DOCSLIB.ORG

Prediction and Analysis of Stochastic Convergence in the Standard And

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Prediction and Analysis of Stochastic Convergence in the Standard And Prediction and Analysis of Stochastic Convergence in the Standard and Extrapolated Power Methods Applied to Monte Carlo Fission Source Iterations by Bryan Elmer Toth A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Nuclear Engineering and Radiological Sciences) in the University of Michigan 2018 Doctoral Committee: Professor William R. Martin, Chair Dr. David P. Griesheimer, Bettis Atomic Power Laboratory Professor James Paul Holloway Professor Edward W. Larsen Professor Divakar Viswanath Bryan Elmer Toth [email protected] ORCID iD: 0000-0003-4219-5531 © Bryan Elmer Toth 2018 Dedication This dissertation is dedicated to the “best family ever” (A. Toth, personal communication, 2014). ii Acknowledgements The author would like to acknowledge that portions of this research were performed under appointment to the Rickover Graduate Fellowship Program sponsored by the Naval Reactors Division of the U.S. Department of Energy and the Nuclear Regulatory Commission Fellowship Program. iii Table of Contents Dedication...........................................................................................................................iii Acknowledgements.............................................................................................................iv List of Tables.....................................................................................................................vii List of Figures.....................................................................................................................ix List of Appendices............................................................................................................xix Abstract..............................................................................................................................xx Chapter 1 . Introduction.......................................................................................................1 1.1 Neutron Transport.......................................................................................................3 1.2 Power Method............................................................................................................8 1.3 Extrapolation..............................................................................................................9 1.4 Prior MC Extrapolation Research............................................................................14 Chapter 2 . Spectral Analysis of Standard and Extrapolated Monte Carlo........................18 2.1 Monte Carlo Eigenvalue Calculations......................................................................19 2.2 Expectations from the Stochastic Power Method.....................................................22 2.3 Expectations from Stochastic Extrapolation............................................................30 2.4 Implementation of Extrapolation..............................................................................33 2.5 Convergence Comparison.........................................................................................42 2.6 Estimating the Initial Source Bank Decomposition Coefficients.............................60 2.6.1 Decomposition Coefficients of Initial Starting Source from Prior Simulation...67 Chapter 3 . Monte Carlo Noise..........................................................................................68 3.1 Normality of the Decomposition Coefficients..........................................................69 3.1.1 Normality Under Analog Transport....................................................................70 3.1.2 Normality Under Implicit Capture......................................................................71 3.1.3 Normality of the Source Bank Decomposition Coefficients...............................75 iv 3.2 Verifying Gaussian Nature of Decomposition Coefficients.....................................75 3.3 Validation of Variance from Constant Multiplier Model..........................................85 3.3.1 Calculating the Representative Variances...........................................................85 3.3.2 Standard Power Method Converged Variances...................................................93 3.3.3 Extrapolated Power Method Converged Variances...........................................103 3.4 Estimating Representative Variances......................................................................112 3.4.1 Estimating Representative Variances Using the Eigenvectors..........................113 3.4.2 Estimating Representative Variances Without Eigenvectors............................130 Chapter 4 . Identifying, Predicting, and Optimizing Convergence..................................140 4.1 Defining Modal Convergence Measure..................................................................141 4.2 Defining a Modal Convergence Diagnostic...........................................................148 4.3 Selecting the Extrapolation Cutoff Cycle...............................................................160 Chapter 5 . Summary and Conclusions............................................................................177 Appendices.......................................................................................................................180 Bibliography....................................................................................................................193 v List of Tables Table 2.6.1: Parameters of 256-bin uniform slab with initial source in left-most bin.......62 Table 2.6.2: Properties of exponential fit to variances of Y/β1 estimates from cycles 401 to 1000 of uniform slab simulations with various numbers of histories per cycle. ........................................................................................................................64 Table 3.1.2.1: Events contributing to the moment expectation value................................73 Table 3.4.2.1: Representative variance ranges of non-fundamental modes from several test problems.................................................................................................135 Table 3.4.2.2: Estimated spectral mean variances of several test problems, which are compared with their empirically determined reference values.....................139 Table 4.2.1: Minimum number of discarded cycles for different modal relative entropy convergence criteria applied to the second mode of the uniform slab in Appendix A...................................................................................................158 Table 4.3.1: Parameters of uniform slab problem with 33-mode starting source that are used to predict the number of discarded cycles............................................171 Table 4.3.2: Number of discard cycles, mn,discard, required to converge various modes of the uniform slab problem for Dmax=3.18E-1 with mn,cutoff=100 along with relevant properties of the SBDC PDFs after their first sufficiently converged cycles. ......................................................................................................................171 Table 4.3.3: Number of discard cycles, mn,discard, required to converge various modes of the uniform slab problem for Dmax=5.0E-3 with mn,cutoff=100 along with relevant properties of the DC PDFs after their first sufficiently converged cycles....172 Table 4.3.4: While holding constant T=100 and α=1.0 the number of discard cycles were calculated for various eigenvalue ratios satisfying Dmax=5.0E-3..................178 Table 4.3.5: While holding constant T=0.0 and α=1.0, the number of discard cycles were vi calculated for various eigenvalue ratios satisfying Dmax=5.0E-3..................178 Table 4.3.6: While holding constant λn /λ1 = 0.99 and α=1.0, the number of discard cycles were calculated for various values of the grouped coefficient satisfying Dmax=5.0E-3..................................................................................................179 Table 4.3.7: While holding constant T=100 and λn /λ1 = 0.99, the number of discarded cycles were calculated for various extrapolation parameters satisfying Dmax=5.0E-3..................................................................................................179 Table A.1: Eigenvalues and eigenvalue ratios of the uniform slab..................................186 Table B.1: Macroscopic cross sections of the uniform slab.............................................188 Table D.1: Eigenvalues and eigenvalue ratios of the double peak slab...........................194 vii List of Figures Figure 1.3.1 The decay coefficient (1.3.10) and convergence rate (1.3.11) versus eigenvalue ratio for several different levels of extrapolation and the standard power method.................................................................................................13 Figure 2.2.1. Empirical data collected from a highly scattering uniform slab for the LHS and RHS of (2.2.21) for several non-fundamental modes..............................26 Figure 2.2.2. Uniform slab second mode expected DC versus cycle as estimated with empirical data, with constant normalization, and with adjusted constant normalization..................................................................................................29 Figure 2.2.3. Uniform slab third mode expected DC versus cycle as estimated with empirical data, with constant normalization, and with adjusted constant normalization..................................................................................................29
Load more

Recommended publications
  • Calculus and Differential Equations II
    Calculus and Differential Equations II MATH 250 B Sequences and series Sequences and series Calculus and Differential Equations II Sequences A sequence is an infinite list of numbers, s1; s2;:::; sn;::: , indexed by integers. 1n Example 1: Find the first five terms of s = (−1)n , n 3 n ≥ 1. Example 2: Find a formula for sn, n ≥ 1, given that its first five terms are 0; 2; 6; 14; 30. Some sequences are defined recursively. For instance, sn = 2 sn−1 + 3, n > 1, with s1 = 1. If lim sn = L, where L is a number, we say that the sequence n!1 (sn) converges to L. If such a limit does not exist or if L = ±∞, one says that the sequence diverges. Sequences and series Calculus and Differential Equations II Sequences (continued) 2n Example 3: Does the sequence converge? 5n 1 Yes 2 No n 5 Example 4: Does the sequence + converge? 2 n 1 Yes 2 No sin(2n) Example 5: Does the sequence converge? n Remarks: 1 A convergent sequence is bounded, i.e. one can find two numbers M and N such that M < sn < N, for all n's. 2 If a sequence is bounded and monotone, then it converges. Sequences and series Calculus and Differential Equations II Series A series is a pair of sequences, (Sn) and (un) such that n X Sn = uk : k=1 A geometric series is of the form 2 3 n−1 k−1 Sn = a + ax + ax + ax + ··· + ax ; uk = ax 1 − xn One can show that if x 6= 1, S = a .
    [Show full text]
  • Topic 7 Notes 7 Taylor and Laurent Series
    Topic 7 Notes Jeremy Orloff 7 Taylor and Laurent series 7.1 Introduction We originally defined an analytic function as one where the derivative, defined as a limit of ratios, existed. We went on to prove Cauchy's theorem and Cauchy's integral formula. These revealed some deep properties of analytic functions, e.g. the existence of derivatives of all orders. Our goal in this topic is to express analytic functions as infinite power series. This will lead us to Taylor series. When a complex function has an isolated singularity at a point we will replace Taylor series by Laurent series. Not surprisingly we will derive these series from Cauchy's integral formula. Although we come to power series representations after exploring other properties of analytic functions, they will be one of our main tools in understanding and computing with analytic functions. 7.2 Geometric series Having a detailed understanding of geometric series will enable us to use Cauchy's integral formula to understand power series representations of analytic functions. We start with the definition: Definition. A finite geometric series has one of the following (all equivalent) forms. 2 3 n Sn = a(1 + r + r + r + ::: + r ) = a + ar + ar2 + ar3 + ::: + arn n X = arj j=0 n X = a rj j=0 The number r is called the ratio of the geometric series because it is the ratio of consecutive terms of the series. Theorem. The sum of a finite geometric series is given by a(1 − rn+1) S = a(1 + r + r2 + r3 + ::: + rn) = : (1) n 1 − r Proof.
    [Show full text]
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • Zeros and Coefficients
    Zeros and coefficients Alexandre Eremenko∗ 10.28.2015 Abstract Asymptotic distribution of zeros of sequences of analytic functions is studied. Let fn be a sequence of analytic functions in a region D in the complex plane. The case of entire functions, D = C will be the most important. Sup- pose that there exist positive numbers Vn → ∞, such that the subharmonic functions 1 log |fn| areuniformlyboundedfromabove (1) Vn on every compact subset of D, and for some z0 ∈ D the sequence 1 log |fn(z0)| isboundedfrombelow. (2) Vn Under these conditions, one can select a subsequence such that for this sub- sequence 1 log |fn| → u, (3) Vn where u is a subharmonic function in D, u =6 −∞. This convergence can be understood in various senses, for example, quasi-everywhere, which means at every point except a set of points of zero logarithmic capacity, and also in ′ p D (Schwartz distributions), or in Lloc, 1 ≤ p< ∞. The corresponding Riesz measures converge weakly. The limit measure, −1 which is (2π) ∆u, describes the limit distribution of zeros of fn. ∗Supported by NSF grant DMS-1361836. 1 For all these facts we refer to [9] or [10]. This setting is useful in many cases when one is interested in asymptotic distribution of zeros of sequences of analytic functions. Let us mention several situations to which our results apply. 1. Let fn be monic polynomials of degree n, and Vn = n. We denote by µn the so-called empirical measures, which are the Riesz measures of (1/n)log |fn|. The measure µn is discrete, with finitely many atoms, with an atom of mass m at every zero of fn of multiplicity m.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Power Series and Convergence Issues
    POWER SERIES AND CONVERGENCE ISSUES MATH 153, SECTION 55 (VIPUL NAIK) Corresponding material in the book: Section 12.8, 12.9. What students should definitely get: The meaning of power series. The relation between Taylor series and power series, the notion of convergence, interval of convergence, and radius of convergence. The theorems for differentiation and integration of power series. Convergence at the boundary and Abel’s theorem. What students should hopefully get: How notions of power series provide an alternative interpre- tation for many of the things we have studied earlier in calculus. How results about power series combine with ideas like the root and ratio tests and facts about p-series. Executive summary Words ... P∞ k (1) The objects of interest here are power series, which are series of the form k=0 akx . Note that for power series, we start by default with k = 0. If the set of values of the index of summation is not specified, assume that it starts from 0 and goes on to ∞. The exception is when the index of summation occurs in the denominator, or some other such thing that forces us to exclude k = 0. 0 (2) Also note that x is shorthand for 1. When evaluating a power series at 0, we simply get a0. We do not actually do 00. (3) If a power series converges for c, it converges absolutely for all |x| < |c|. If a power series diverges for c, it diverges for all |x| > |c|. P k (4) Given a power series akx , the set of values where it converges is either 0, or R, or an interval that (apart from the issue of inclusion of endpoints) is symmetric about 0.
    [Show full text]
  • Calculus 1120, Fall 2012
    Calculus 1120, Fall 2012 Dan Barbasch November 13, 2012 Dan Barbasch () Calculus 1120, Fall 2012 November 13, 2012 1 / 5 Absolute and Conditional Convergence We need to consider series with arbitrary terms not just positive (nonnegative) ones. For such series the partial sums sn = a1 + ··· + an are not necessarily increasing. So we cannot use the bounded monotone convergence theorem. We cannot use the root/ratio test or the integral test directly on such series. The notion of absolute convergence allows us to deal with the problem. Dan Barbasch () Calculus 1120, Fall 2012 November 13, 2012 2 / 5 Absolute and Conditional Convergence P Definition: A series an is said to converge absolutely, if the series P janj converges. For convergence without absolute values, we say plain convergent. For plain convergence, but the series with absolute values does not converge, we say conditionally convergent. Theorem: If a series is absolutely convergent, then it is convergent. Note the difference between if and only if and if then. Typically we apply the root/ratio integral test and comparison test for absolute convergence, and use the theorem. REMARK: NOTE the precise definitions of the terms and the differences. Dan Barbasch () Calculus 1120, Fall 2012 November 13, 2012 2 / 5 Alternating Series Test P1 n−1 Theorem: (Alternating Series Test) Suppose n=1(−1) an satisfies 1 an ≥ 0; (series is alternating) 2 an is decreasing 3 limn!1 an = 0: Then the series is (possibly only conditionally) convergent. Error Estimate: jS − snj ≤ an+1. P1 i−1 S − sn = i=n+1(−1) ai . an+1 − an+2 + an+3 − · · · =(an+1 − an+2) + (an+3 − an+4) + ··· = =an+1 − (an+2 − an+3) − (an+4 − an+5) − ::: So 0 ≤ jS − snj = an+1 − an+2 + an+3 − · · · ≤ an+1 The proof is in the text and in the previous slides (also every text on calculus).
    [Show full text]
  • 8.6 Power Series Convergence
    Power Series Week 8 8.6 Power Series Convergence Definition A power series about c is a function defined by the infinite sum 1 X n f(x) = an(x − c) n=0 1 where the terms of the sequence fangn=0 are called the coefficients and c is referred to as the center of the power series. Theorem / Definition Every power series converges absolutely for the value x = c. For every power series that converges at more than a single point there is a value R, called the radius of convergence, such that the series converges absolutely whenever jx − cj < R and diverges when jx − cj > R. A series may converge or diverge when jx − cj = R. The interval of x values for which a series converges is called the interval of convergence. By convention we allow R = 1 to indicate that a power series converges for all x values. If the series converges only at its center, we will say that R = 0. Examples f (n)(c) (a) Every Taylor series is also a power series. Their centers coincide and a = . n n! (b) We saw one example of a power series weeks ago. Recall that for jrj < 1, the geometric series converges and 1 X a arn = a + ar + ar2 + ar3 ::: = 1 − r n=0 We can view this as a power series where fang = a and c = 0, 1 X a axn = 1 − x n=0 This converges for jx − 0j = jxj < 1 so the radius of converges is 1. The series does not converge for x = 1 or x = −1.
    [Show full text]
  • 9.4 Radius of Convergence
    AP Calculus BC 9.4 Radius of Convergence Objective: able to use the n th – Term Test, the Direct Comparison Test, and the Ratio Test to determine the convergence or divergence of a series of numbers or the radius of convergence of a power series. Do you think these series converge? Why or why not? (hint: write out a few terms) 1. ∑ 2. 3. 1 4. 5. 6. Convergence of a series means the total of the added terms comes to a finite value The Convergence Theorem for Power Series The number R is the radius of convergence , and the set of all values of x for which the series converges is the interval of convergence . The nth – Term Test for Divergence If the limit equals zero that does not mean that the series converges . It means that it could converge. If the limit equals zero, more tests have to be performed to prove whether or not the series converges. However, if the limit is not zero, we know the series diverges. This test is used as a _____________ 7. 8. 9. 1 10. 11. 12. An inconclusive result means that you need to do more work. The Direct Comparison Test Similar to the comparison test for improper integrals, Section 8.4, p. 464. Hint: Geometric series are your friend, the trick is finding the right one Does the series converge or diverge? 13. 14. 15. ! When your series fails the Divergence ( nth - Term) Test, you have to do MORE WORK!! The Direct Comparison test is nice, especially when you can find a friendly geometric series as your comparison series.
    [Show full text]
  • Power Series, Taylor and Maclaurin Series
    4.5. POWER SERIES 97 4.5. Power Series A power series is a series of the form ∞ c xn = c + c x + c x2 + + c xn + 0 0 1 2 · · · n · · · n=0 X where x is a variable of indeterminate. It can be interpreted as an infinite polynomial. The cn’s are the coefficients of the series. The sum of the series is a function ∞ n f(x) = c0x n=0 X For instance the following series converges to the function shown for 1 < x < 1: − ∞ 1 xn = 1 + x + x2 + + xn + = . · · · · · · 1 x n=0 X − More generally given a fix number a, a power series in (x a), or centered in a, or about a, is a series of the form − ∞ c (x a)n = c + c (x a) + c (x a)2 + + c (x a)n + 0 − 0 1 − 2 − · · · n − · · · n=0 X 4.5.1. Convergence of Power Series. For a given power series ∞ c (x a)n there are only three possibilities: n − n=1 X (1) The series converges only for x = a. (2) The series converges for all x. (3) There is a number R, called radius of convergence, such that the series converges if x a < R and diverges if x a > R. | − | | − | The interval of convergence is the set of values of x for which the series converges. Example: Find the radius of convergence and interval of convergence of the series ∞ (x 3)n − . n n=0 X 4.5. POWER SERIES 98 Answer: We use the Ratio Test: a (x 3)n+1/(n + 1) n n+1 = − = (x 3) x 3 , a (x 3)n/n − n + 1 n−→ − ¯ n ¯ →∞ ¯ ¯ − So the ¯power¯ series converges if x 3 < 1 and diverges if x 3 > 1.
    [Show full text]
  • Fundamental Domains and Analytic Continuation of General Dirichlet Series
    FUNDAMENTAL DOMAINS AND ANALYTIC CONTINUATION OF GENERAL DIRICHLET SERIES Dorin Ghisa1 with an introduction by Les Ferry2 AMS subject classification: 30C35, 11M26 Abstract: Fundamental domains are found for functions defined by general Dirichlet series and by using basic properties of conformal mappings the Great Riemann Hypothesis is studied. Introduction My interest in the topic of this paper appeared when I became aware that John Derbyshare, with whom I was in the same class as an undergraduate, had written a book on the Riemann Hypothesis and that this had won plaudits for its remarkable success in representing an abstruse topic to a lay audience. On discovering this area of mathematics for myself through different other sources, I found it utterly fascinating. More than a quarter of century before, I was taken a strong interest in the work of Peitgen, Richter, Douady and Hubbard concerning Julia, Fatou and Mandelbrot sets. From this I had learned the basic truth that, however obstruse might be the method of definition, the nature of analytic function is very tightly controlled in practice. I sensed intuitively at that time the Riemann Zeta function would have to be viewed in this light if a proof of Riemann Hypothesis was to be achieved. Many mathematicians appear to require that any fundamental property of the Riemann Zeta function and in general of any L-function must be derived from the consideration of the Euler product and implicitly from some mysterious properties of prime numbers. My own view is that it is in fact the functional equation which expresses the key property with which any method of proof of the Riemann Hypothesis and its extensions must interact.
    [Show full text]
  • Power Series and Their Convergence
    Power Series and their Convergence Professor Je¤ Stuart Paci…c Lutheran University c 2008 Introduction A power series in x is a series whose terms are all constant multiples of integer powers of (x a) for some …xed real number a: We say that the power series is centered at x= a; or that it is expanded about x = a: Thus S(x) = 1 + x + x2 + x3 + x4 + + xn + and e e e e T (x) = e (x 2)2 + (x 2)4 (x 2)6 + +( 1)n (x 2)2n + 2 24 720 n! are both power series, the …rst one centered at x = 0; and the second one centered at x = 2: Note also that some or even many positive integer powers of (x a) can have zero as a coe¢ cient. In the power series T (x); every odd power of (x 2) has coe¢ cient 0. For an arbitrary real number a; a generic power series expanded about x = a can be written as 2 3 n U(x) = a0 + a1(x a) + a2(x a) + a3(x a) + + an(x a) + ( ) where a0; a1; a2; a3 and an are all …xed real numbers. The …rst natural question to ask about a power series is where does it con- verge. That is, for what real numbers x is the series equal to a real number? Observe that S(0) = 1; that T (2) = e; and that in general, 2 3 n U(a) = a0 + a1(a a) + a2(a a) + a3(a a) + + an(a a) + 2 3 n = a0 + a1(0) + a2(0) + a3(0) + + an(0) + = a0 + 0 + 0 + 0 + + 0 + = a0 Thus a power series must converge at its center.
    [Show full text]