Alternative Implementations of the Monte Carlo Power Method

Blomquist, R.N.; Gelbard, E.M.

doi:10.13182/nse01-30

Cited by 18 publications

(10 citation statements)

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“…In order to solve the previously described eigenvalue/eigenvector problem, the power iteration method [9,25] is usually applied. Its convergence is ensured in the previous aformentionned conditions.…”

Section: The Monte Carlo Resolution Of a Deterministic Eigenvalue Pro...mentioning

confidence: 99%

“…Its convergence is ensured in the previous aformentionned conditions. Several 'versions' of the power iteration method exist for k eff computations, see [25]. We suggest recalling one of these versions (which can be found in [25]) in the following lines.…”

Section: The Monte Carlo Resolution Of a Deterministic Eigenvalue Pro...mentioning

confidence: 99%

“…The paper is organized as follows: in section 2, we recall one specific deterministic k eff MC algorithm allowing to solve (1) (power iteration). Several algorithms exist, see [25], but we can not go through every of them. Following the descriptions of the next sections should allow the interested reader to perform the relevant modifications to its own eigenvalue solver.…”

Section: Introductionmentioning

confidence: 99%

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Efficient uncertain keff computations with the Monte Carlo resolution of generalised Polynomial Chaos based reduced models

Poëtte

Brun

2022

Journal of Computational Physics

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In this paper, we are interested in taking into account uncertainties for k eff computations in neutronics. More generally, the material of this paper can be applied to propagate uncertainties in eigenvalue/eigenvector computations for the linear Boltzmann equation. In [1,2], an intrusive MC solver for the gPC based reduced model of the instationary linear Boltzmann equation has been put forward. The MC-gPC solver presents interesting characteristics (mainly a better efficiency than non-intrusive strategies and spectral convergence): our aim is to recover these characteristics in an eigenvalue/eigenvector estimation context. This is done in practice at the price of few well identified modifications of an existing Monte Carlo implementation.

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Section: The Monte Carlo Resolution Of a Deterministic Eigenvalue Pro...mentioning

confidence: 99%

Section: The Monte Carlo Resolution Of a Deterministic Eigenvalue Pro...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Efficient uncertain keff computations with the Monte Carlo resolution of generalised Polynomial Chaos based reduced models

Poëtte

Brun

2022

Journal of Computational Physics

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“…The relations R n = (A, Y n )/(e, Y n ) = (A, U n )/(e, U n ); (6) K n = S n+1 /S n = (e, U n+1 ) (7) are, respectively, the rate of a neutron reaction of type A and the neutron multiplication coefficient in the generation n. It is obvious that Eq. (7) is a particular case of expression (6) with A = H + e (H + is the conjugate operator).…”

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confidence: 99%

Estimates of the Bias in the Results of Monte Carlo Calculations of Reactors and Storage Sites for Nuclear Fuel

Maiorov

2005

At Energy

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The bias in Monte Carlo calculations of the neutron-physical characteristics of full-scale models of reactors and storage sites for nuclear fuel is investigated. The bias is shown to arise directly from the nature of the Monte Carlo solution of the eigenvalue problem and estimates of similar bias without loss of generality are proposed.As computers become more powerful, the Monte Carlo method is used more and more for calculating full-scale models of storage sites for spent nuclear fuel and of nuclear-reactor cores. This is increasing interest in understanding the specific features of the Monte Carlo solution of the homogeneous stationary neutron-transport equation.In Monte Carlo modeling, the solution and all its linear functionals are determined to within a factor, and therefore ratios of the functionals, i.e., linear-fractional functionals, are meaningful. When such functionals are calculated, biases arise in their estimates and not their values.The questions studied in the present paper have been attracting the attention of many specialists (see, for example, [1-7]). A set of model problems has been developed to check the proposed algorithms (for example, [8]).The most important result in this field is an asymptotic formula for determining the bias of the estimate of k eff , which is valid for large values of the number of neutrons in a generation. The formula makes it possible to estimate efficiently the systematic error in k eff in practical calculations, though it has been derived only for a method of simulation with a constant number of neutrons in a generation, a finite-dimensional model of a reactor, and the simplest analog estimate of k eff from the number of neutrons produced in the fissioning of a nucleus.The question of whether or not the systematic errors in the reaction rates calculated by the methods of generations can be estimated and the range of application of the formulas for the bias of the estimates of k eff can be expanded has thus far remained open. In the present paper, without loss of generality, general and asymptotic formulas for determining the bias of any estimates of any reaction rates calculated by simulation according to generations are obtained for real neutron-multiplication systems. It is important that all proposed proofs are technically simple and give a clear picture of the mechanism leading to the appearance of biases and the possibilities for estimating them.Formulation of the Problem. The problem is to calculate by the method of generations the main intrinsic value of k eff and the normalized reaction rates of neutrons, determined for the main eigenfunction Ψ(x) of the equation for the generation density of fission neutrons in a conditionally critical reactor. This equation can be written in operator form as follows:(1)

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“…In its general form, normalization reduces to the following: a set of neutrons obtained in modeling nuclear fission and having a random sum of statistical weights transforms into a set with the standard sum of weights within the admissible interval [1,3]. We shall use the term secondary neutrons for the set of particles which is obtained by modeling and we shall term the transformed set the next generation of neutrons.…”

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confidence: 99%

Statistical check of the correctness of the generation normalization algorithm implementation in monte-carlo calculations of nuclear power facilities

2011

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The statistical properties of the operation of a generation normalization algorithm used in the MCU program package are checked. A problem arises when a large packet of particles is used in a generation; this is associated with full-scale calculations of nuclear power facilities. It is shown that the algorithm used is correct.The processes occurring inside the core of a nuclear power reactor are often modeled using homogeneous solutions obtained for the neutron transport equation by the Monte Carlo method. This is done using the method of generations, which includes, as do all algorithms for maximum eigenvalue problems, a normalization process consisting of preserving the norm of a vector, for example, the sums of the absolute values of the components of the result of each iteration step. Among the different normalization methods used in the Monte Carlo method, the algorithm proposed by Frank-Kamenetskii [1] is easy to implement with minimum dispersion of the secondary particles obtained. As a result, it enters into the MCU package of problems [2].When working with a large number of particles in a single generation (this number can reach one million) it becomes necessary to analyze the quality of the number generators which are used. For such a large number of particles, it is necessary to check the statistical characteristics of the normalization algorithm.A simplification is used in the present article -the weight of the secondary particles is the same, which corresponds to the implementation of the algorithm in [1].In its general form, normalization reduces to the following: a set of neutrons obtained in modeling nuclear fission and having a random sum of statistical weights transforms into a set with the standard sum of weights within the admissible interval [1,3]. We shall use the term secondary neutrons for the set of particles which is obtained by modeling and we shall term the transformed set the next generation of neutrons. New phase coordinates of the particles do not arise in the normalization process -any particle from a generation possesses phase coordinates taken from some particle from the set of secondary neutrons. We shall assume that a new particle is obtained from an old particle.Equality of secondary particles is maintained in the following sense. Let w 1 and w 2 be the weights of the particles before normalization and W i the total weight of the particles obtained from the ith particle. Then the expectation of the random variables W i satisfies M(W 1 )/M(W 2 ) = w 1 /w 2 .(1)The algorithm of [1] can be described as follows.

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Alternative Implementations of the Monte Carlo Power Method

Cited by 18 publications

References 3 publications

Efficient uncertain keff computations with the Monte Carlo resolution of generalised Polynomial Chaos based reduced models

Efficient uncertain keff computations with the Monte Carlo resolution of generalised Polynomial Chaos based reduced models

Estimates of the Bias in the Results of Monte Carlo Calculations of Reactors and Storage Sites for Nuclear Fuel

Statistical check of the correctness of the generation normalization algorithm implementation in monte-carlo calculations of nuclear power facilities

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