Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): I. Mathematical Framework

This work presents the “nth-Order Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (abbreviated as “nth-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 nth-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 nth-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 nth-FASAM-N methodology becomes identical to the extant nth CASAM-N (“nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems”) methodology. Both the nth-FASAM-N and the nth-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 nth-FASAM-N and the nth-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.

Introducing the nth-Order Features Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-FASAM-N): II. Illustrative Example

This work highlights the unparalleled efficiency of the “nth-Order Function/ Feature Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-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 nth-FASAM-N methodology is compared to the extant “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) showing that: (i) the 1st-FASAM-N and the 1st-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” (1st-LASS);(ii) the 2nd-FASAM-N methodology is considerably more efficient than the 2nd-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 2nd-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 2nd-CASAM-N methodology requires 7 large-scale computations for obtaining these 28 second-order sensitivities;(iii) the 3rd-FASAM-N methodology is even more efficient than the 3rd-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 3rd-FASAM-N methodology, while the application of the 3rd-CASAM-N methodology requires at least 22 large-scale computations for computing the same 84 distinct third-order sensitivities. Together, the nth-FASAM-N and the nth-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.

STUDY ON THE ADJOINT METHOD IN DATA ASSIMILATION AND THE RELATED PROBLEMS

It is not reasonable that one can only use the adjoint of model in data assimilation. The simulated numerical experiment shows that for the tidal model, the result of the adjoint of equation is almost the same as that of the adjoint of model: the averaged absolute difference of the amplitude between observations and simulation is less than 5.0 cm and that of the phase-lag is less than 5.0°. The results are both in good agreement with the observed M2 tide in the Bohai Sea and the Yellow Sea. For comparison, the traditional methods also have been used to simulate M2 tide in the Bohai Sea and the Yellow Sea. The initial guess values of the boundary conditions are given first, and then are adjusted to acquire the simulated results that are as close as possible to the observations. As the boundary conditions contain 72 values, which should be adjusted and how to adjust them can only be partially solved by adjusting them many times. The satisfied results are hard to acquire even gigantic efforts are done. Here, the automation of the treatment of the open boundary conditions is realized. The method is unique and superior to the traditional methods. It is emphasized that if the adjoint of equation is used, tedious and complicated mathematical deduction can be avoided. Therefore the adjoint of equation should attract much attention.

提高基于Adjoint方法翼型优化设计鲁棒性的研究

通过引入线搜索方法,提高了基于Adjoint方法翼型优化设计的鲁棒性。针对给定的目标函数,推导了贴体坐标系下相应的Adjoint方程与边界条件的具体表达形式,以及梯度表达式。通过数值求解流动控制方程和Adjoint方程,得到目标函数对设计变量的梯度,并采用线搜索方法获得最优步长,由此提高了优化算法的鲁棒性。算例表明,线搜索方法可以自动寻找最优的步长,有效解决了传统的取常数步长优化步长选取受到限制,优化结果受步长影响的问题,使得优化方法对步长的依赖性变小,提高了基于Adjoint方法翼型优化设计的鲁棒性。

Assimilated Tidal Results of Tide Gauge and TOPEX/POSEIDON Data over the China Seas Using a Variational Adjoint Approach with a Nonlinear Numerical Model

In order to obtain an accurate tide description in the China Seas, the 2-dimensional nonlinear numerical Princeton Ocean Model （POM） is employed to incorporate in situ tidal measurements both from tide gauges and TOPEX/POSEIDON （T/P） derived datasets by means of the variational adjoint approach in such a way that unknown internal model parameters, bottom topography, friction coefficients and open boundary conditions, for example, are adjusted during the process. The numerical model is used as a forward model. After the along-track T/P data are processed, two classical methods, i.e. harmonic and response analysis, are implemented to estimate the tide from such datasets with a domain covering the model area extending from 0° to 41°N in latitude and from 99°E to 142°E in longitude. And the results of these two methods are compared and interpreted. The numerical simulation is performed for 16 major constituents. In the data assimilation experiments, three types of unknown parameters （water depth, bottom friction and tidal open boundary conditions in the model equations） are chosen as control variables. Among the various types of data assimilation experiments, the calibration of water depth brings the most promising results. By comparing the results with selected tide gauge data, the average absolute errors are decreased from 7.9 cm to 6.8 cm for amplitude and from 13.0° to 9.0° for phase with respect to the semidiurnal tide M2 constituent, which is the largest tidal constituent in the model area. After the data assimilation experiment is performed, the comparison between model results and tide gauge observation for water levels shows that the RMS errors decrease by 9 cm for a total of 14 stations, mostly selected along the coast of China's Mainland, when a one-month period is considered, and the correlation coefficients improve for most tidal stations among these stations.

Estimation of eddy viscosity on the South China Sea shelf with adjoint assimilation method

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.

Adjoint-based Sensitivity Analysis of a Mesoscale Low on the Mei-yu Front and Its Implications for Adaptive Observation

An adjoint sensitivity analysis of one mesoscale low on the mei-yu Front is presented in this paper. The sensitivity gradient of simulation error dry energy with respect to initial analysis is calculated. And after verifying the ability of a tangent linear and adjoint model to describe small perturbations in the nonlinear model, the sensitivity gradient analysis is implemented in detail. The sensitivity gradient with respect to different physical fields are not uniform in intensity, simulation error is most sensitive to the vapor mixed ratio. The localization and consistency are obvious characters of horizontal distribution of the sensitivity gradient, which is useful for the practical implementation of adaptive observation. The sensitivity region tilts to the northwest with height increasing; the singular vector calculation proves that this tilting characterizes a quick-growing structure, which denotes that using the leading singular vectors to decide the adaptive observation region is proper. When connected with simulation of a mesoscale low on the mei-yu Front, the sensitivity gradient has the following physical characters： the obvious sensitive region is mesoscale, concentrated in the middle-upper troposphere, and locates around the key system; and the sensitivity gradient of different physical fields correlates dynamically.

Mesoscale Moist Adjoint Sensitivity Study of a Mei-yu Heavy Rainfall Event

The mesoscale moist adjoint sensitivities related to the initiation of mesoscale convective systems （MCSs） are evaluated for a mei-yu heavy rainfall event. The sensitivities were calculated on a realistic background gained from a four-dimensional variational data assimilation of precipitation experiment to make the sensitivity computation possible and reasonable within a strong moist convective event at the mesoscale. The results show that the computed sensitivities at the mesoscale were capable of capturing the factors affecting MCS initiation. The sensitivities to the initial temperature and moisture are enhanced greatly by diabatic processes, especially at lower levels, and these sensitivities are much larger than those stemming from the horizontal winds, which implies that initiation of MCSs is more sensitive to low-level temperature and moisture perturbations rather than the horizontal winds. Moreover, concentration of sensitivities at low levels reflects the characteristics of the mei-yu front. The results provide some hints about how to improve quantitative precipitation forecasts of mei-yu heavy rainfall, such as by conducting mesoscale targetted observations via the adjoint-based method to reduce the low-level errors in the initial temperature and moisture.

Multi-objective optimization strategies using adjoint method and game theory in aerodynamics

There are currently three different game strategies originated in economics： （1） Cooperative games （Pareto front）, （2） Competitive games （Nash game） and （3） Hierarchical games （Stackelberg game）. Each game achieves different equilibria with different performance, and their players play different roles in the games. Here, we introduced game concept into aerodynamic design, and combined it with adjoint method to solve multi- criteria aerodynamic optimization problems. The performance distinction of the equilibria of these three game strategies was investigated by numerical experiments. We computed Pareto front, Nash and Stackelberg equilibria of the same optimization problem with two conflicting and hierarchical targets under different parameterizations by using the deterministic optimization method. The numerical results show clearly that all the equilibria solutions are inferior to the Pareto front. Non-dominated Pareto front solutions are obtained, however the CPU cost to capture a set of solutions makes the Pareto front an expensive tool to the designer.

典型李超代数的极大Adjoint二次迷向空间的维数

对复数域上的典型李超代数定义了极大Adjoint二次迷向空间,这是典型李超代数的内蕴特征;对每一类典型李超代数的极大Adjoint二次迷向空间做了细致的刻画,并确定它们各自的维数.

Adjoint Sensitivity Experiments of a Meso-β-scale Vortex in the Middle Reaches of the Yangtze River

A relatively independent and small-scale heavy rainfall event occurred to the south of a slow eastwardmoving meso-α-scale vortex. The analysis shows that a meso-β-scale system is heavily responsible for the intense precipitation. An attempt to simulate it met with some failures. In view of its small scale, short lifetime and relatively sparse observations at the initial time, an adjoint model was used to examine the sensitivity of the meso-β-scale vortex simulation with respect to initial conditions. The adjoint sensitivity indicates how small perturbations of initial model variables anywhere in the model domain can influence the central vorticity of the vortex. The largest sensitivity for both the wind and temperature perturbation is located below 700 hPa, especially at the low level. The largest sensitivity for the water vapor perturbation is located below 500 hPa, especially at the middle and low levels. The horizontal adjoint sensitivity for all variables is mainly located toward the upper reaches of the Yangtze River with respect to the simulated meso-β-scale system in Hunan and Jiangxi provinces with strong locality. The sensitivity shows that warm cyclonic perturbations in the upper reaches can have a great effect on the development of the meso-β-scale vortex. Based on adjoint sensitivity, forward sensitivity experiments were conducted to identify factors influencing the development of the meso-β-scale vortex and to explore ways of improving the prediction. A realistic prediction was achieved by using adjoint sensitivity to modify the initial conditions and implanting a warm cyclone at the initial time in the upper reaches of the river with respect to the meso-β-scale vortex, as is commonly done in tropical cyclone prediction.

Inversion Study on Pollutant Discharges in the Bohai Sea with the Adjoint Method

The adjoint method is presented which could be used to estimate the time-varying contamination concentration(CC) from pollution source(PS). Then the pollutant flux is calculated. In order to guarantee the continuity of pollutant distribution and make the calculated results more accurate, the independent point(IP) scheme is proposed. The contamination concentrations(CCs) at some time steps are selected as the independent points(IPs), and only CCs at these IPs are optimized while CCs at other points are calculated through linear interpolation of the independent CCs. In twin numerical experiments, all the given distributions are successfully inverted with the adjoint method. The cost functions and the mean absolute errors(MAEs) in concentrations and pollutant fluxes decrease greatly after assimilation, and the cost functions are reduced by about 5 orders of magnitude compared with their initial values. The results indicate that the adjoint method is computationally efficient to recover CCs from PS. It is easier to invert the given distribution which is less complex. The inversion efficiency with IP scheme is raised compared to that without this scheme. The IP scheme is significant for the inversion result, in which appropriate IP number could yield better inversion results. More work will be done to apply this method to real experiment.

Adjoint Matching Condition for Parameterized Discontinuities—A Derivation Using Lagrangian-form Costfunction

The generalized adjoint property and adjoint matching condition for systems that contain discontinuous on/off switches are derived by a perturbation analysis of the Lagranging-form costfunction.

Design optimization of unsteady airfoils with continuous adjoint method

In this paper, a new unsteady aerodynamic design method is presented based on the Navier-Stokes equations and a continuous adjoint approach. A basic framework of time-accurate unsteady airfoil optimization which adopts time-averaged aerodynamic coefficients as objective functions is presented. The time-accurate continuous adjoint equation and its boundary conditions are derived. The flow field and the adjoint equation are simulated numerically by the finite volume method （FVM）. Feasibility and accuracy of the approach are perfectly validated by the design optimization results of the plunging NACA0012 airfoil.

An Application of the Adjoint Method to a Statistical-Dynamical Tropical-Cyclone Prediction Model （SD-90）Ⅱ：Real Tropical Cyclone Cases

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.

Theoretical Aspect of Suitable Spatial Boundary Condition Specified for Adjoint Model on Limited Area

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.

Second-Order Adjoint Sensitivity Analysis Methodology for Computing Exactly Response Sensitivities to Uncertain Parameters and Boundaries of Linear Systems: Mathematical Framework

This work presents the “Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology (2nd-CASAM)” for the efficient and exact computation of 1st- and 2nd-order response sensitivities to uncertain parameters and domain boundaries of linear systems. The model’s response (i.e., 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 2nd-CASAM requires a single large-scale computation using the First-Level Adjoint Sensitivity System (1st-LASS) for obtaining all of the first-order response sensitivities, and at most N large-scale computations using the Second-Level Adjoint Sensitivity System (2nd-LASS) for obtaining exactly all of the second-order response sensitivities. In contradistinction, forward other methods would require (N2/2 + 3 N/2) large-scale computations for obtaining all of the first- and second-order sensitivities. This work also shows that constructing and solving the 2nd-LASS requires very little additional effort beyond the construction of the 1st-LASS needed for computing the first-order sensitivities. Solving the equations underlying the 1st-LASS and 2nd-LASS requires the same computational solvers as needed for solving (i.e., “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 1st-LASS and the 2nd-LASS. Since neither the 1st-LASS nor the 2nd-LASS involves any differentials of the operators underlying the original system, the 1st-LASS is designated as a “first-level” (as opposed to a “first-order”) adjoint sensitivity system, while the 2nd-LASS is designated as a “second-level” (rather than a “second-order”) adjoint sensitivity system. Mixed second-order response sensitivities involving boundary parameters may arise from all source terms of the 2nd-LASS that involve the imprecisely known boundary parameters. Notably, the 2nd-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.

Fourth-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: II. Mathematical Expressions and CPU-Time Comparisons for Computing 4^th-Order Sensitivities

This work extends to fourth-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. Previous works showed that the third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections are far larger than the corresponding 1st-order and 2nd-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark’s response. This finding has motivated the development of the original 4th-order formulas presented in this work, which are valid not only for the PERP benchmark but can also be used for computing the 4th-order sensitivities of response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of the largest fourth-order sensitivities of the PERP benchmark’s response to the total microscopic cross sections, and use them for a pioneering fourth-order uncertainty analysis of the PERP benchmark’s response.

Third Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Sensitivities

This work presents the results of the exact computation of (180)3 = 5,832,000 third-order mixed sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental benchmark with respect to the benchmark’s 180 microscopic total cross sections. This computation was made possible by applying the Third-Order Adjoint Sensitivity Analysis Methodology developed by Cacuci. The numerical results obtained in this work revealed that many of the 3rd-order sensitivities are significantly larger than their corresponding 1st- and 2nd-order ones, which is contrary to the widely held belief that higher-order sensitivities are all much smaller and hence less important than the first-order ones, for reactor physics systems. In particular, the largest 3rd-order relative sensitivity is the mixed sensitivity of the PERP leakage response with respect to the lowest energy-group (30) total cross sections of 1H (“isotope 6”) and 239Pu (“isotope 1”). These two isotopes are shown in this work to be the two most important parameters affecting the PERP benchmark’s leakage response. By comparison, the largest 1st-order sensitivity is that of the PERP leakage response with respect to the lowest energy-group total cross section of isotope 1H, having the value , while the largest 2nd-order sensitivity is . The 3rd-order sensitivity analysis presented in this work is the first ever such analysis in the field of reactor physics. The consequences of the results presented in this work on the uncertainty analysis of the PERP benchmark’s leakage response will be presented in a subsequent work.

Illustrative Application of the 2^nd-Order Adjoint Sensitivity Analysis Methodology to a Paradigm Linear Evolution/Transmission Model: Point-Detector Response

This work illustrates the application of the “Second Order Comprehensive Adjoint Sensitivity Analysis Methodology” (2nd-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 1st- and 2nd-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 2nd-CASAM.