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E-raamat: Second-Order Adjoint Sensitivity Analysis Methodology

(University of South Carolina)
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Second-Order Adjoint Sensitivity Analysis MethodologySuurem pilt
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The Second-Order Adjoint Sensitivity Analysis Methodology generalizes the First-Order Theory presented in the authors previous books published by CRC Press. This breakthrough has many applications in sensitivity and uncertainty analysis, optimization, data assimilation, model calibration, and reducing uncertainties in model predictions. The book has many illustrative examples that will help readers understand the complexity of the subject and will enable them to apply this methodology to problems in their own fields.

Highlights:

Covers a wide range of needs, from graduate students to advanced researchers

Provides a text positioned to be the primary reference for high-order sensitivity and uncertainty analysis

Applies to all fields involving numerical modeling, optimization, quantification of sensitivities in direct and inverse problems in the presence of uncertainties.

About the Author:

Dan Gabriel Cacuci is a South Carolina SmartState Endowed Chair Professor and the Director of the Center for Nuclear Science and Energy, Department of Mechanical Engineering at the University of South Carolina. He has a Ph.D. in Applied Physics, Mechanical and Nuclear Engineering from Columbia University. He is also the recipient of many awards including four honorary doctorates, the Ernest Orlando Lawrence Memorial award from the U.S. Dept. of Energy and the Arthur Holly Compton, Eugene P. Wigner and the Glenn Seaborg Awards from the American Nuclear Society.
Preface ix
Acknowledgments xvii
Author xix
1 Motivation for Computing First- and Second-Order Sensitivities of System Responses to the System's Parameters
1(26)
1.1 The Fundamental Role of Response Sensitivities for Uncertainty Quantification
6(6)
1.2 The Fundamental Role of Response Sensitivities for Predictive Modeling
12(11)
1.3 Advantages and Disadvantages of Statistical and Deterministic Methods for Computing Response Sensitivities
23(4)
2 Illustrative Application of the 2nd-ASAM to a Linear Evolution Problem
27(32)
2.1 Exact Computation of the First-Order Response Sensitivities
29(4)
2.2 Exact Computation of the Second-Order Response Sensitivities
33(26)
2.2.1 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ(t1)/∂βi
33(7)
2.2.2 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ(t1)/∂wi
40(4)
2.2.3 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ(t1)/∂q
44(3)
2.2.4 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ(t1)/∂ρin
47(3)
2.2.5 Discussion of the Essential Features of the 2nd-ASAM
50(2)
2.2.6 Illustrative Use of Response Sensitivities for Predictive Modeling
52(7)
3 The 2nd-ASAM for Linear Systems
59(22)
3.1 Mathematical Modeling of a General Linear System
60(2)
3.2 The 1st-LASS for Computing Exactly and Efficiently First-Order Sensitivities of Scalar-Valued Responses for Linear Systems
62(7)
3.3 The 2nd-LASS for Computing Exactly and Efficiently First-Order Sensitivities of Scalar-Valued Responses for Linear Systems
69(11)
3.4 Concluding Remarks
80(1)
4 Application of the 2nd-ASAM to a Linear Heat Conduction and Convection Benchmark Problem
81(62)
4.1 Heat Transport Benchmark Problem: Mathematical Modeling
84(4)
4.2 Computation of First-Order Sensitivities
88(24)
4.2.1 Computation of First-Order Sensitivities of the Heated Rod Temperature, T(r, z), at an Arbitrary Location (r0, z0)
94(5)
4.2.2 Computation of First-Order Sensitivities of the Heated Rod Temperature, Tmax(zmax), at the Location zmax
99(3)
4.2.3 Computation of First-Order Sensitivities of the Heated Rod Temperature, Ts(z1), at an Arbitrary Location z1
102(5)
4.2.4 Computation of First-Order Sensitivities of the Coolant Temperature
107(4)
4.2.5 Verification of the ANSYS/FLUENT Adjoint Solver
111(1)
4.3 Applying the 2nd-ASAM to Compute the Second-Order Sensitivities and Uncertainties for the Heat Transport Benchmark Problem
112(29)
4.3.1 Computation of the Second-Order Sensitivities and Uncertainties of the Heated Rod Temperature, T(r, z), at an Arbitrary Location (r0, z0)
113(1)
4.3.1.1 Computation of the Second-Order Response Sensitivities ∂2T(r0, z0)/(∂α1 ∂αj), α1 ≡ q, and j = 1, ..., Nα = 6
114(3)
4.3.1.2 Computation of the Second-Order Response Sensitivities 9∂2T(r0, z0)/(∂α2 ∂αj), α2 ≡ k, and j = 1, ..., Nα = 6
117(1)
4.3.1.3 Computation of the Second-Order Response Sensitivities ∂2T(r0, z0)/(∂α3 ∂αj), α3 ≡ h, and j = 1, ..., Nα = 6
118(2)
4.3.1.4 Computation of the Second-Order Response Sensitivities ∂2T(r0, z0)/(∂α4 ∂αj), α4 ≡ W, and j = 1, ..., Nα = 6
120(1)
4.3.1.5 Computation of Second-Order Response Sensitivities ∂2T(r0, z0)/(∂α5 ∂αj), α5 ≡ cp, and j = 1, ..., Nα = 6
121(1)
4.3.1.6 Computation of Second-Order Response Sensitivities ∂2T(r0, z0)/(∂α6 ∂αj), α6 ≡ Tinlet, and j = 1, ..., Nα = 6
122(1)
4.3.1.7 Quantitative Comparison of Second-Order Sensitivities of the Rod Temperature Distribution to G4M Reactor Model Parameters
122(5)
4.3.1.8 Quantitative Contributions of Second-Order Sensitivities to the Uncertainty in the Rod Temperature Distribution for G4M Reactor Model Parameters
127(10)
4.3.2 Computation of Second-Order Sensitivities of the Coolant Temperature, Tfi(z)
137(4)
4.4 Concluding Remarks
141(2)
5 Application of the 2nd-ASAM to a Linear Particle Diffusion Problem
143(50)
5.1 Problem Description
144(1)
5.2 Applying the 2nd-ASAM to Compute the First-Order Response Sensitivities to Model Parameters
145(8)
5.3 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities to Model Parameters
153(14)
5.3.1 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S4i Δ ∂2R/(∂Σd∂αi)
154(2)
5.3.2 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S3i Δ ∂2R/(∂Q∂αi)
156(2)
5.3.3 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S1i Δ ∂2R/(∂Σa∂αi)
158(4)
5.3.4 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S2i Δ ∂2R/(∂D∂αi)
162(5)
5.4 Role of Second-Order Response Sensitivities for Quantifying Non-Gaussian Features of the Response Uncertainty Distribution
167(5)
5.5 Illustrative Application of First-Order Response Sensitivities for Predictive Modeling
172(21)
5.5.1 Assimilating an Imprecise but Consistent Measurement
173(4)
5.5.2 Assimilating a Precise and Consistent Measurement
177(4)
5.5.3 Assimilating Two Consistent Measurements
181(5)
5.5.4 Assimilating Four Consistent Measurements
186(7)
6 Application of the 2nd-ASAM for Computing Sensitivities of Detector Responses to Uncollided Radiation Transport
193(22)
6.1 The Ray-Tracing Form of the Forward and Adjoint Boltzmann Transport Equations
193(2)
6.2 Application of the 2nd-ASAM to Compute the First-Order Response Sensitivities to Variations in Model Parameters
195(6)
6.3 Application of the 2nd-ASAM to Compute the Second-Order Response Sensitivities to Variations in Model Parameters
201(12)
6.3.1 Computation of the Second-Order Sensitivities S(2)i,j Δ ∂S(1)i∂Ni ∂αj, i = 1, ..., Nm, j = 1, ..., Nα
206(2)
6.3.2 Computation of the Second-Order Sensitivities S(2)i+3Nm,j Δ ∂S(1)i∂μi ∂αj, i = 1, ..., Nm, j = 1, ..., Nα
208(2)
6.3.3 Computation of the Second-Order Sensitivities S(2)i+2Nm,j Δ ∂S(1)∂σi ∂αj, i = 1, ..., Nm, j = 1, ..., Nα
210(1)
6.3.4 Computation of the Second-Order Sensitivities S(2)i+2Nm,j Δ ∂S(1)∂qi ∂αj, i = 1, ..., Nd, j = 1, ..., Nα
211(2)
6.4 Concluding Remarks
213(2)
7 The 2nd-ASAM for Nonlinear Systems
215(22)
7.1 Mathematical Modeling of a General Nonlinear System
215(1)
7.2 The 1st-LASS for Computing Exactly and Efficiently the First-Order Sensitivities
216(8)
7.3 The 2nd-LASS for Computing Exactly and Efficiently the Second-Order Sensitivities of Scalar-Valued Responses for Nonlinear Systems
224(11)
7.4 Concluding Remarks
235(2)
8 Application of the 2nd-ASAM to a Nonlinear Heat Conduction Problem
237(58)
8.1 Mathematical Modeling of Heated Cylindrical Test Section
237(2)
8.2 Application of the 2nd-ASAM for Computing the First-Order Sensitivities
239(7)
8.3 Application of the 2nd-ASAM to Compute the Second-Order Sensitivities
246(46)
8.3.1 Computation of the Second-Order Sensitivities R(2)1i Δ ∂2T(zr)/(∂Q∂αi)
247(8)
8.3.2 Computation of the Second-Order Sensitivities R(2)2i(α0) Δ ∂2T(zr)/(∂q ∂αi)
255(6)
8.3.3 Computation of the Second-Order Sensitivities R(2)3i(alpha;0) Δ ∂2T(zr)/(∂Ta ∂αi)
261(5)
8.3.4 Computation of the Second-Order Sensitivities R(2)4i(alpha;0) Δ ∂2T(zr)/(∂k0 ∂αi)
266(4)
8.3.5 Computation of the Second-Order Sensitivities R(2)5i(α0) Δ ∂2T(zr)/(∂c ∂αi)
270(4)
8.3.6 Computation of Standard Deviation and Skewness of the Temperature Distribution
274(18)
8.4 Concluding Remarks
292(3)
References 295(4)
Index 299
Dan Gabriel Cacuci is a South Carolina SmartState Endowed Chair Professor and the Director of the Center for Nuclear Science and Energy, Department of Mechanical Engineering at the University of South Carolina. He has a Ph.D. in Applied Physics, Mechanical and Nuclear Engineering from Columbia University. He is also the recipient of many awards including four honorary doctorates, the Ernest Orlando Lawrence Memorial award from the U.S. Dept. of Energy and the Arthur Holly Compton, Eugene P. Wigner and the Glenn Seaborg Awards from the American Nuclear Society.
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