A computational complexity analysis of Tienstra’s solution to equality-constrained adjustment

Tienstra’s method was developed to solve parameter adjustment with linear equality constraints, which has been otherwise often carried out by directly applying the conventional (or standard) method of Lagrange multipliers to quadratic optimization problems with a positive definite matrix. We analyze the computational complexity of the celebrated Tienstra’s method and compare it with that of the method of Lagrange multipliers. We show that Tienstra’s method is not only statistically elegant but also interestingly of significant computational advantage over the method of Lagrange multipliers to solve weighted least squares adjustment with linear equality constraints, with the saving of computational cost by a minimum of about 38% to a maximum of 100%. Tienstra’s method can be very important for large scale problems of adjustment and inversion with linear equality constraints.     

Fail-safe optimization of viscous dampers for seismic retrofitting

This paper presents a new optimization approach for designing minimum-cost fail-safe distributions of fluid viscous dampers for seismic retrofitting. Failure is modeled as either complete damage of the dampers or partial degradation of the dampers' properties. In general, this leads to optimization problems with large number of constraints. This may result in high computational costs if all the constraints are simultaneously considered during the optimization analysis. Thus, to reduce the computational effort, the use of a working-set optimization algorithm is proposed in this paper. The main idea is to solve a sequence of relaxed optimization subproblems with a small subset of all constraints. The algorithm terminates once a solution of a subproblem is found that satisfies all the constraints of the problem. The retrofitting cost is minimized with constraints on the interstory drifts at the peripheries of frame structures. The structures considered are subjected to a realistic ensemble of ground motions, and their response is evaluated with time-history analyses. The transient optimization problem is efficiently solved with a gradient-based sequential linear programming algorithm. The gradients of the response functions are calculated with a consistent adjoint sensitivity analysis procedure. Promising results attained for 3-D irregular frames are presented and discussed. The numerical results highlight the fact that the optimized layout and size of the dampers can change significantly even for moderate levels of damage.     

Line fitting in Euclidean 3D space

Over a century ago Pearson solved the problem of fitting lines in 2D space to points with noisy coordinates in both dimensions. Surprisingly, however, the case of fitting lines in 3D space has seen little attention, though Adcock long ago published a brief (one page) article claiming that the solution that minimized orthogonal distances is the most probable. We solve this problem using a new algorithm for the Total Least-Squares (TLS) solution within an Errors-In-Variables Model, respectively an equivalent nonlinear Gauss-Helmert Model. Following Roberts, only four parameters are estimated, thereby avoiding over-parametrization that may lead to unnecessary singularities and, hence, require the introduction of constraints to the model. The current pervasiveness of Global Navigation Satellite Systems, robotic total stations, and digital laser scanners as sources of geodetic observations means that geodetic engineers and scientists now commonly work with observational models in 3D space as opposed to classical geodetic methods that often separated horizontal and vertical observational models. And while several papers have been written describing a TLS solution for line fitting problems in 2D space, the extension to 3D space is not readily apparent from these works. This further motivates the treatment of the 3D problem in some detail in this contribution.     

A method for reliability-based optimization with multiple non-normal stochastic parameters: a simplified airshed management study

We develop methodologies to enable applications of reliability-based design optimization (RBDO) to environmental policy setting problems. RBDO considers uncertainty as random variables and parameters in an optimization framework with probabilistic constraints. Three challenges in environmental decision-making problems not addressed by current RBDO methods are efficient methods in handling: (1) non-normally distributed random parameters, (2) discrete random parameters, and (3) joint reliability constraints (e.g., meeting constraints simultaneously with a single reliability). We propose a modified sequential quadratic programming algorithm to address these challenges. An active set strategy is combined with a reliability contour formulation to solve problems with multiple non-normal random parameters. The reliability contour formulation can also handle discrete random parameters by converting them to equivalent continuous ones. Joint reliability constraints are estimated by their theoretical upper bounds using reliability indexes and angles of normal vectors between active constraints. To demonstrate the methods, we consider a simplified airshed example where CO and NOx standards are violated and are brought into compliance by reducing the speed limits of two nearby highways. This analytical example is based on the CALINE4 model. Results show the potential of this approach to handle complex large-scale environmental regulation problems.     

Partial eigensolution of damped structural systems by Arnoldi's method

An efficient numerical algorithm is developed to solve the quadratic eigenvalue problems arising in the dynamic analysis of damped structural systems. The algorithm can even be applied to structural systems with non-symmetric matrices. The algorithm is based on the use of Arnoldi's method to generate a Krylov subspace of trial vectors, which is then used to reduce a large eigenvalue problem to a much smaller one. The reduced eigenvalue problem is solved and the solutions are used to construct approximate solutions to the original large system. In the process, the algorithm takes full advantage of the sparseness and symmetry of the system matrices and requires no complex arithmetic, therefore, making it very economical for use in solving large problems. The numerical results from test examples are presented to demonstrate that a large fraction of the approximate solutions calculated are very accurate, indicating that the algorithm is highly effective for extracting a number of vibration modes for a large dynamic system, whether it is lightly or heavily damped.     

Total least square method applied to rating curves

The ordinary least square method (OLS) has been the most frequently used least square method in hydrological data analysis. Its computational algorithm is simple, and the error analysis is also simple and clear. However, the primary assumption of the OLS method, which states that the dependent variable is the only error‐contaminated variable and all other variables are error free, is often violated in hydrological data analyses. Recently, a matrix algorithm using the singular value decomposition for the total least square (TLS) method has been developed and used in data analyses as errors‐in‐variables model where several variables could be contaminated with observational errors. In our study, the algorithm of the TLS is introduced in the evaluation of rating curves between the flow discharge and the water level. Then, the TLS algorithm is applied to real data set for rating curves. The evaluated TLS rating curves are compared with the OLS rating curves, and the result indicates that the TLS rating curve and the OLS rating curve are in good agreement. The TLS and OLS rating curves are discussed about their algorithms and error terms in the study. Copyright © 2013 John Wiley & Sons, Ltd.     

基于输出反馈的建筑结构闭开环次优控制

对传统的结构抗震闭开环控制算法进行改进。基于地面运动自回归模型,采用Kalman滤波利用可以量测到的地面加速度激励对未来时段即将发生的地面加速度激励进行预估,并在微分方程的求解中引入精确高效的精细积分算法。考虑到实际控制中量测全部状态变量的困难,改进算法仅需量测部分状态变量。数值仿真表明,基于输出反馈的闭开环次优控制策略能大大降低结构的地震响应。     

Method of Nonlinear Filtering of Observations from Systems of Vector and Scalar Magnetometers

A method of nonlinear filtering of observations from systems of vector and scalar magnetometers is proposed. The method is based on the calculation of sliding local approximating model functions and weighted averaging. An algorithm intended for the filtering of observations from magnetometer systems based on sliding local approximating piecewise linear model functions (reduction of initial functionals to quadratic forms, calculation of parameters of sliding local models, and implementation of their weighted averaging) has been developed. Test results are presented for the filtering algorithm on model and observatory INTERMAGNET observations with a 1-min discretization of the magnetometer system. The efficiency of the developed filtering algorithm is estimated with statistical modeling.     

Optimizing wastewater pumping system with data-driven models and a greedy electromagnetism-like algorithm

Optimizing a pumping system in the wastewater treatment process by improving its operational schedules is presented. The energy consumption and outflow rate of the pumping system are modeled by a data-driven approach. A mixed-integer nonlinear programming (MINLP) model containing data-driven components and pump operational constraints is developed to minimize the energy consumption of the pumping system while maintaining the required pumping workload. A greedy electromagnetism-like (GEM) algorithm is designed to solve the MINLP model for optimized operational schedules and pump speeds. Three computational cases are studied to demonstrate the effectiveness of the proposed data-driven modeling and GEM algorithm. The computational results show that significant energy saving can be obtained.     

Kalman filter-based algorithms for monitoring the ionosphere and plasmasphere with GPS in near-real time

Data collected from a GPS receiver located at low latitudes in the American sector are used to investigate the performance of the WinTEC algorithm [Anghel et al., 2008a, Kalman filter-based algorithm for near realtime monitoring of the ionosphere using dual frequency GPS data. GPS Solutions, accepted for publication; for different ionospheric modeling techniques: the single-shell linear, quadratic, and cubic approaches, and the multi-shell linear approach. Our results indicate that the quadratic and cubic approaches perform much better than the single-shell and multi-shell linear approaches in terms of post-fit residuals. The performance of the algorithm for the cubic approach is then further tested by comparing the vertical TEC predicted by WinTEC and USTEC [Spencer et al., 2004. Ionospheric data assimilation methods for geodetic applications. In: Proceedings of IEEE PLANS, Monterey, CA, 26–29 April, pp. 510–517] at five North American stations. In addition, since the GPS-derived total electron content (TEC) contains contributions from both ionospheric and plasmaspheric sections of the GPS ray paths, in an effort to improve the accuracy of the TEC retrievals, a new data assimilation module that uses background information from an empirical plasmaspheric model [Gallagher et al., 1988. An empirical model of the Earth's plasmasphere. Advances in Space Research 8, (8)15–(8)24] has been incorporated into the WinTEC algorithm. The new Kalman filter-based algorithm estimates both the ionospheric and plasmaspheric electron contents, the combined satellite and receiver biases, and the estimation error covariance matrix, in a single-site or network solution. To evaluate the effect of the plasmaspheric component on the estimated biases and total TEC and to assess the performance of the newly developed algorithm, we compare the WinTEC results, with and without the plasmaspheric term included, at three GPS receivers located at different latitudes in the American sector, during a solar minimum period characterized by quiet and moderate geomagnetic conditions. We also investigate the consistency of our plasmaspheric results by taking advantage of the specific donut-shaped geometry of the plasmasphere and applying the technique at 12 stations distributed roughly over four geomagnetic latitudes and three longitude sectors.     

A structured and constrained Total Least-Squares solution with cross-covariances

A new proof is presented of the desirable property of the weighted total least-squares (WTLS) approach in preserving the structure of the coefficient matrix in terms of the functional independent elements. The WTLS considers the full covariance matrix of observed quantities in the observation vector and in the coefficient matrix; possible correlation between entries in the observation vector and the coefficient matrix are also considered. The WTLS approach is then equipped with constraints in order to produce the constrained structured TLS (CSTLS) solution. The proposed approach considers the correlation between the observation vector and the coefficient matrix of an Error-In-Variables model, which is not considered in other, recently proposed approaches. A rigid transformation problem is done by preservation of the structure and satisfying the constraints simultaneously.     

非线性二次规划贝叶斯叠前反演

叠前反演的目的是基于弹性波理论从地震数据中获得地层参数的可靠估计,进而用于描述地层的流体和岩性特征.然而叠前反演问题都是高维的和非适定的,并且容易受各种噪声和采集过程中不确定因素的影响,因此,为了获得稳定可靠的解必需对反演过程加以合理的约束.本文提出了一种基于非线性二次规划的叠前三参数反演方法.首先基于贝叶斯参数估计理论,假设似然函数服从高斯分布,并使待反演的参数服从于改进的Cauchy分布,从而提高了反演结果的分辨率;其次用协方差矩阵来描述参数间的相关程度,进一步提高了反演结果的稳定性;最后将问题转化为一个非线性二次规划的求解问题,并在多种约束下得到问题的解.仿真实验和实际应用皆已表明,本文提出的反演方法运算速度快捷,既使在信噪比很低的情况下也可获得较好的反演结果,为储层的进一步识别提供更多的物性参数.     

LINEAR QUADRATIC REGULATOR FOR STRUCTURE UNDER ON-LINE PREDICTED FUTURE SEISMIC EXCITATION

A predictive-adaptive (PA) control algorithm has been developed for a structure under a seismic excitation. This algorithm analyses information of an observed seismic excitation, estimates future structural responses and determines the control force for the structure, based on the linear quadratic regulator. That is, at a given moment t_k: (1) seismic excitation information is converted to an autoregressive model, which forms the state equation for the excitation; (2) the identification model is combined with the structural model to build a state equation in an augmented space; (3) the weighted quadratic norm of the state vector and the future control force is formed as a cost function for estimating future responses; (4) the Ricatti equation is solved to find the optimum value of the cost function; and (5) the optimum gain matrix is obtained, and the control force is determined. The PA algorithm is not restricted to one type of control system, but can be applied to both an active driver system and an active tendon system. Its effectiveness is confirmed by numerical experiments for 1DOF and 3DOF structural models under sine and seismic excitations.     

Amplification and energy transformation of the magnetized Rossby waves in the ionosphere with an inhomogeneous zonal wind. I. A theory

The generation and further dynamics of the planetary magnetized Rossby waves and inertial waves in a dissipative ionosphere in the presence of a smooth inhomogeneous zonal wind (shear flow) have been studied. The magnetized Rossby waves are caused by the interaction with the spatially inhomogeneous geomagnetic field and represent the ionospheric manifestations of usual tropospheric Rossby waves. The effective linear mechanism of amplification and mutual transformation of the Rossby and inertial waves has been revealed. For shear flows, the operators of linear problems are not self-adjoint, and the corresponding eigenfunctions are non-orthogonal; therefore, a canonical modal approach is of little use in studying such motions. It becomes necessary to apply the so-called nonmodal mathematical analysis, which has actively been developed for the last years. The nonmodal approach makes it possible to reveal that the transformation of wave-like disturbances in shear flows is caused by the nonorthogonality of eigenfunctions in the problem of linear dynamics. Thus, there appear a new degree of the system freedom and a new way of disturbance evolution in the medium.     

基于混合差分进化算法的地球物理线性反演

地球物理反问题线性化处理之后, 各种反演算法归结为对病态线性方程组的求解. 为了快速准确地计算出地球物理参数, 本文提出了一种全新的基于LSQR算法的混合差分进化算法(Hybrid Differential Evolution Algorithm, HDE). 该算法利用LSQR算法给出DE算法的初始种群, 提高DE算法的计算速度和稳定性. 在不同噪声水平下, 对四种正则化方法Tikhonov、TSVD、LSQR和HDE的反演结果进行详细比较. 理论模型和实际数据反演的结果都表明: 改进的HDE算法应用于地球物理反问题的求解是成功的: 反演结果与原设定模型具有较高的相关性, 在稳定性和准确性上较常规的反演算法都具有一定的优势; 而且不需要给定正则化参数, 具有更强的实用性.     

Hybrid-optimization algorithm for the management of a conjunctive-use project and well field design

Hi-Desert Water District (HDWD), the primary water-management agency in the Warren Groundwater Basin, California, plans to construct a waste water treatment plant to reduce future septic-tank effluent from reaching the groundwater system. The treated waste water will be reclaimed by recharging the groundwater basin via recharge ponds as part of a larger conjunctive-use strategy. HDWD wishes to identify the least-cost conjunctive-use strategies for managing imported surface water, reclaimed water, and local groundwater. As formulated, the mixed-integer nonlinear programming (MINLP) groundwater-management problem seeks to minimize water-delivery costs subject to constraints including potential locations of the new pumping wells, California State regulations, groundwater-level constraints, water-supply demand, available imported water, and pump/recharge capacities. In this study, a hybrid-optimization algorithm, which couples a genetic algorithm and successive-linear programming, is developed to solve the MINLP problem. The algorithm was tested by comparing results to the enumerative solution for a simplified version of the HDWD groundwater-management problem. The results indicate that the hybrid-optimization algorithm can identify the global optimum. The hybrid-optimization algorithm is then applied to solve a complex groundwater-management problem. Sensitivity analyses were also performed to assess the impact of varying the new recharge pond orientation, varying the mixing ratio of reclaimed water and pumped water, and varying the amount of imported water available. The developed conjunctive management model can provide HDWD water managers with information that will improve their ability to manage their surface water, reclaimed water, and groundwater resources.     

A robust weighted total least squares algorithm and its geodetic applications

Total least squares (TLS) can solve the issue of parameter estimation in the errors-invariables (EIV) model, however, the estimated parameters are affected or even severely distorted when the observation vector and coefficient matrix are contaminated by gross errors. Currently, the use of existing robust TLS (RTLS) methods for the EIV model is unreasonable. Original residuals are directly used in most studies to construct the weight factor function, thus the robustness for the structure space is not considered. In this study, a robust weighted total least squares (RWTLS) algorithm for the partial EIV model is proposed based on Newton-Gauss method and the equivalent weight principle of general robust estimation. The algorithm utilizes the standardized residuals to construct the weight factor function and employs the median method to obtain a robust estimator of the variance component. Therefore, the algorithm possesses good robustness in both the observation and structure spaces. To obtain standardized residuals, we use the linearly approximate cofactor propagation law for deriving the expression of the cofactor matrix of WTLS residuals. The iterative procedure and precision assessment approach for RWTLS are presented. Finally, the robustness of RWTLS method is verified by two experiments involving line fitting and plane coordinate transformation. The results show that RWTLS algorithm possesses better robustness than the general robust estimation and the robust total least squares algorithm directly constructed with original residuals.     

Solving nonlinear water management models using a combined genetic algorithm and linear programming approach

Gradient-based nonlinear programming (NLP) methods can solve problems with smooth nonlinear objectives and constraints. However, in large and highly nonlinear models, these algorithms can fail to find feasible solutions, or converge to local solutions which are not global. Evolutionary search procedures in general, and genetic algorithms (GAs) specifically, are less susceptible to the presence of local solutions. However, they often exhibit slow convergence, especially when there are many variables, and have problems finding feasible solutions in constrained problems with “narrow” feasible regions. In this paper, we describe strategies for solving large nonlinear water resources models management, which combine GAs with linear programming. The key idea is to identify a set of complicating variables in the model which, when fixed, render the problem linear in the remaining variables. The complicating variables are then varied by a GA. This GA&LP approach is applied to two nonlinear models: a reservoir operation model with nonlinear hydropower generation equations and nonlinear reservoir topologic equations, and a long-term dynamic river basin planning model with a large number of nonlinear relationships. For smaller instances of the reservoir model, the CONOPT2 nonlinear solver is more accurate and faster, but for larger instances, the GA&LP approach finds solutions with significantly better objective values. The multiperiod river basin model is much too large to be solved in its entirety. The complicating variables are chosen here so that, when they are fixed, each period's model is linear, and these models can be solved sequentially. This approach allows sufficient model detail to be retained so that long-term sustainability issues can be explored.     

Non‐parametric–parametric model for random uncertainties in non‐linear structural dynamics: application to earthquake engineering

This paper deals with the transient response of a non‐linear dynamical system with random uncertainties. The non‐parametric probabilistic model of random uncertainties recently published and extended to non‐linear dynamical system analysis is used in order to model random uncertainties related to the linear part of the finite element model. The non‐linearities are due to restoring forces whose parameters are uncertain and are modeled by the parametric approach. Jayne's maximum entropy principle with the constraints defined by the available information allows the probabilistic model of such random variables to be constructed. Therefore, a non‐parametric–parametric formulation is developed in order to model all the sources of uncertainties in such a non‐linear dynamical system. Finally, a numerical application for earthquake engineering analysis is proposed concerning a reactor cooling system under seismic loads. Copyright © 2003 John Wiley & Sons, Ltd.     

Limiting performance of shock isolation systems by a modal approach

The problem of determining the limiting performance of vibrating systems under shock loading is solved by replacing portions of the system by control forces which can represent any design. For the class of problems treated here, the performance index and the constraints are linear combinations of system response variables such as displacements, velocities and accelerations. Furthermore, the equations of motion are linear, so that it is possible to formulate the optimization procedure as a linear programming problem. In expressing the performance index and the constraints as linear functions of the unknown control forces, a modal approach is used to simplify and improve previous treatments of this problem. In spite of these linearity requirements, the control forces are not required to be linear functions of the response variables.