Reparameterization for statistical state estimation applied to differential equations

The ensemble Kalman filter is a widely applied data assimilation technique useful for improving the forecast of computational models. The main computational cost of the ensemble Kalman filter comes from the numerical integration of each ensemble member forward in time. When the computational model involves a partial differential equation, the degrees of freedom of the solution in the discretization of the spatial domain are oftentimes used for the representation of the state of the system, and the filter is applied to this state vector. We propose a method of approximating the state of a partial differential equation in a representation space developed separately from the numerical method. This representation space represents a reparameterization of the state vector and can be chosen to retain desirable physical features of the solutions. We apply the ensemble Kalman filter to this representation of the state, and numerically demonstrate that acceptable results are obtained with substantially smaller ensemble sizes.     

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

Data assimilation is an iterative approach to the problem of estimating the state of a dynamical system using both current and past observations of the system together with a model for the system’s time evolution. Rather than solving the problem from scratch each time new observations become available, one uses the model to “forecast” the current state, using a prior state estimate (which incorporates information from past data) as the initial condition, then uses current data to correct the prior forecast to a current state estimate. This Bayesian approach is most effective when the uncertainty in both the observations and in the state estimate, as it evolves over time, are accurately quantified. In this article, we describe a practical method for data assimilation in large, spatiotemporally chaotic systems. The method is a type of “ensemble Kalman filter”, in which the state estimate and its approximate uncertainty are represented at any given time by an ensemble of system states. We discuss both the mathematical basis of this approach and its implementation; our primary emphasis is on ease of use and computational speed rather than improving accuracy over previously published approaches to ensemble Kalman filtering. We include some numerical results demonstrating the efficiency and accuracy of our implementation for assimilating real atmospheric data with the global forecast model used by the US National Weather Service.     

New data assimilation system DNDAS for high-dimensional models

The tangent linear(TL) models and adjoint(AD) models have brought great difficulties for the development of variational data assimilation system. It might be impossible to develop them perfectly without great efforts, either by hand, or by automatic differentiation tools. In order to break these limitations, a new data assimilation system, dual-number data assimilation system(DNDAS), is designed based on the dual-number automatic differentiation principles. We investigate the performance of DNDAS with two different optimization schemes and subsequently give a discussion on whether DNDAS is appropriate for high-dimensional forecast models. The new data assimilation system can avoid the complicated reverse integration of the adjoint model, and it only needs the forward integration in the dual-number space to obtain the cost function and its gradient vector concurrently. To verify the correctness and effectiveness of DNDAS, we implemented DNDAS on a simple ordinary differential model and the Lorenz-63 model with different optimization methods. We then concentrate on the adaptability of DNDAS to the Lorenz-96 model with high-dimensional state variables. The results indicate that whether the system is simple or nonlinear, DNDAS can accurately reconstruct the initial condition for the forecast model and has a strong anti-noise characteristic. Given adequate computing resource, the quasi-Newton optimization method performs better than the conjugate gradient method in DNDAS.     

一类离散时间广义系统的迭代学习控制

针对一类离散时间广义系统,提出了一种离散迭代学习控制算法.首先,通过非奇异变换将离散时间广义系统分解为正常离散状态方程和代数方程的形式.然后,利用上一次迭代学习获得的前一时刻误差和当前时刻误差来修正上一次的控制量,从而获得下一次迭代学习的新控制量,并对算法的收敛性进行了理论证明,给出了算法收敛的充分条件.研究结果表明,所提算法能够在有限时间区间内实现系统状态对期望状态的完全跟踪.最后,通过仿真算例进一步验证了所提算法的有效性.     

Topics in data assimilation: Stochastic processes

Stochastic models with varying degrees of complexity are increasingly widespread in the oceanic and atmospheric sciences. One application is data assimilation, i.e., the combination of model output with observations to form the best picture of the system under study. For any given quantity to be estimated, the relative weights of the model and the data will be adjusted according to estimated model and data error statistics, so implementation of any data assimilation scheme will require some assumption about errors, which are considered to be random. For dynamical models, some assumption about the evolution of errors will be needed. Stochastic models are also applied in studies of predictability.

The formal theory of stochastic processes was well developed in the last half of the twentieth century. One consequence of this theory is that methods of simulation of deterministic processes cannot be applied to random processes without some modification. In some cases the rules of ordinary calculus must be modified.

The formal theory was developed in terms of mathematical formalism that may be unfamiliar to many oceanic and atmospheric scientists. The purpose of this article is to provide an informal introduction to the relevant theory, and to point out those situations in which that theory must be applied in order to model random processes correctly.     

Variational assimilation in combination with the regularization method for sea level pressure retrieval from QuikSCAT scatterometer data I: Theoretical frame construction

A new method of constructing a sea level pressure field from satellite microwave scatterometer measurements is presented. It is based on variational assimilation in combination with a regularization method using geostrophic vorticity to construct a sea level pressure field from scatterometer data that are given in this paper, which offers a new idea for the application of scatterometer measurements. Firstly, the geostrophic vorticity from the scatterometer data is computed to construct the observation field, and the vorticity field in an area and the sea level pressure on the borders are assimilated. Secondly, the gradient of sea level pressure (semi-norm) is used as the stable functional to educe the adjoint system, the adjoint boundary condition and the gradient of the cost functional in which a weight parameter is introduced for the harmony of the system and the Tikhonov regularization techniques in inverse problem are used to overcome the ill-posedness of the assimilation. Finally, the iteration method of the sea level pressure field is developed.     

Data assimilation by field alignment

Classical formulations of data assimilation, whether sequential, ensemble-based or variational, are amplitude adjustment methods. Such approaches can perform poorly when forecast locations of weather systems are displaced from their observations. Compensating position errors by adjusting amplitudes can produce unacceptably “distorted” states, adversely affecting analysis, verification and subsequent forecasts.

There are many sources of position error. It is non-trivial to decompose position error into constituent sources and yet correcting position errors during assimilation can be essential for operationally predicting strong, localized weather events such as tropical cyclones.

In this paper, we propose a method that accounts for both position and amplitude errors. The proposed method assimilates observations in two steps. The first step is field alignment, where the current model state is aligned with observations by adjusting a continuous field of local displacements, subject to certain constraints. The second step is amplitude adjustment, where contemporary assimilation approaches are used. We demonstrate with 1D and 2D examples how applying field alignment produces better analyses with sparse and uncertain observations.     

An iterative method to solve acoustic scattering problems using a boundary integral equation

In this work, a simple iterative method to solve the acoustic scattering/radiation problems using the boundary integral equation (BIE) formulation is presented. The operator equation obtained in the BIE formulation is converted into a matrix equation using the well-known method of moments solution procedure. The present method requires much fewer mathematical operations per iteration when compared to other available iterative methods. Further, the present iterative method can easily handle multiple incident fields, a highly desirable feature not available in any other iterative method, much the same way as direct solution techniques. Several numerical examples are presented to illustrate the efficiency and accuracy of the method.     

基于Bi PLS结合Si PLS的组合权值COD浓度预测模型

水体中过高浓度的有机物含量危害巨大,不仅会造成严重的环境污染,而且危害人类身体健康,传统化学法检测水体化学需氧量(COD)的步骤繁琐且时效性差,不利于水体中COD的快速定量检测。针对这些问题,提出了一种将紫外光谱与组合权值模型相结合的快速定量检测COD方法,该组合权值模型是基于反向区间偏最小二乘法(BiPLS)结合组合区间偏最小二乘法(SiPLS)算法对紫外光谱的特征子区间筛选组合,然后依据特征子区间的权值建立的预测模型。首先按照一定的浓度梯度配制45份COD标准液样本,通过实验获取标准液的紫外光谱数据;对获取到的COD紫外光谱数据做一阶导数和S-G滤波(Savitzky-Golay)的预处理,消除基线漂移和环境干扰噪声;应用SPXY(Sample set partitioning based on jiont X-Y)算法将实验样本数据组划分成校正集和预测集。然后基于BiPLS算法对全光谱区间进行波长筛选,在BiPLS筛选过程中,目标区间的划分数量会对建模产生较大影响,于是对子区间划分数量进行优化,把子区间分成15~25个,在不同区间数下都进行偏最小二乘(PLS)建模,通过交互验证均方根误差(RMSECV)来筛选最优子区间数,得到区间数为18时,模型效果最佳。从18个波长区间筛选出了6个特征波长子区间,入选的子区间为2,1,3,11,7和6,对应波长为234~240,262~268,269~275,290~296,297~303和304~310 nm,这6个特征波长区间涵盖了大量的光谱信息,对最终预测模型的贡献度大;接下来通过SiPLS算法对这6个初选区间进行进一步的筛选组合,采用不同的组合数构建不同特征区间上的PLS模型,在相同组合数下,筛选出一个区间组合数最优的结果,对比不同组合数下预测模型的误差与相关性,将6个区间筛选组合为3个特征波长区间,分别为234~240,262~275和290~310 nm,这三个特征区间最佳因子数分别为4,4和3。对传统SiPLS的特征区间组合方法进行改进,基于权值的大小来对这3个特征区间进行线性组合,代替过去特征区间直接组合的方法。通过权值公式计算出这3个特征区间的权重大小分别为0.509,0.318和0.173,最终建立线性组合权值COD浓度预测模型。为了验证组合权重预测模型的精度,另外建立了全波长范围内的PLS预测模型、单个特征波长区间的PLS预测模型、直接组合特征波长区间的PLS模型,并使用评价参数相关系数的平方(R2)、预测值与真实浓度值的均方根误差(RMSEP)和预测回收率(T)来对模型评价。验证结果表明,相比其他预测模型,组合权值模型相关系数的平方达到了0.999 7,明显优于直接组合特征区间建模的0.968 0,预测均方根误差为0.532,比直接组合特征区间的预测模型误差降低了29.3%,预测回收率为96.4%~103.1%,显著地提高了预测精度。该方法简单可行,不会产生二次污染,可为在线监测水体中COD浓度提供一定的技术支持。     

A framework for iterative analysis of non-classically damped dynamical systems

In this paper, we propose a general iterative framework to solve the dynamic problem for linear systems with non-classical viscous damping. A systematic approach is used to derive families of stationary iterative schemes that, as an instance of particular interest, decouple the equations of motion for numerical study of the system response. For such schemes, we present a detailed convergence analysis and propose several solution strategies suitable for a broad class of systems. These techniques are based on spectral analysis of particular iteration matrices arising in the derivation and aim at optimizing the convergence performance of the method. We demonstrate that the proposed systematic framework, based on a novel application of the homotopy analysis method, generalizes iterative schemes previously reported in the literature and, importantly, provides a unified perspective for the study of iterative solutions of dynamic problems. Further, we establish a connection between our results and the theory of iterative schemes for algebraic linear systems, thus providing insights on convergence results and applicability of the method. Numerical examples illustrate the effectiveness of the approach and indicate future research directions.     

PREDICTION OF DYNAMIC CHARACTERISTICS USING UPDATED FINITE ELEMENT MODELS

Model updating techniques are used to update a finite element model of a structure so that an updated model predicts the dynamics of a structure more accurately. The application of such an updated model in dynamic design demands that it also predict the effects of structural modifications with a reasonable accuracy. This paper deals with updating of a finite element model of a structure and its subsequent use for predicting the effects of structural modifications. Updated models have been obtained by a direct model updating method and by an iterative method of model updating based on the frequency response function (FRF) data. The suitability of updated models for predicting the effect of structural modifications is evaluated by some computer and laboratory experiments. First a study is performed using a simulated fixed-fixed beam. Cases of complete, incomplete and noisy data are considered. Updated models are obtained by the direct and the FRF-based method in each of these cases. These models are then used for predicting the changes in the dynamic characteristics brought about due to a mass and a beam modification. The simulated study is followed by a study involving actual measured data for the case of an F-shape test structure. The updated finite element models for this structure are obtained again by the direct and the FRF-based method. Structural modifications in terms of mass and beam modifications are then introduced to evaluate the updated model for its usefulness in dynamic design. It is found that the predictions based on the iterative method based updated model are reasonably accurate and, therefore, this updated model can be used with reasonable accuracy to perform dynamic design. The predictions on the basis of the direct method based updated model are found to be reasonably accurate in the lower portion of the updating frequency range but the predictions are in a significant error in the remaining portion of the updating frequency range. It is concluded that the updated models that are closer to the structure physically are likely to perform better in predicting the effects of structural modification.     

Analysis of multiple scattering from two-dimensional dielectric sea surface with iterative Kirchhoff approximation

<正>An iterative method in the Kirchhoff approximation is proposed for high frequency multiple electromagnetic scattering from two-dimensional dielectric sea surface.The multiple interaction of the scattering field is characterized with the corrected electromagnetic currents of the wind-driven sea surface.The actual surface currents are approximated with the iterative solution of the corrected currents.A newly developed sea spectrum,Elfouhaily spectrum,is utilized to build the sea surface model.The shadowing correction is improved by the Depth-Buffer algorithm.The validity of the iterative Kirchhoff approximation is verified by the agreement of backscattering coefficients with the measured data.     

Simultaneous estimation of unwrapped phase and phase derivative from a closed fringe pattern

We propose a new approach for the direct estimation of the unwrapped phase from a single closed fringe pattern. The fringe analysis is performed along a given row/column at a time by approximating the phase with a weighted linear combination of linearly independent basis functions. Gaussian radial basis functions with equally distributed centers and a fixed variance are considered for the phase approximation. A state space model is defined with the weights of the basis functions as the state vector elements. Extended Kalman filter is effectively utilized for the accurate state estimation. A fringe density estimation based criteria is established to select whether the phase estimation is performed in a row by row or column by column manner. In the seed row/column decided based on this criteria, the optimal basis dimension is computed. The proposed method effectively renders itself in the simultaneous estimation of the phase and the phase derivative. The proposed phase modeling approach also allows us to successfully demodulate the low density fringe patterns. Simulation and experimental results validate the practical applicability of the proposed method.     

Model calculations in reconstructions of measured fields

The state of technological systems, such as reactions in a confined volume, are usually monitored with sensors within as well as outside the volume. To achieve the level of precision required by regulators, these data often need to be supplemented with the solution to a mathematical model of the process. The present work addresses an observed, and until now unexplained, convergence problem in the iterative solution in the application of the finite element method to boundary value problems. We use point group theory to clarify the cause of the non-convergence, and give rule problems. We use the appropriate and consistent orders of approximation on the boundary and within the volume so as to avoid non-convergence.     

A completely iterative method for the infinite domain electrostatic problem with nonlinear dielectric media

We present an iterative method for the solution of the exterior all-space electrostatic problem for nonlinear dielectric media. The electric potential is specified on interior boundaries and the electric field decays at infinity. Our approach uses a natural variational formulation based on the total energy of the nonlinear dielectric medium subject to boundary conditions. The problem is decomposed into an exterior calculation and an interior calculation with the boundary-specified electric potentials imposed as constraints between them. Together, these enable an iterative method that is based on the variational formulation. In contrast to direct solution of the electrostatic problems, we avoid the construction, storage and solution of dense and large linear systems. This provides important advantages for multiphysics problems that couple the linear electrostatic Poisson problem to nonlinear physics: the latter necessarily involves iterative approaches, and our approach replaces a large number of direct solves for the electrostatics with an iterative algorithm that can be coupled to the iterations of the nonlinear problem. We present examples applying the method to inhomogeneous, anisotropic nonlinear dielectrics. A key advantage of our variational formulation is that we require only the free-space, isotropic, homogeneous Greens function for all these settings.     

Enhancement of electromagnetically induced transparency in room temperature alkali metal vapor

Electromagnetically induced transparency (EIT) has led to several quantum optics effects such as lasing without inversion or squeezed light generation. More recently quantum memories based on EIT have been experimentally implemented in different systems such as alkali metal atoms. In this system the excited state of the optical transition splits into several sublevels due to the hyperfine interaction. However, most of the theoretical models used to describe the experimental results are based on a Λ-system with only one excited state. In this article, we present a theoretical model for the Λ-type interaction of two light, fields and an atomic system with multiple excited state. In particular we show that if the control and probe fields are orthogonally circularly polarized the EIT effect in an alkali-metal vapor can almost disappears. We also identify the reasons of this reduction and propose a method to recover the transparency via velocity selective optical pumping.     

A path integral method for data assimilation

Described here is a path integral, sampling-based approach for data assimilation, of sequential data and evolutionary models. Since it makes no assumptions on linearity in the dynamics, or on Gaussianity in the statistics, it permits consideration of very general estimation problems. The method can be used for such tasks as computing a smoother solution, parameter estimation, and data/model initialization.Speedup in the Monte Carlo sampling process is essential if the path integral method has any chance of being a viable estimator on moderately large problems. Here a variety of strategies are proposed and compared for their relative ability to improve the sampling efficiency of the resulting estimator. Provided as well are details useful for its implementation and testing.The method is applied to a problem in which standard methods are known to fail, an idealized flow/drifter problem, which has been used as a testbed for assimilation strategies involving Lagrangian data. It is in this kind of context that the method may prove to be a useful assimilation tool in oceanic studies.     

Unified Generative Adversarial Networks for Multidomain Fingerprint Presentation Attack Detection

With the rapid growth of fingerprint-based biometric systems, it is essential to ensure the security and reliability of the deployed algorithms. Indeed, the security vulnerability of these systems has been widely recognized. Thus, it is critical to enhance the generalization ability of fingerprint presentation attack detection (PAD) cross-sensor and cross-material settings. In this work, we propose a novel solution for addressing the case of a single source domain (sensor) with large labeled real/fake fingerprint images and multiple target domains (sensors) with only few real images obtained from different sensors. Our aim is to build a model that leverages the limited sample issues in all target domains by transferring knowledge from the source domain. To this end, we train a unified generative adversarial network (UGAN) for multidomain conversion to learn several mappings between all domains. This allows us to generate additional synthetic images for the target domains from the source domain to reduce the distribution shift between fingerprint representations. Then, we train a scale compound network (EfficientNetV2) coupled with multiple head classifiers (one classifier for each domain) using the source domain and the translated images. The outputs of these classifiers are then aggregated using an additional fusion layer with learnable weights. In the experiments, we validate the proposed methodology on the public LivDet2015 dataset. The experimental results show that the proposed method improves the average classification accuracy over twelve classification scenarios from 67.80 to 80.44% after adaptation.     

Direct Strong Measurement of a High-Dimensional Quantum State

It is of great importance to determine an unknown quantum state for fundamental studies of quantum mechanics,yet it is still difficult to characterize systems of large dimensions in practice. Although the scan-free direct measurement approach based on a weak measurement scheme was proposed to measure a high-dimensional photonic state, how weak the interaction should be to give a correct estimation remains unclear. Here we propose and experimentally demonstrate a technique that measures a high-di...     

A geometric approach to direct minimization

The approach presented, geometric direct minimization (GDM), is derived from purely geometrical arguments, and is designed to minimize a function of a set of orthonormal orbitals. The optimization steps consist of sequential unitary transformations of the orbitals, and convergence is accelerated using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) approach in the iterative subspace, together with a diagonal approximation to the Hessian for the remaining degrees of freedom. The approach is tested by implementing the solution of the self-consistent field (SCF) equations and comparing results with the standard direct inversion in the iterative subspace (DIIS) method. It is found that GDM is very robust and converges in every system studied, including several cases in which DIIS fails to find a solution. For main group compounds, GDM convergence is nearly as rapid as DIIS, whereas for transition metal-containing systems we find that GDM is significantly slower than DIIS. A hybrid procedure where DIIS is used for the first several iterations and GDM is used thereafter is found to provide a robust solution for transition metal-containing systems.