row reduction algorithm conceptually similar to the Thomas algorithm, Askar and Karawia (2015), Benkert and Fischer (2007). The method can be accelerated by pre-computing the algorithm factors σ, ϕ, ω, ρ and ψ and reusing these during each iteration.…”

Section: D Multilayer Numerical Methodologymentioning

confidence: 99%

1D analytic and numerical analysis of multilayer laminates and thin film heat transfer gauges

Baker

Rosic

2022

J. Glob. Power Propuls. Soc.

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The impulse response method is widely used for heat transfer analysis in turbomachinery applications. Traditionally, the 1D method assumes a linear time invariant, isotropic, semi-infinite block and does not accurately model the behaviour of laminated materials. This paper evaluates the error introduced by the single layer assumption and outlines the required modifications for multilayer analysis. The analytic solution for an N layer, semi-infinite laminate is presented. Adapted multilayer basis functions are derived for the impulse response method and used to evaluate the impact of uniform, isotropic assumptions. A numerical solution to the laminate problem is also presented. A penta-diagonal inversion algorithm, for a modified Crank-Nicolson scheme, is evaluated for fast stable implementation of multilayer simulation. The scheme shows comparable performance to the impulse response, whilst removing the requirement for linear time invariance. The methods are demonstrated in the case of analysing a thin film gauge, used in laboratory analysis of heat transfer in a turbine nozzle guide vane. Thin film gauge manufacturing techniques have advanced significantly in recent years. Advanced multilayer constructions are now used however, post-processing commonly relies on outdated single layer methods. This paper provides a universal methodology, required to analyse modern-day multilayer heat transfer measurements.

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“…(In our Matlab implementation, this was achieved with cubic-spline interpolation-but not necessarily so with the other available methods.) The resulting linear system to be solved at each time step has a block pentadiagonal structure (there are linear algebra routines tailored to such systems, see [BF07]).…”

Section: Let the Spatial Domainmentioning

confidence: 99%

Volatility Uncertainty Quantification in a Stochastic Control Problem Applied to Energy

Bernal¹,

Gobet²,

Printems³

2019

Methodol Comput Appl Probab

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This work designs a methodology to quantify the uncertainty of a volatility parameter in a stochastic control problem arising in energy management. The difficulty lies in the non-linearity of the underlying scalar Hamilton-Jacobi-Bellman equation. We proceed by decomposing the unknown solution on a Hermite polynomial basis (of the unknown volatility), whose different coefficients are solution to a system of non-linear PDEs of the same kind. Numerical tests show that computing the first basis elements may be enough to get an accurate approximation with respect to the uncertain volatility parameter. We experiment the methodology in the context of swing contract (energy contract with flexibility in purchasing energy power), this allows to introduce the concept of Uncertainty Value Adjustment (UVA), whose aim is to value the risk of misspecification of the volatility model.

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An Efficient Implementation of the Thomas-Algorithm for Block Penta-diagonal Systems on Vector Computers

Cited by 14 publications

References 8 publications

Multigrid second-order accurate solution of parabolic control-constrained problems

Multigrid second-order accurate solution of parabolic control-constrained problems

1D analytic and numerical analysis of multilayer laminates and thin film heat transfer gauges

Volatility Uncertainty Quantification in a Stochastic Control Problem Applied to Energy

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