Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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D. Kernel Distortion, Importance Sampling 96IV. Collision Density Equations • 98A. Definition of the Collision Densities 99B. Definition of the Transition and Collision Kernels 100C. The Equations Connecting the Collision Densities 101D. The Theory of the Step-By-Step Solution of the CollisionDensity Equations 103E. Normalizations of the Transition and Collision Kernels 104F. Termination of the Monte Carlo Cycle 107V. Scoring 108A. General Formulation of the Reaction Rates 108B. Estimation of More than One Response 108C. Monte Carlo Estimation of the Responses 109D. Examples of Pay-Off Functions 110VI. Three Special Problems 119A. Path Stretching (Exponential Transformation) 119B. Perturbation Monte Carlo 121C Criticality Studies 125VII. Adjoint Monte Carlo 126A. The Value Equations • 127B. Solution of the Value Equations (Adjoint Monte Carlo) — 129C. Sampling the Adjoint Source 130D. The Collision Kernel of the Value Equation 131E. Scoring in the Adjoint Monte Carlo 134F. Contributions of the Uncollided Particles 137VIII. Variances 138A. Variance Estimates by the Moment Equations 139B. The Value Used as Importance Function — 141References. 141Chapter 5The Moment Equations 143I. Introductory Remarks • • • • • 143A. Relation of the Expected Score to the AdjointCollision Density 145B. Conditions of Existence and Uniqueness 146C. Analog and Nonanolog Simulation — 149D. Definitions and Notations 151E. Heuristic Interpretation of the Moment Equations 155II. Moment Equations in Nonmultiplying Games 158A. Score Probability Equations — . — 158B. Moment of a General Score Function 160C Special Cases: Expectation and Second Moment ofthe Score 163D. An Analytical Example 166III. Extension to Multiplying Games 169A. Score Probability Equation 170B. Expectation and Second Moment 171C. An Equivalent Nonmultiplying Game 173D. Splitting: When a Nonmultiplying Game is Played asa Multiplying One. 178E. Alternative Forms of the Collision Kernel 182
IV. Further Generalizations — 183A. Interruption and Restart of a Free Flight .... — 1.84B. Geometrical Splitting — ...186C. Score Probabilities in a General Time-Independent Game 192D. Inclusion of Time Dependence — — 193V. Analysis of the First-Moment Equation 196A. Unbiased Estimators — 197B. Weight Generation Rules 199C. A Nonanolog Game Without Statistical Weights:Importance Sampling 203D. Generalized Exponential Transformation 207E. Path Stretching ,. 212F. Computing Time and Number of Events per History .213G. Feasibility of a Nonanolog Game .216H. Delta Scattering 222VI. Partially Unbiased Estimators 226A. Transformation Theorems — .. 228B. Commonly Used Estimators ,231C. Analysis of Variances in the Straight-AheadScattering Model — . 236VII. Approximate Solutions of the Moment Equations 239A. The Simplified Model 240B. The Separation Assumption 241C. On the Quality of the Approximation 244D. Effect of Surroundings 246VIII. Analysis of Second Moment Equations —.. 249A. Zero-Variance Schemes — ............................ 250B. On the Boundedness of the Variance 258C Sufficient Conditions of Variance Reduction byNonanolog Games — 260D. Examples: Survival Biasing and ELP and MELP Methods 263E. Variance and Efficiency of the Equivalent NonmultiplyingGame 265F. Zero-Variance Partially Unbiased Estimators:The Minimum-Variance Composed Estimator 271G. Relative Merits of the Common Estimators..... — 275H. The Self-Improving Estimator 280I. Variance Versus Efficiency in a Nonanalog Game 283J. Optimization of Source Distribution ....... 284IX. Miscellaneous Specific Moment Equations 286A. Estimation of Bilinear Forms 287B. Correlation of Estimators — 289C Moment-Generating Equation 290D. Coupled Multiparticle Simulation — 290Appendix 5A:Solution of the Moment Equations in the Forward/Backward Model 294Appendix 5 B:Second Moments of Multiple Convolutions 297
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Page 17 and 18: Chapter 2INTRODUCTIONWhen we starte
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100 Monte Carlo Panicle Transport M
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102 Monte Carlo Particle Transport
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25!) Monte Carlo Particle Transport
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305Chapter 6SPECIAL GAMESIn the maj
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3©7A. CORRELATED MOMENT EQUATIONSI
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309Equation (6.1) will be referred
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311The second moment of the score d
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313andC S(P',P")/C S(P',P") = y(P',
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315introduced in Equation (6.1) for
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317andL„(P 0,P;,...,Pi + 1) = L n
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319whereB„(P„,P; P:, ,) = A*(P
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E. EXAMPLES AND SPECIAL TECHNIQUESL
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323The weight factors associated wi
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325It is seen from Equation (6.37)
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327G. PARAMETRIC PERTURBATIONS: INT
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129Hall 31 gave a constructive deri
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331kernels, which remain unchanged
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333and from Equation (6.51)w';,,(i)
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335According to Equations (6,55) th
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337Minimization is performed by set
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339and letdsdaLet us consider a cor
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341whereis the length of the flight
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and from Equation (6.79)I / ds \ 2
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345Let us assume that we are intere
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347Introduction of this factor also
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3492. Let i|/ n (P) denote the solu
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3SJNow, if ( "> > 0, then k = i an
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353andk„ = S l "VS'" •" (6. if;
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3SSLet k„ denote the normalizatio
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3572. It is then proved that with t
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and start new histories until N new
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361The application of source iterat
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363we haveD 2 [k] = < 2 (k, - k,) )
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365neutron density, S 0 for every
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367if the same relation holds for t
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369Then the multiplication factors
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371in the unperturbed system, the w
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373The perturbation of the effectiv
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375where P" = (r',E'). Multiplying
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377iterate of the derivative of the
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579where P* = (r*,w*,E) = (r*,E*).
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381This estimator, unlike f, in Equ
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383tends to some stable attracting
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3§SIo the case of the next-event e
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387LDFIGURE 6.1.Geometry in scoring
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389Let us denoteThen Equation (6.18
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391is to be increased in order to o
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393Now, since bl(P) = g(P) is bound
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395is scored at every collision wit
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397and its expectation isR — R 1n
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3994. If O > 0 m, then the particle
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401plays a distinguished role), and
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403Theorem 6.7 — If x and s denot
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this a priori information, the best
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40?Thus, the bias introduced into t
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409In view of Equation (6.226) and
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41.1This means that if we use the k
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413Then the whole-sample and rare-s
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41.5According to the results of Sec
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41?(Note that vD 2 [|i|v] is indepe
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419withkOn the other hand, § is an
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421Thus, the expected ratio reads
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423St follows from Equations (6.265
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425Wc start from the observation th
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427Nowi'6.77K)kj„,and therefore(V
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42*3Similarly, for the fourth momen
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43 sOn the other hand, multiplying
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43.¾This will be proven in the lem
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435ThenT.m = EJ.Iy.y.Rjeyiy,,,and s
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6. Brown. F. B. and Martin, W. R.,
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67. Rief, H. and Fioretti, A., Mont
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Q = iy.M 2 2 (1/T, - 1/1,..,)/1, £
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48© Monte Carlo Particle Transport
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a ( X506 Monte Carlo Particle Trans
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513INDEXAAbsorptiondefined, 25photo
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515score probability in general tim
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517path stretching, 447—455splitt
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Monte Carlo Particle Transport Methods: Neutron and Photon - gnssn

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