# The Second-Order Adjoint Sensitivity Analysis Methodology.

Cacuci, Dan Gabriel.

The Second-Order Adjoint Sensitivity Analysis Methodology. - 1st ed. - 1 online resource (327 pages) - Advances in Applied Mathematics Ser. . - Advances in Applied Mathematics Ser. .

Cover -- Title Page -- Copyright Page -- Contents -- Preface -- Acknowledgments -- Author -- Chapter 1: Motivation for Computing First- and Second-Order Sensitivities of System Responses to the System's Parameters -- 1.1 The Fundamental Role of Response Sensitivities for Uncertainty Quantification -- 1.2 The Fundamental Role of Response Sensitivities for Predictive Modeling -- 1.3 Advantages and Disadvantages of Statistical and Deterministic Methods for Computing Response Sensitivities -- Chapter 2: Illustrative Application of the 2nd-ASAM to a Linear Evolution Problem -- 2.1 Exact Computation of the First-Order Response Sensitivities -- 2.2 Exact Computation of the Second-Order Response Sensitivities -- 2.2.1 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ( t 1 )/∂β i -- 2.2.2 Computing the Second - Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ ( t 1 )/ ∂w i -- 2.2.3 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ( t 1 )/∂ q -- 2.2.4 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ( t 1 )/∂ρ in -- 2.2.5 Discussion of the Essential Features of the 2nd-ASAM -- 2.2.6 Illustrative Use of Response Sensitivities for Predictive Modeling -- Chapter 3: The 2nd-ASAM for Linear Systems -- 3.1 Mathematical Modeling of a General Linear System -- 3.2 The 1st-LASS for Computing Exactly and Efficiently First-Order Sensitivities of Scalar-Valued Responses for Linear Systems -- 3.3 The 2nd-LASS for Computing Exactly and Efficiently First-Order Sensitivities of Scalar-Valued Responses for Linear Systems -- 3.4 Concluding Remarks. Chapter 4: Application of the 2nd-ASAM to a Linear Heat Conduction and Convection Benchmark Problem -- 4.1 Heat Transport Benchmark Problem: Mathematical Modeling -- 4.2 Computation of First-Order Sensitivities -- 4.2.1 Computation of First-Order Sensitivities of the Heated Rod Temperature, T ( r , z ), at an Arbitrary Location ( r 0 , z 0 ) -- 4.2.2 Computation of First-Order Sensitivities of the Heated Rod Temperature, T max ( z max ), at the Location z max -- 4.2.3 Computation of First-Order Sensitivities of the Heated Rod Temperature, T s ( z 1 ), at an Arbitrary Location z 1 -- 4.2.4 Computation of First-Order Sensitivities of the Coolant Temperature -- 4.2.5 Verification of the ANSYS/FLUENT Adjoint Solver -- 4.3 Applying the 2nd-ASAM to Compute the Second-Order Sensitivities and Uncertainties for the Heat Transport Benchmark Problem -- 4.3.1 Computation of the Second-Order Sensitivities and Uncertainties of the Heated Rod Temperature, T ( r , z ), at an Arbitrary Location ( r 0 , z 0 ) -- 4.3.1.1 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 1 ∂ α j ), α 1 ≡ q , and j = 1, …, N α = 6 -- 4.3.1.2 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 2 ∂ α j ), α 2 ≡ k , and j = 1, …, N α = 6 -- 4.3.1.3 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 3 ∂ α j ), α 3 ≡ h , and j = 1, …, N α = 6 -- 4.3.1.4 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 4 ∂ α j ), α 4 ≡ W , and j = 1, …, N α = 6 -- 4.3.1.5 Computation of Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 5 ∂ α j ), α 5 ≡ c p , and j = 1, …, N α = 6. 4.3.1.6 Computation of Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 6 ∂ α j ), α 6 ≡ T inlet , and j = 1, …, N α = 6 -- 4.3.1.7 Quantitative Comparison of Second-Order Sensitivities of the Rod Temperature Distribution to G4M Reactor Model Parameters -- 4.3.1.8 Quantitative Contributions of Second-Order Sensitivities to the Uncertainty in the Rod Temperature Distribution for G4M Reactor Model Parameters -- 4.3.2 Computation of Second-Order Sensitivities of the Coolant Temperature, T fl ( z ) -- 4.4 Concluding Remarks -- Chapter 5: Application of the 2nd-ASAM to a Linear Particle Diffusion Problem -- 5.1 Problem Description -- 5.2 Applying the 2nd-ASAM to Compute the First-Order Response Sensitivities to Model Parameters -- 5.3 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities to Model Parameters -- 5.3.1 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 4 i ≜ ∂ 2 R /(∂∑ d ∂ α i ) -- 5.3.2 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 3 i ≜ ∂ 2 R /(∂ Q ∂ α i ) -- 5.3.3 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 1 i ≜ ∂ 2 R /(∂∑ a ∂ α i ) -- 5.3.4 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 2 i ≜ ∂ 2 R /(∂ D ∂ α i ) -- 5.4 Role of Second-Order Response Sensitivities for Quantifying Non-Gaussian Features of the Response Uncertainty Distribution -- 5.5 Illustrative Application of First-Order Response Sensitivities for Predictive Modeling -- 5.5.1 Assimilating an Imprecise but Consistent Measurement -- 5.5.2 Assimilating a Precise and Consistent Measurement -- 5.5.3 Assimilating Two Consistent Measurements -- 5.5.4 Assimilating Four Consistent Measurements. Chapter 6: Application of the 2nd-ASAM for Computing Sensitivities of Detector Responses to Uncollided Radiation Transport -- 6.1 The Ray-Tracing Form of the Forward and Adjoint Boltzmann Transport Equations -- 6.2 Application of the 2nd-ASAM to Compute the First-Order Response Sensitivities to Variations in Model Parameters -- 6.3 Application of the 2nd-ASAM to Compute the Second-Order Response Sensitivities to Variations in Model Parameters -- 6.3.1 Computation of the Second-Order Sensitivities -- 6.3.2 Computation of the Second-Order Sensitivities -- 6.3.3 Computation of the Second-Order Sensitivities -- 6.3.4 Computation of the Second-Order Sensitivities -- 6.4 Concluding Remarks -- Chapter 7: The 2nd-ASAM for Nonlinear Systems -- 7.1 Mathematical Modeling of a General Nonlinear System -- 7.2 The 1st-LASS for Computing Exactly and Efficiently the First-Order Sensitivities -- 7.3 The 2nd-LASS for Computing Exactly and Efficiently the Second-Order Sensitivities of Scalar-Valued Responses for Nonlinear Systems -- 7.4 Concluding Remarks -- Chapter 8: Application of the 2nd-ASAM to a Nonlinear Heat Conduction Problem -- 8.1 Mathematical Modeling of Heated Cylindrical Test Section -- 8.2 Application of the 2nd-ASAM for Computing the First-Order Sensitivities -- 8.3 Application of the 2nd-ASAM to Compute the Second-Order Sensitivities -- 8.3.1 Computation of the Second-Order Sensitivities -- 8.3.2 Computation of the Second-Order Sensitivities -- 8.3.3 Computation of the Second-Order Sensitivities -- 8.3.4 Computation of the Second-Order Sensitivities -- 8.3.5 Computation of the Second-Order Sensitivities -- 8.3.6 Computation of Standard Deviation and Skewness of the Temperature Distribution -- 8.4 Concluding Remarks -- References -- Index -- A -- B -- C -- D -- E. F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W.

ISBN: 9781498726498

Subjects--Topical Terms:

Sensitivity theory (Mathematics).

Large scale systems..

Nonlinear systems.

Index Terms--Genre/Form:

Electronic books.

LC Class. No.: QA402 .C338 2018

Dewey Class. No.: 003.71

The Second-Order Adjoint Sensitivity Analysis Methodology. - 1st ed. - 1 online resource (327 pages) - Advances in Applied Mathematics Ser. . - Advances in Applied Mathematics Ser. .

Cover -- Title Page -- Copyright Page -- Contents -- Preface -- Acknowledgments -- Author -- Chapter 1: Motivation for Computing First- and Second-Order Sensitivities of System Responses to the System's Parameters -- 1.1 The Fundamental Role of Response Sensitivities for Uncertainty Quantification -- 1.2 The Fundamental Role of Response Sensitivities for Predictive Modeling -- 1.3 Advantages and Disadvantages of Statistical and Deterministic Methods for Computing Response Sensitivities -- Chapter 2: Illustrative Application of the 2nd-ASAM to a Linear Evolution Problem -- 2.1 Exact Computation of the First-Order Response Sensitivities -- 2.2 Exact Computation of the Second-Order Response Sensitivities -- 2.2.1 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ( t 1 )/∂β i -- 2.2.2 Computing the Second - Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ ( t 1 )/ ∂w i -- 2.2.3 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ( t 1 )/∂ q -- 2.2.4 Computing the Second-Order Response Sensitivities Corresponding to the First-Order Sensitivities ∂ρ( t 1 )/∂ρ in -- 2.2.5 Discussion of the Essential Features of the 2nd-ASAM -- 2.2.6 Illustrative Use of Response Sensitivities for Predictive Modeling -- Chapter 3: The 2nd-ASAM for Linear Systems -- 3.1 Mathematical Modeling of a General Linear System -- 3.2 The 1st-LASS for Computing Exactly and Efficiently First-Order Sensitivities of Scalar-Valued Responses for Linear Systems -- 3.3 The 2nd-LASS for Computing Exactly and Efficiently First-Order Sensitivities of Scalar-Valued Responses for Linear Systems -- 3.4 Concluding Remarks. Chapter 4: Application of the 2nd-ASAM to a Linear Heat Conduction and Convection Benchmark Problem -- 4.1 Heat Transport Benchmark Problem: Mathematical Modeling -- 4.2 Computation of First-Order Sensitivities -- 4.2.1 Computation of First-Order Sensitivities of the Heated Rod Temperature, T ( r , z ), at an Arbitrary Location ( r 0 , z 0 ) -- 4.2.2 Computation of First-Order Sensitivities of the Heated Rod Temperature, T max ( z max ), at the Location z max -- 4.2.3 Computation of First-Order Sensitivities of the Heated Rod Temperature, T s ( z 1 ), at an Arbitrary Location z 1 -- 4.2.4 Computation of First-Order Sensitivities of the Coolant Temperature -- 4.2.5 Verification of the ANSYS/FLUENT Adjoint Solver -- 4.3 Applying the 2nd-ASAM to Compute the Second-Order Sensitivities and Uncertainties for the Heat Transport Benchmark Problem -- 4.3.1 Computation of the Second-Order Sensitivities and Uncertainties of the Heated Rod Temperature, T ( r , z ), at an Arbitrary Location ( r 0 , z 0 ) -- 4.3.1.1 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 1 ∂ α j ), α 1 ≡ q , and j = 1, …, N α = 6 -- 4.3.1.2 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 2 ∂ α j ), α 2 ≡ k , and j = 1, …, N α = 6 -- 4.3.1.3 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 3 ∂ α j ), α 3 ≡ h , and j = 1, …, N α = 6 -- 4.3.1.4 Computation of the Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 4 ∂ α j ), α 4 ≡ W , and j = 1, …, N α = 6 -- 4.3.1.5 Computation of Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 5 ∂ α j ), α 5 ≡ c p , and j = 1, …, N α = 6. 4.3.1.6 Computation of Second-Order Response Sensitivities ∂ 2 T ( r 0 , z 0 )/(∂ α 6 ∂ α j ), α 6 ≡ T inlet , and j = 1, …, N α = 6 -- 4.3.1.7 Quantitative Comparison of Second-Order Sensitivities of the Rod Temperature Distribution to G4M Reactor Model Parameters -- 4.3.1.8 Quantitative Contributions of Second-Order Sensitivities to the Uncertainty in the Rod Temperature Distribution for G4M Reactor Model Parameters -- 4.3.2 Computation of Second-Order Sensitivities of the Coolant Temperature, T fl ( z ) -- 4.4 Concluding Remarks -- Chapter 5: Application of the 2nd-ASAM to a Linear Particle Diffusion Problem -- 5.1 Problem Description -- 5.2 Applying the 2nd-ASAM to Compute the First-Order Response Sensitivities to Model Parameters -- 5.3 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities to Model Parameters -- 5.3.1 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 4 i ≜ ∂ 2 R /(∂∑ d ∂ α i ) -- 5.3.2 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 3 i ≜ ∂ 2 R /(∂ Q ∂ α i ) -- 5.3.3 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 1 i ≜ ∂ 2 R /(∂∑ a ∂ α i ) -- 5.3.4 Applying the 2nd-ASAM to Compute the Second-Order Response Sensitivities S 2 i ≜ ∂ 2 R /(∂ D ∂ α i ) -- 5.4 Role of Second-Order Response Sensitivities for Quantifying Non-Gaussian Features of the Response Uncertainty Distribution -- 5.5 Illustrative Application of First-Order Response Sensitivities for Predictive Modeling -- 5.5.1 Assimilating an Imprecise but Consistent Measurement -- 5.5.2 Assimilating a Precise and Consistent Measurement -- 5.5.3 Assimilating Two Consistent Measurements -- 5.5.4 Assimilating Four Consistent Measurements. Chapter 6: Application of the 2nd-ASAM for Computing Sensitivities of Detector Responses to Uncollided Radiation Transport -- 6.1 The Ray-Tracing Form of the Forward and Adjoint Boltzmann Transport Equations -- 6.2 Application of the 2nd-ASAM to Compute the First-Order Response Sensitivities to Variations in Model Parameters -- 6.3 Application of the 2nd-ASAM to Compute the Second-Order Response Sensitivities to Variations in Model Parameters -- 6.3.1 Computation of the Second-Order Sensitivities -- 6.3.2 Computation of the Second-Order Sensitivities -- 6.3.3 Computation of the Second-Order Sensitivities -- 6.3.4 Computation of the Second-Order Sensitivities -- 6.4 Concluding Remarks -- Chapter 7: The 2nd-ASAM for Nonlinear Systems -- 7.1 Mathematical Modeling of a General Nonlinear System -- 7.2 The 1st-LASS for Computing Exactly and Efficiently the First-Order Sensitivities -- 7.3 The 2nd-LASS for Computing Exactly and Efficiently the Second-Order Sensitivities of Scalar-Valued Responses for Nonlinear Systems -- 7.4 Concluding Remarks -- Chapter 8: Application of the 2nd-ASAM to a Nonlinear Heat Conduction Problem -- 8.1 Mathematical Modeling of Heated Cylindrical Test Section -- 8.2 Application of the 2nd-ASAM for Computing the First-Order Sensitivities -- 8.3 Application of the 2nd-ASAM to Compute the Second-Order Sensitivities -- 8.3.1 Computation of the Second-Order Sensitivities -- 8.3.2 Computation of the Second-Order Sensitivities -- 8.3.3 Computation of the Second-Order Sensitivities -- 8.3.4 Computation of the Second-Order Sensitivities -- 8.3.5 Computation of the Second-Order Sensitivities -- 8.3.6 Computation of Standard Deviation and Skewness of the Temperature Distribution -- 8.4 Concluding Remarks -- References -- Index -- A -- B -- C -- D -- E. F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- R -- S -- T -- U -- V -- W.

Sensitivity theory (Mathematics).

Large scale systems..

Nonlinear systems.

Electronic books.