In recent years,incidents of simultaneous exceedance of PM_(2.5)and O_(3) concentrations,termed PM_(2.5)and O_(3) co-pollution events,have frequently occurred in China.This study conducted atmospheric circulation anal...In recent years,incidents of simultaneous exceedance of PM_(2.5)and O_(3) concentrations,termed PM_(2.5)and O_(3) co-pollution events,have frequently occurred in China.This study conducted atmospheric circulation analysis on two typical co-pollution events in Beijing,occurring from July 22 to July 28,2019,and from April 25 to May 2,2020.These events were categorized into pre-trough southerly airflow type(Type 1)and post-trough northwest flow type(Type 2).Subsequently,sensitivity analyses using the GRAPES-CUACE adjoint model were performed to quantify the contributions of precursor emissions from Beijing and surrounding areas to PM_(2.5)and O_(3) concentrations in Beijing for two types of co-pollution.The results indicated that the spatiotemporal distribution of sensitive source region varied among different circulation types.Primary PM_(2.5)(PPM_(2.5))emissions from Hebei contributed the most to the 24-hour average PM_(2.5)(24-h PM_(2.5))peak concentration(41.6%-45.4%),followed by Beijing emissions(31%-35.7%).The maximum daily 8-hour average ozone peak concentration was primarily influenced by the emissions from Hebei and Beijing,with contribution ratios respectively of 32.8%-44.8% and 29%-42.1%.Additionally,NO_(x)emissions were the main contributors in Type 1,while both NO_(x)and VOCs emissions contributed similarly in Type 2.The iterative emission reduction experiments for two types of co-pollution indicated that Type 1 required emission reductions in NO_(x)(52.4%-71.8%)and VOCs(14.1%-33.8%)only.In contrast,Type 2 required combined emission reductions in NO_(x)(37.0%-65.1%),VOCs(30.7%-56.2%),and PPM_(2.5)(31%-46.9%).This study provided a reference for controlling co-pollution events and improving air quality in Beijing.展开更多
This paper aims at a review of the work carried out to date on the adjoint assimilation of data in marine ecosys-tem models since 1995. The structure and feature of the adjoint assimilation in marine ecosystem models ...This paper aims at a review of the work carried out to date on the adjoint assimilation of data in marine ecosys-tem models since 1995. The structure and feature of the adjoint assimilation in marine ecosystem models are also introduced. To illustrate the application of the adjoint technique and its merits, a 4-variable ecosystem model coupled with a 3-D physical model is established for the Bohai Sea and the Yellow Sea. The chlorophyll concentration data derived from the SeaWiFS o-cean colour data are assimilated in the model with the technique. Some results are briefly presented.展开更多
In the first paper in this series, a variational data assimilation of ideal tropical cyclone （TC） tracks was performed for the statistical-dynamical prediction model SD-90 by the adjoint method, and a prediction of ...In the first paper in this series, a variational data assimilation of ideal tropical cyclone （TC） tracks was performed for the statistical-dynamical prediction model SD-90 by the adjoint method, and a prediction of TC tracks was made with good accuracy for tracks containing no sharp turns. In the present paper, the cases of real TC tracks are studied. Due to the complexity of TC motion, attention is paid to the diagnostic research of TC motion. First, five TC tracks are studied. Using the data of each entire TC track, by the adjoint method, five TC tracks are fitted well, and the forces acting on the TCs are retrieved. For a given TC, the distribution of the resultant of the retrieved force and Coriolis force well matches the corresponding TC track, i.e., when a TC turns, the resultant of the retrieved force and Coriolis force acts as a centripetal force, which means that the TC indeed moves like a particle; in particular, for TC 9911, the clockwise looping motion is also fitted well. And the distribution of the resultant appears to be periodic in some cases. Then, the present method is carried out for a portion of the track data for TC 9804, which indicates that when the amount of data for a TC track is sufficient, the algorithm is stable. And finally, the same algorithm is implemented for TCs with a double-eyewall structure, namely Bilis （2000） and Winnie （1997）, and the results prove the applicability of the algorithm to TCs with complicated mesoscale structures if the TC track data are obtained every three hours.展开更多
Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary ...Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary condition resulting from an adjoint model on a limited-area is no longer an issue, and yet preserve its well-poseness and optimal character in the boundary setting. The ill-poseness of over-specified spatial boundary condition is in a sense, inevitable from an adjoint model since data assimilation processes have to adapt prescribed observations that used to be over-specified at the spatial boundaries of the modeling domain. In the view of pragmatic implement, the theoretical framework of our proposed condition for spatial boundaries indeed can be reduced to the hybrid formulation of nudging filter, radiation condition taking account of ambient forcing, together with Dirichlet kind of compatible boundary condition to the observations prescribed in data assimilation procedure. All of these treatments, no doubt, are very familiar to mesoscale modelers. Key words Variational data assimilation - Adjoint model - Over-specified partial boundary condition This research work is sponsored by the National Key Programme for Developing Basic Sciences (G1998040907), the Project of Natural Science Foundation of Jiangsu Province (BK99020), the President Foundation of Nanjing University (985) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.展开更多
Melt ponds significantly affect Arctic sea ice thermodynamic processes.The melt pond parameterization scheme in the Los Alamos sea ice model(CICE6.0) can predict the volume,area fraction(the ratio between melt pond ar...Melt ponds significantly affect Arctic sea ice thermodynamic processes.The melt pond parameterization scheme in the Los Alamos sea ice model(CICE6.0) can predict the volume,area fraction(the ratio between melt pond area to sea ice area in a model grid),and depth of melt ponds.However,this scheme has some uncertain parameters that affect melt pond simulations.These parameters could be determined through a conventional parameter estimation method,which requires a large number of sensitivity simulations.The adjoint model can calculate the parameter sensitivity efficiently.In the present research,an adjoint model was developed for the CESM(Community Earth System Model) melt pond scheme.A melt pond parameter estimation algorithm was then developed based on the CICE6.0 sea ice model,melt pond adjoint model,and L-BFGS(Limited-memory Broyden-Fletcher-Goldfard-Shanno) minimization algorithm.The parameter estimation algorithm was verified under idealized conditions.By using MODIS(Moderate Resolution Imaging Spectroradiometer)melt pond fraction observation as a constraint and the developed parameter estimation algorithm,the melt pond aspect ratio parameter in CESM scheme,which is defined as the ratio between pond depth and pond area fraction,was estimated every eight days during summertime for two different regions in the Arctic.One region was covered by multi-year ice(MYI) and the other by first-year ice(FYI).The estimated parameter was then used in simulations and the results show that:(1) the estimated parameter varies over time and is quite different for MYI and FYI;(2) the estimated parameter improved the simulation of the melt pond fraction.展开更多
This work illustrates the application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2<sup>nd</sup>-CASAM) to a mathematical model that can simulate the evolution and/or tr...This work illustrates the application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2<sup>nd</sup>-CASAM) to a mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response is the value of the model’s state function (particle concentration or particle flux) at a point in phase-space, which would simulate a pointwise measurement of the respective state function. This paradigm model admits exact closed-form expressions for all of the 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to the model’s uncertain parameters and domain boundaries. These closed-form expressions can be used to verify the numerical results of production and/or commercial software, e.g., particle transport codes. Furthermore, this paradigm model comprises many uncertain parameters which have relative sensitivities of identical magnitudes. Therefore, this paradigm model could serve as a stringent benchmark for inter-comparing the performances of all deterministic and statistical sensitivity analysis methods, including the 2<sup>nd</sup>-CASAM.展开更多
This work continues the illustrative application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2<sup>nd</sup>-CASAM) to a benchmark mathematical model that can simulate th...This work continues the illustrative application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2<sup>nd</sup>-CASAM) to a benchmark mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response considered in this work is a reaction-rate detector response, which provides the average interactions of particles with the respective detector or, alternatively, the time-average of the concentration of a mixture of substances in a medium. The definition of this model response includes both uncertain boundary points of the benchmark, thereby providing both direct and indirect contributions to the response sensitivities stemming from the boundaries. The exact expressions for the 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to the boundary and model parameters obtained in this work can serve as stringent benchmarks for inter-comparing the performances of all (deterministic and statistical) sensitivity analysis methods.展开更多
The strong nonlinearity of boundary layer parameterizations in atmospheric and oceanic models can cause difficulty for tangent linear models in approximating nonlinear perturbations when the time integration grows lon...The strong nonlinearity of boundary layer parameterizations in atmospheric and oceanic models can cause difficulty for tangent linear models in approximating nonlinear perturbations when the time integration grows longer. Consequently, the related 4—D variational data assimilation problems could be difficult to solve. A modified tangent linear model is built on the Mellor-Yamada turbulent closure (level 2.5) for 4-D variational data assimilation. For oceanic mixed layer model settings, the modified tangent linear model produces better finite amplitude, nonlinear perturbation than the full and simplified tangent linear models when the integration time is longer than one day. The corresponding variational data assimilation performances based on the adjoint of the modified tangent linear model are also improved compared with those adjoints of the full and simplified tangent linear models.展开更多
The eddy viscosity of the ocean is an important parameter indicating the small-scale mixing process in the oceanic interior water column. Ekman wind-driven current model and adjoint assimilation technique are used to ...The eddy viscosity of the ocean is an important parameter indicating the small-scale mixing process in the oceanic interior water column. Ekman wind-driven current model and adjoint assimilation technique are used to calculate the vertical profiles of eddy viscosity by fitting model results to the observation data. The data used in the paper include observed wind data and ADCP data obtained at Wenchang Oil Rig on the SCS （the South China Sea） shelf in August 2002. Different simulations under different wind conditions are analyzed to explore how the eddy viscosity develops with varying wind field. The results show that the eddy viscosity endured gradual variations in the range of 10^-3 -10^-2 m^2 /s during the periods of wind changes. The mean eddy viscosity undergoing strong wind could rise by about 25% as compared to the value under weak wind.展开更多
The design optimization taking into account the impact of uncertainties favors improving the robustness of the design.A Surrogate-Assisted Gradient-Based(SAGB)method for the robust aerodynamic design optimization of t...The design optimization taking into account the impact of uncertainties favors improving the robustness of the design.A Surrogate-Assisted Gradient-Based(SAGB)method for the robust aerodynamic design optimization of turbomachinery blades considering large-scale uncertainty is introduced,verified and validated in the study.The gradient-based method is employed due to its high optimization efficiency and any one surrogate model with sufficient response accuracy can be employed to quantify the nonlinear performance changes.The gradients of objective performance function to the design parameters are calculated first for all the training samples,from which the gradients of cost function can be fast determined.To reveal the high efficiency and high accuracy of SAGB on gradient calculation,the number of flow computations needed is evaluated and compared with three other methods.Through the aerodynamic design optimization of a transonic turbine cascade minimizing total pressure loss at the outlet,the SAGB-based gradients of the base and optimized blades are compared with those obtained by the Monte Carlo-assisted finite difference method.Moreover,the results of both the robust and deterministic aerodynamic design optimizations are presented and compared to demonstrate the practicability of SAGB on improving the aerodynamic robustness of turbomachinery blades.展开更多
Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for it to obtain the gradients of all design variables. However, the computational cost of adjoint vector is app...Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for it to obtain the gradients of all design variables. However, the computational cost of adjoint vector is approximately equal to that of flow computation. In order to accelerate the solution of adjoint vector and improve the efficiency of adjoint-based optimization, machine learning for adjoint vector modeling is presented. Deep neural network (DNN) is employed to construct the mapping between the adjoint vector and the local flow variables. DNN can efficiently predict adjoint vector and its generalization is examined by a transonic drag reduction of NACA0012 airfoil. The results indicate that with negligible computational cost of the adjoint vector, the proposed DNN-based adjoint method can achieve the same optimization results as the traditional adjoint method.展开更多
Constructing sophisticated refractivity models is one of the key problems for the RFC(refractivity from clutter)technology. If prior knowledge of the local refractivity environment is available, more accurate paramete...Constructing sophisticated refractivity models is one of the key problems for the RFC(refractivity from clutter)technology. If prior knowledge of the local refractivity environment is available, more accurate parameterized model can be constructed from the statistical information, which in turn can be used to improve the quality of the local refractivity retrievals. The validity of this proposal was demonstrated by range-dependent refractivity profile inversions using the adjoint parabolic equation method to the Wallops’ 98 experimental data.展开更多
The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this a...The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields.The framework consists of a Reynolds-averaged Navier–Stokes(RANS)solver and a neuralnetwork-represented turbulence model,each accompanied by its derivative computations.For computing the sensitivities of the indirect observations to the Reynolds stress field,we use the continuous adjoint equations for the RANS equations,while the gradient of the neural network is obtained via its built-in automatic differentiation capability.We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures.We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists.The trained deep-neural-network turbulence model showed predictive capability on similar flows.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
Economic development has caused a lot of environmental problems,in turn,environmental pollution restricts economic development.Considering the influence of wind direction and speed,temperature and humidity on pollutan...Economic development has caused a lot of environmental problems,in turn,environmental pollution restricts economic development.Considering the influence of wind direction and speed,temperature and humidity on pollutants,as well as the influence of epidemic,war and exchange rate on economic development.In this paper,we develop a stochastic economic-environment model with pollution control strategies.Furthermore,sufficient and necessary conditions for the near-optimality are established.Finally,we perform some numerical simulations to demonstrate the correctness of the theoretical results,which shows that some control strategies could decrease the environmental pollution,and therefore,could alleviate economic losses caused by environmental pollution.展开更多
An HIV model was considered. The parameters of the model are estimated by adjoint dada assimilation method. The results showed the method is valid. This method has potential application to a wide variety of models in ...An HIV model was considered. The parameters of the model are estimated by adjoint dada assimilation method. The results showed the method is valid. This method has potential application to a wide variety of models in biomathematics.展开更多
This work highlights the unparalleled efficiency of the “n<sup>th</sup>-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-FASAM-N) by con...This work highlights the unparalleled efficiency of the “n<sup>th</sup>-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-FASAM-N) by considering the well-known Nordheim-Fuchs reactor dynamics/safety model. This model describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This nonlinear paradigm model is sufficiently complex to model realistically self-limiting power excursions for short times yet admits closed-form exact expressions for the time-dependent neutron flux, temperature distribution and energy released during the transient power burst. The n<sup>th</sup>-FASAM-N methodology is compared to the extant “n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-CASAM-N) showing that: (i) the 1<sup>st</sup>-FASAM-N and the 1<sup>st</sup>-CASAM-N methodologies are equally efficient for computing the first-order sensitivities;each methodology requires a single large-scale computation for solving the “First-Level Adjoint Sensitivity System” (1<sup>st</sup>-LASS);(ii) the 2<sup>nd</sup>-FASAM-N methodology is considerably more efficient than the 2<sup>nd</sup>-CASAM-N methodology for computing the second-order sensitivities since the number of feature-functions is much smaller than the number of primary parameters;specifically for the Nordheim-Fuchs model, the 2<sup>nd</sup>-FASAM-N methodology requires 2 large-scale computations to obtain all of the exact expressions of the 28 distinct second-order response sensitivities with respect to the model parameters while the 2<sup>nd</sup>-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3<sup>rd</sup>-FASAM-N methodology is even more efficient than the 3<sup>rd</sup>-CASAM-N methodology: only 2 large-scale computations are needed to obtain the exact expressions of the 84 distinct third-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters when applying the 3<sup>rd</sup>-FASAM-N methodology, while the application of the 3<sup>rd</sup>-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-CASAM-N methodologies are the most practical methodologies for computing response sensitivities of any order comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis.展开更多
This work presents the “n<sup>th</sup>-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “n<sup>th</sup>-FASAM-N”), which will be shown to be the...This work presents the “n<sup>th</sup>-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “n<sup>th</sup>-FASAM-N”), which will be shown to be the most efficient methodology for computing exact expressions of sensitivities, of any order, of model responses with respect to features of model parameters and, subsequently, with respect to the model’s uncertain parameters, boundaries, and internal interfaces. The unparalleled efficiency and accuracy of the n<sup>th</sup>-FASAM-N methodology stems from the maximal reduction of the number of adjoint computations (which are considered to be “large-scale” computations) for computing high-order sensitivities. When applying the n<sup>th</sup>-FASAM-N methodology to compute the second- and higher-order sensitivities, the number of large-scale computations is proportional to the number of “model features” as opposed to being proportional to the number of model parameters (which are considerably more than the number of features).When a model has no “feature” functions of parameters, but only comprises primary parameters, the n<sup>th</sup>-FASAM-N methodology becomes identical to the extant n<sup>th</sup> CASAM-N (“n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-CASAM-N methodologies are formulated in linearly increasing higher-dimensional Hilbert spaces as opposed to exponentially increasing parameter-dimensional spaces thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems. Both the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-CASAM-N are incomparably more efficient and more accurate than any other methods (statistical, finite differences, etc.) for computing exact expressions of response sensitivities of any order with respect to the model’s features and/or primary uncertain parameters, boundaries, and internal interfaces.展开更多
基金supported by the National Key Research and Development Program of China(No.2022YFC3701205)the National Natural Science Foundation of China(No.41975173)the Science and Technology Development Fund of the Chinese Academy of Meteorological Sciences(No.2021KJ011)。
文摘In recent years,incidents of simultaneous exceedance of PM_(2.5)and O_(3) concentrations,termed PM_(2.5)and O_(3) co-pollution events,have frequently occurred in China.This study conducted atmospheric circulation analysis on two typical co-pollution events in Beijing,occurring from July 22 to July 28,2019,and from April 25 to May 2,2020.These events were categorized into pre-trough southerly airflow type(Type 1)and post-trough northwest flow type(Type 2).Subsequently,sensitivity analyses using the GRAPES-CUACE adjoint model were performed to quantify the contributions of precursor emissions from Beijing and surrounding areas to PM_(2.5)and O_(3) concentrations in Beijing for two types of co-pollution.The results indicated that the spatiotemporal distribution of sensitive source region varied among different circulation types.Primary PM_(2.5)(PPM_(2.5))emissions from Hebei contributed the most to the 24-hour average PM_(2.5)(24-h PM_(2.5))peak concentration(41.6%-45.4%),followed by Beijing emissions(31%-35.7%).The maximum daily 8-hour average ozone peak concentration was primarily influenced by the emissions from Hebei and Beijing,with contribution ratios respectively of 32.8%-44.8% and 29%-42.1%.Additionally,NO_(x)emissions were the main contributors in Type 1,while both NO_(x)and VOCs emissions contributed similarly in Type 2.The iterative emission reduction experiments for two types of co-pollution indicated that Type 1 required emission reductions in NO_(x)(52.4%-71.8%)and VOCs(14.1%-33.8%)only.In contrast,Type 2 required combined emission reductions in NO_(x)(37.0%-65.1%),VOCs(30.7%-56.2%),and PPM_(2.5)(31%-46.9%).This study provided a reference for controlling co-pollution events and improving air quality in Beijing.
文摘This paper aims at a review of the work carried out to date on the adjoint assimilation of data in marine ecosys-tem models since 1995. The structure and feature of the adjoint assimilation in marine ecosystem models are also introduced. To illustrate the application of the adjoint technique and its merits, a 4-variable ecosystem model coupled with a 3-D physical model is established for the Bohai Sea and the Yellow Sea. The chlorophyll concentration data derived from the SeaWiFS o-cean colour data are assimilated in the model with the technique. Some results are briefly presented.
基金This work was supported jointly by the Typhoon Foundation of Shanghaiby LASC of the Institute of Atmospheric Physics of the Chinese Academy of Sciencesby the National Natural Science Foundation of China under Grant No. 40633030.
文摘In the first paper in this series, a variational data assimilation of ideal tropical cyclone （TC） tracks was performed for the statistical-dynamical prediction model SD-90 by the adjoint method, and a prediction of TC tracks was made with good accuracy for tracks containing no sharp turns. In the present paper, the cases of real TC tracks are studied. Due to the complexity of TC motion, attention is paid to the diagnostic research of TC motion. First, five TC tracks are studied. Using the data of each entire TC track, by the adjoint method, five TC tracks are fitted well, and the forces acting on the TCs are retrieved. For a given TC, the distribution of the resultant of the retrieved force and Coriolis force well matches the corresponding TC track, i.e., when a TC turns, the resultant of the retrieved force and Coriolis force acts as a centripetal force, which means that the TC indeed moves like a particle; in particular, for TC 9911, the clockwise looping motion is also fitted well. And the distribution of the resultant appears to be periodic in some cases. Then, the present method is carried out for a portion of the track data for TC 9804, which indicates that when the amount of data for a TC track is sufficient, the algorithm is stable. And finally, the same algorithm is implemented for TCs with a double-eyewall structure, namely Bilis （2000） and Winnie （1997）, and the results prove the applicability of the algorithm to TCs with complicated mesoscale structures if the TC track data are obtained every three hours.
基金the National Key Programme for Developing Basic Sciences(G1998040907)the Project of Natural Science Foundation of Jiangsu Pr
文摘Theoretical argumentation for so-called suitable spatial condition is conducted by the aid of homotopy framework to demonstrate that the proposed boundary condition does guarantee that the over-specification boundary condition resulting from an adjoint model on a limited-area is no longer an issue, and yet preserve its well-poseness and optimal character in the boundary setting. The ill-poseness of over-specified spatial boundary condition is in a sense, inevitable from an adjoint model since data assimilation processes have to adapt prescribed observations that used to be over-specified at the spatial boundaries of the modeling domain. In the view of pragmatic implement, the theoretical framework of our proposed condition for spatial boundaries indeed can be reduced to the hybrid formulation of nudging filter, radiation condition taking account of ambient forcing, together with Dirichlet kind of compatible boundary condition to the observations prescribed in data assimilation procedure. All of these treatments, no doubt, are very familiar to mesoscale modelers. Key words Variational data assimilation - Adjoint model - Over-specified partial boundary condition This research work is sponsored by the National Key Programme for Developing Basic Sciences (G1998040907), the Project of Natural Science Foundation of Jiangsu Province (BK99020), the President Foundation of Nanjing University (985) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
基金funded by the National Key R&D Program of China (Grant No.2018YFA0605904)。
文摘Melt ponds significantly affect Arctic sea ice thermodynamic processes.The melt pond parameterization scheme in the Los Alamos sea ice model(CICE6.0) can predict the volume,area fraction(the ratio between melt pond area to sea ice area in a model grid),and depth of melt ponds.However,this scheme has some uncertain parameters that affect melt pond simulations.These parameters could be determined through a conventional parameter estimation method,which requires a large number of sensitivity simulations.The adjoint model can calculate the parameter sensitivity efficiently.In the present research,an adjoint model was developed for the CESM(Community Earth System Model) melt pond scheme.A melt pond parameter estimation algorithm was then developed based on the CICE6.0 sea ice model,melt pond adjoint model,and L-BFGS(Limited-memory Broyden-Fletcher-Goldfard-Shanno) minimization algorithm.The parameter estimation algorithm was verified under idealized conditions.By using MODIS(Moderate Resolution Imaging Spectroradiometer)melt pond fraction observation as a constraint and the developed parameter estimation algorithm,the melt pond aspect ratio parameter in CESM scheme,which is defined as the ratio between pond depth and pond area fraction,was estimated every eight days during summertime for two different regions in the Arctic.One region was covered by multi-year ice(MYI) and the other by first-year ice(FYI).The estimated parameter was then used in simulations and the results show that:(1) the estimated parameter varies over time and is quite different for MYI and FYI;(2) the estimated parameter improved the simulation of the melt pond fraction.
文摘This work illustrates the application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2<sup>nd</sup>-CASAM) to a mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response is the value of the model’s state function (particle concentration or particle flux) at a point in phase-space, which would simulate a pointwise measurement of the respective state function. This paradigm model admits exact closed-form expressions for all of the 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to the model’s uncertain parameters and domain boundaries. These closed-form expressions can be used to verify the numerical results of production and/or commercial software, e.g., particle transport codes. Furthermore, this paradigm model comprises many uncertain parameters which have relative sensitivities of identical magnitudes. Therefore, this paradigm model could serve as a stringent benchmark for inter-comparing the performances of all deterministic and statistical sensitivity analysis methods, including the 2<sup>nd</sup>-CASAM.
文摘This work continues the illustrative application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2<sup>nd</sup>-CASAM) to a benchmark mathematical model that can simulate the evolution and/or transmission of particles in a heterogeneous medium. The model response considered in this work is a reaction-rate detector response, which provides the average interactions of particles with the respective detector or, alternatively, the time-average of the concentration of a mixture of substances in a medium. The definition of this model response includes both uncertain boundary points of the benchmark, thereby providing both direct and indirect contributions to the response sensitivities stemming from the boundaries. The exact expressions for the 1<sup>st</sup>- and 2<sup>nd</sup>-order response sensitivities to the boundary and model parameters obtained in this work can serve as stringent benchmarks for inter-comparing the performances of all (deterministic and statistical) sensitivity analysis methods.
基金Acknowledgments. The authors would like to thank Prof. Z. Yuan for her numerous suggestions in the writing of this paper. This work is supported by the National Natural Science Foundation of China (Grant No.40176009), the National Key Programme for Devel
文摘The strong nonlinearity of boundary layer parameterizations in atmospheric and oceanic models can cause difficulty for tangent linear models in approximating nonlinear perturbations when the time integration grows longer. Consequently, the related 4—D variational data assimilation problems could be difficult to solve. A modified tangent linear model is built on the Mellor-Yamada turbulent closure (level 2.5) for 4-D variational data assimilation. For oceanic mixed layer model settings, the modified tangent linear model produces better finite amplitude, nonlinear perturbation than the full and simplified tangent linear models when the integration time is longer than one day. The corresponding variational data assimilation performances based on the adjoint of the modified tangent linear model are also improved compared with those adjoints of the full and simplified tangent linear models.
基金The National Key Basic Research Program of China under contract No. 2005CB422303the International Cooperation Program Project under contract No. 2004DFB02700the National Natural Science Foundation of China under contract No. 40552002
文摘The eddy viscosity of the ocean is an important parameter indicating the small-scale mixing process in the oceanic interior water column. Ekman wind-driven current model and adjoint assimilation technique are used to calculate the vertical profiles of eddy viscosity by fitting model results to the observation data. The data used in the paper include observed wind data and ADCP data obtained at Wenchang Oil Rig on the SCS （the South China Sea） shelf in August 2002. Different simulations under different wind conditions are analyzed to explore how the eddy viscosity develops with varying wind field. The results show that the eddy viscosity endured gradual variations in the range of 10^-3 -10^-2 m^2 /s during the periods of wind changes. The mean eddy viscosity undergoing strong wind could rise by about 25% as compared to the value under weak wind.
基金National Natural Science Foundation of China(Nos.51676003,51976183)National Science and Technology Major Project of China(No.J2019II-0012-0032)。
文摘The design optimization taking into account the impact of uncertainties favors improving the robustness of the design.A Surrogate-Assisted Gradient-Based(SAGB)method for the robust aerodynamic design optimization of turbomachinery blades considering large-scale uncertainty is introduced,verified and validated in the study.The gradient-based method is employed due to its high optimization efficiency and any one surrogate model with sufficient response accuracy can be employed to quantify the nonlinear performance changes.The gradients of objective performance function to the design parameters are calculated first for all the training samples,from which the gradients of cost function can be fast determined.To reveal the high efficiency and high accuracy of SAGB on gradient calculation,the number of flow computations needed is evaluated and compared with three other methods.Through the aerodynamic design optimization of a transonic turbine cascade minimizing total pressure loss at the outlet,the SAGB-based gradients of the base and optimized blades are compared with those obtained by the Monte Carlo-assisted finite difference method.Moreover,the results of both the robust and deterministic aerodynamic design optimizations are presented and compared to demonstrate the practicability of SAGB on improving the aerodynamic robustness of turbomachinery blades.
基金This work was supported by the National Numerical Wind tunnel Project(Grant NNW2018-ZT1B01)the National Natural Science Foundation of China(Grant 91852115).
文摘Adjoint method is widely used in aerodynamic design because only once solution of flow field is required for it to obtain the gradients of all design variables. However, the computational cost of adjoint vector is approximately equal to that of flow computation. In order to accelerate the solution of adjoint vector and improve the efficiency of adjoint-based optimization, machine learning for adjoint vector modeling is presented. Deep neural network (DNN) is employed to construct the mapping between the adjoint vector and the local flow variables. DNN can efficiently predict adjoint vector and its generalization is examined by a transonic drag reduction of NACA0012 airfoil. The results indicate that with negligible computational cost of the adjoint vector, the proposed DNN-based adjoint method can achieve the same optimization results as the traditional adjoint method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.41775027 and 41405025)
文摘Constructing sophisticated refractivity models is one of the key problems for the RFC(refractivity from clutter)technology. If prior knowledge of the local refractivity environment is available, more accurate parameterized model can be constructed from the statistical information, which in turn can be used to improve the quality of the local refractivity retrievals. The validity of this proposal was demonstrated by range-dependent refractivity profile inversions using the adjoint parabolic equation method to the Wallops’ 98 experimental data.
文摘The emerging push of the differentiable programming paradigm in scientific computing is conducive to training deep learning turbulence models using indirect observations.This paper demonstrates the viability of this approach and presents an end-to-end differentiable framework for training deep neural networks to learn eddy viscosity models from indirect observations derived from the velocity and pressure fields.The framework consists of a Reynolds-averaged Navier–Stokes(RANS)solver and a neuralnetwork-represented turbulence model,each accompanied by its derivative computations.For computing the sensitivities of the indirect observations to the Reynolds stress field,we use the continuous adjoint equations for the RANS equations,while the gradient of the neural network is obtained via its built-in automatic differentiation capability.We demonstrate the ability of this approach to learn the true underlying turbulence closure when one exists by training models using synthetic velocity data from linear and nonlinear closures.We also train a linear eddy viscosity model using synthetic velocity measurements from direct numerical simulations of the Navier–Stokes equations for which no true underlying linear closure exists.The trained deep-neural-network turbulence model showed predictive capability on similar flows.
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.
基金supported by the National Natural Science Foundation of China(12071407,12171193)the Natural Science Foundation of Hubei Province(2024AFB170)。
文摘Economic development has caused a lot of environmental problems,in turn,environmental pollution restricts economic development.Considering the influence of wind direction and speed,temperature and humidity on pollutants,as well as the influence of epidemic,war and exchange rate on economic development.In this paper,we develop a stochastic economic-environment model with pollution control strategies.Furthermore,sufficient and necessary conditions for the near-optimality are established.Finally,we perform some numerical simulations to demonstrate the correctness of the theoretical results,which shows that some control strategies could decrease the environmental pollution,and therefore,could alleviate economic losses caused by environmental pollution.
文摘An HIV model was considered. The parameters of the model are estimated by adjoint dada assimilation method. The results showed the method is valid. This method has potential application to a wide variety of models in biomathematics.
文摘This work highlights the unparalleled efficiency of the “n<sup>th</sup>-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-FASAM-N) by considering the well-known Nordheim-Fuchs reactor dynamics/safety model. This model describes a short-time self-limiting power excursion in a nuclear reactor system having a negative temperature coefficient in which a large amount of reactivity is suddenly inserted, either intentionally or by accident. This nonlinear paradigm model is sufficiently complex to model realistically self-limiting power excursions for short times yet admits closed-form exact expressions for the time-dependent neutron flux, temperature distribution and energy released during the transient power burst. The n<sup>th</sup>-FASAM-N methodology is compared to the extant “n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (n<sup>th</sup>-CASAM-N) showing that: (i) the 1<sup>st</sup>-FASAM-N and the 1<sup>st</sup>-CASAM-N methodologies are equally efficient for computing the first-order sensitivities;each methodology requires a single large-scale computation for solving the “First-Level Adjoint Sensitivity System” (1<sup>st</sup>-LASS);(ii) the 2<sup>nd</sup>-FASAM-N methodology is considerably more efficient than the 2<sup>nd</sup>-CASAM-N methodology for computing the second-order sensitivities since the number of feature-functions is much smaller than the number of primary parameters;specifically for the Nordheim-Fuchs model, the 2<sup>nd</sup>-FASAM-N methodology requires 2 large-scale computations to obtain all of the exact expressions of the 28 distinct second-order response sensitivities with respect to the model parameters while the 2<sup>nd</sup>-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3<sup>rd</sup>-FASAM-N methodology is even more efficient than the 3<sup>rd</sup>-CASAM-N methodology: only 2 large-scale computations are needed to obtain the exact expressions of the 84 distinct third-order response sensitivities with respect to the Nordheim-Fuchs model’s parameters when applying the 3<sup>rd</sup>-FASAM-N methodology, while the application of the 3<sup>rd</sup>-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-CASAM-N methodologies are the most practical methodologies for computing response sensitivities of any order comprehensively and accurately, overcoming the curse of dimensionality in sensitivity analysis.
文摘This work presents the “n<sup>th</sup>-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “n<sup>th</sup>-FASAM-N”), which will be shown to be the most efficient methodology for computing exact expressions of sensitivities, of any order, of model responses with respect to features of model parameters and, subsequently, with respect to the model’s uncertain parameters, boundaries, and internal interfaces. The unparalleled efficiency and accuracy of the n<sup>th</sup>-FASAM-N methodology stems from the maximal reduction of the number of adjoint computations (which are considered to be “large-scale” computations) for computing high-order sensitivities. When applying the n<sup>th</sup>-FASAM-N methodology to compute the second- and higher-order sensitivities, the number of large-scale computations is proportional to the number of “model features” as opposed to being proportional to the number of model parameters (which are considerably more than the number of features).When a model has no “feature” functions of parameters, but only comprises primary parameters, the n<sup>th</sup>-FASAM-N methodology becomes identical to the extant n<sup>th</sup> CASAM-N (“n<sup>th</sup>-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-CASAM-N methodologies are formulated in linearly increasing higher-dimensional Hilbert spaces as opposed to exponentially increasing parameter-dimensional spaces thus overcoming the curse of dimensionality in sensitivity analysis of nonlinear systems. Both the n<sup>th</sup>-FASAM-N and the n<sup>th</sup>-CASAM-N are incomparably more efficient and more accurate than any other methods (statistical, finite differences, etc.) for computing exact expressions of response sensitivities of any order with respect to the model’s features and/or primary uncertain parameters, boundaries, and internal interfaces.
