In this paper, we study the directional derivative, subderivative, and subdifferential of sup-type functions without any compactness assumption on the index set. As applications, we provide an estimate of the Lipschit...In this paper, we study the directional derivative, subderivative, and subdifferential of sup-type functions without any compactness assumption on the index set. As applications, we provide an estimate of the Lipschitz modulus for sup-type functions.展开更多
This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2<sup>nd</sup>-CASAM)” for the efficient and exact computation of 1<sup>st</sup>- and 2<sup>...This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2<sup>nd</sup>-CASAM)” for the efficient and exact computation of 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (<em>i.e.</em>, model result of interest) is a generic nonlinear function of the model’s forward and adjoint state functions, and also depends on the imprecisely known boundaries and model parameters. In the practically important particular case when the response is a scalar-valued functional of the forward and adjoint state functions characterizing a model comprising N parameters, the 2<sup>nd</sup>-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1<sup>st</sup>-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2<sup>nd</sup>-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (<em>N</em>2/2 + 3 <em>N</em>/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2<sup>nd</sup>-LASS requires very little additional effort beyond the construction of the 1<sup>st</sup>-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1<sup>st</sup>-LASS and 2<sup>nd</sup>-LASS requires the same computational solvers as needed for solving (<em>i.e.</em>, “inverting”) either the forward or the adjoint linear operators underlying the initial model. Therefore, the same computer software and “solvers” used for solving the original system of equations can also be used for solving the 1<sup>st</sup>-LASS and the 2<sup>nd</sup>-LASS. Since neither the 1<sup>st</sup>-LASS nor the 2<sup>nd</sup>-LASS involves any differentials of the operators underlying the original system, the 1<sup>st</sup>-LASS is designated as a “<u>first-level</u>” (as opposed to a “first-order”) adjoint sensitivity system, while the 2<sup>nd</sup>-LASS is designated as a “<u>second-level</u>” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2<sup>nd</sup>-LASS that involve the imprecisely known boundary parameters. Notably, the 2<sup>nd</sup>-LASS encompasses an automatic, inherent, and independent “solution verification” mechanism of the correctness and accuracy of the 2nd-level adjoint functions needed for the efficient and exact computation of the second-order sensitivities.展开更多
We study Chaney's and Ben-Tal-Zowe's second-order directional derivatives with applications in minimization problem for max-functions of the formh(x): = max {f(x, τ); τ ∈T},where T is a compact metricspace....We study Chaney's and Ben-Tal-Zowe's second-order directional derivatives with applications in minimization problem for max-functions of the formh(x): = max {f(x, τ); τ ∈T},where T is a compact metricspace. We improve Kawasaki's result on necessary condition for such functions in the minimization problem.展开更多
Based on the empirical electron theory of solids and molecules of S. H. Yu, this paper proposes (i) the calculating model of valence electron structures in L_2~'-type substitutional and interstitial complex solid ...Based on the empirical electron theory of solids and molecules of S. H. Yu, this paper proposes (i) the calculating model of valence electron structures in L_2~'-type substitutional and interstitial complex solid solutions; (ii) the bond length difference analysis (BLD) method of unknown bond length structure solid solutions; (iii) the treatment of uncertainty of BLD analysis solution.展开更多
From systemic research of microstructure, geochemistry, uranium-series and 10Be isotope dating on a new-type deepwater ferromanganese crust from the East Philippine Sea, the paleoenvironment evolu-tion of the target a...From systemic research of microstructure, geochemistry, uranium-series and 10Be isotope dating on a new-type deepwater ferromanganese crust from the East Philippine Sea, the paleoenvironment evolu-tion of the target area since the terminal Late Miocene was recovered. The vertical section changes of microstructure and chemical composition are consistent in the studied crust, which indicate three major accretion periods and corresponding paleoenvironment evolution of the crust. The bottom crust zone was formed in the terminal Late Miocene (5.6 Ma) with loose microstructure, high detritus content and high growth rate. Reductions of mineral element content, accretion rate and positive Ce-anomaly degree at 4.6 Ma indicate temporal warming, which went against the crust accretion and finally formed an accretion gap in the terminal Middle Pliocene (2.8―2.7 Ma). The more active Antarctic bottom sea-waters in the Late Pliocene (2.7 Ma) facilitated the fast transfer to the top pure crust zone. Hereafter, with the further apart of volcanic source and the keeping increase of eolian material (1.0 Ma), although surrounding conditions were still favorable, mineral element content still shows an obvious reducing trend. It thereby offers new carrier and data for the unclear paleoceanographic research of the target area since the terminal Late Miocene.展开更多
基金Supported by the National Natural Science Foundation of China(No.10571106,10671010)
文摘In this paper, we study the directional derivative, subderivative, and subdifferential of sup-type functions without any compactness assumption on the index set. As applications, we provide an estimate of the Lipschitz modulus for sup-type functions.
文摘This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2<sup>nd</sup>-CASAM)” for the efficient and exact computation of 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (<em>i.e.</em>, model result of interest) is a generic nonlinear function of the model’s forward and adjoint state functions, and also depends on the imprecisely known boundaries and model parameters. In the practically important particular case when the response is a scalar-valued functional of the forward and adjoint state functions characterizing a model comprising N parameters, the 2<sup>nd</sup>-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1<sup>st</sup>-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2<sup>nd</sup>-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (<em>N</em>2/2 + 3 <em>N</em>/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2<sup>nd</sup>-LASS requires very little additional effort beyond the construction of the 1<sup>st</sup>-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1<sup>st</sup>-LASS and 2<sup>nd</sup>-LASS requires the same computational solvers as needed for solving (<em>i.e.</em>, “inverting”) either the forward or the adjoint linear operators underlying the initial model. Therefore, the same computer software and “solvers” used for solving the original system of equations can also be used for solving the 1<sup>st</sup>-LASS and the 2<sup>nd</sup>-LASS. Since neither the 1<sup>st</sup>-LASS nor the 2<sup>nd</sup>-LASS involves any differentials of the operators underlying the original system, the 1<sup>st</sup>-LASS is designated as a “<u>first-level</u>” (as opposed to a “first-order”) adjoint sensitivity system, while the 2<sup>nd</sup>-LASS is designated as a “<u>second-level</u>” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2<sup>nd</sup>-LASS that involve the imprecisely known boundary parameters. Notably, the 2<sup>nd</sup>-LASS encompasses an automatic, inherent, and independent “solution verification” mechanism of the correctness and accuracy of the 2nd-level adjoint functions needed for the efficient and exact computation of the second-order sensitivities.
文摘We study Chaney's and Ben-Tal-Zowe's second-order directional derivatives with applications in minimization problem for max-functions of the formh(x): = max {f(x, τ); τ ∈T},where T is a compact metricspace. We improve Kawasaki's result on necessary condition for such functions in the minimization problem.
基金Project supported by the National Natural Science Foundation of China.
文摘Based on the empirical electron theory of solids and molecules of S. H. Yu, this paper proposes (i) the calculating model of valence electron structures in L_2~'-type substitutional and interstitial complex solid solutions; (ii) the bond length difference analysis (BLD) method of unknown bond length structure solid solutions; (iii) the treatment of uncertainty of BLD analysis solution.
基金the Key Program of the National Natural Science Foundation of China (Grant No. 90411014)the National Natural Science Foundation of China (Grant Nos. 40576032 and 40506016).
文摘From systemic research of microstructure, geochemistry, uranium-series and 10Be isotope dating on a new-type deepwater ferromanganese crust from the East Philippine Sea, the paleoenvironment evolu-tion of the target area since the terminal Late Miocene was recovered. The vertical section changes of microstructure and chemical composition are consistent in the studied crust, which indicate three major accretion periods and corresponding paleoenvironment evolution of the crust. The bottom crust zone was formed in the terminal Late Miocene (5.6 Ma) with loose microstructure, high detritus content and high growth rate. Reductions of mineral element content, accretion rate and positive Ce-anomaly degree at 4.6 Ma indicate temporal warming, which went against the crust accretion and finally formed an accretion gap in the terminal Middle Pliocene (2.8―2.7 Ma). The more active Antarctic bottom sea-waters in the Late Pliocene (2.7 Ma) facilitated the fast transfer to the top pure crust zone. Hereafter, with the further apart of volcanic source and the keeping increase of eolian material (1.0 Ma), although surrounding conditions were still favorable, mineral element content still shows an obvious reducing trend. It thereby offers new carrier and data for the unclear paleoceanographic research of the target area since the terminal Late Miocene.
