Combining multiple arcs for orbit determination using normal equations

<p>The normal equations are widely used to combine elementary least squares solutions, to solve very large problems which are not possible to handle directly. The principle is to reduce each problem to a minimal set of parameters present in the global problem, without removing the corresponding information, and connect them. For instance, one important application is the combination over years of daily network solutions, as performed for the ITRF (Altamimi et al., 2016) [1].</p><p>The approach can also be used in orbit determination to connect arcs solutions in order to construct the solution of a global arc. This was applied for example for GPS constellation solutions as in the article written by Beutler et al. (1996) [2]. Due to the size of the problems, it is interesting to divide for example a three days solution into three one day solutions. Another advantage is that the one day solutions are usually efficiently processed by the orbit determination software. For rapid or ultra-rapid GNSS products this is also very interesting, as the solutions are needed very often for small shifts of the global arc (for example 24 hours arcs, shifted every 6 hours in the case of ultra-rapid products). A further extension is to construct recursive solutions from these elementary arcs, leading to a filter similar to a Kalman filter.</p><p>We propose a unified methodology, associated with an efficient implementation compatible with our least squares software GINS, allowing us to solve the various problems ranging from arc connection to sequential filtering. The final objective is to construct efficient GNSS ultra-rapid products.</p><p>The application on a simple problem consisting in connecting different SLR arcs is shown, as a test case to develop and implement the methodology. In this case, the global solution can also be directly constructed for validation purposes. This study includes the construction of the solution at the end points of the elementary arcs, and also the recovery of the global solution state vectors at every epoch.</p><p>The next step will be to implement more complex parameterizations (including measurement parameters, which are not present in the SLR test case), and to apply this for GNSS constellation solutions.</p>

ON THE ALL ORDER SOLUTIONS OF SEIBERG-WITTEN MAP FOR NONCOMMUTATIVE GAUGE THEORIES

We review the recursive solutions of the Seiberg–Witten map to all orders in θ for gauge, matter and ghost fields. We also present the general structure of the homogeneous solutions of the defining equations. Moreover, we show that the contribution of the first order homogeneous solution to the second order can be written recursively similar to inhomogeneous solutions.

Power Systems Investments

The study employs a real options approach that doesn’t need to capture all the uncertainty and proposes a process that directly determines the uncertainty associated with the first period. The results support that its use can be considered fair. However, it shows that long periods of operation and poor adhesion to the geometric Brownian motion by the project returns might call into question its use in the energy market. The values for option pricing have remained inside acceptable ranges, but some shortfalls could be found. First, the study employs Monte Carlo simulations, which can be viewed as forward-looking processes, and option pricing problems need backward recursive solutions. Second, the study shows that its simplicity produces results as accurate as those gathered from approaches with added complexity and computational needs.

A Two-Stage Kalman Estimation Approach for the Identification of Structural Parameters under Unknown Inputs

Detection of structural damages is critical to ensure the reliability and safety of structures. So far, some progresses in structural identification have been made. The extended Kalman filter (EKF) has been one of the classic time-domain approaches for the identification of structural parameters. However, since the extended state vector contains both the state vector and the structural parameters, EKF approach can identify limited numbers of nonlinear structural parameters due to computational convergence difficulty. To overcome such problem, a two-stage Kalman estimation approach, which is not available in the previous literature, is proposed for the identification of structural parameters. In the first stage, state vector of structures is considered as an implicit function of the structural parameters, and the parametric vector is estimated directly based on the Kalman estimator. In the second stage, state vector of the structure is updated by applying the Kalman estimator with the structural parameters being estimated in the first stage. Therefore, analytical recursive solutions for the structural parameters and state vector are respectively derived and presented, by using the Kalman estimator method in a two-stage approach. The proposed approach is straightforward. Moreover, it can identify more numbers of nonlinear structural parameters with less time of iteration calculation compared with the conventional EKF. A numerical example of identifying the parameters of an 8-storey shear-frame structure is conducted. Simulation results show that the proposed approach is effective and accurate.