A normal equation-based extreme learning machine for solving linear partial differential equations

This paper develops an extreme learning machine for solving linear partial differential equations (PDE) by extending the normal equations approach for linear regression. The normal equations method is typically used when the amount of available data is small. In PDEs, the only available ground truths are the boundary and initial conditions (BC and IC). We use the physics-based cost function used in state-of-the-art deep neural network-based PDE solvers called physics informed neural network (PINN) to compensate for the small data. However, unlike PINN, we derive the normal equations for PDEs and directly solve them to compute the network parameters. We demonstrate our method's feasibility and efficiency by solving several problems like function approximation, solving ordinary differential equations (ODEs), steady and unsteady PDEs on regular and complicated geometries. We also highlight our method's limitation in capturing sharp gradients and propose its domain distributed version to overcome this issue. We show that this approach is much faster than traditional gradient descent-based approaches and offers an alternative to conventional numerical methods in solving PDEs in complicated geometries.

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>

APPLICATION OF SVD METHOD IN SOLVING INCORRECT GEODESIC PROBLEMS

The most reliable method for calculating linear equations of the least squares principle, which can be used to solve incorrect geodetic problems, is based on matrix factorization, which is called a singular expansion. There are other methods that require less machine time and memory. But they are less effective in taking into account the errors of the source information, rounding errors and linear dependence. The methodology of such research is that for any matrix A and any two orthogonal matrices U and V there is a matrix Σ, which is determined from the ratio. The idea of a singular decomposition is that by choosing the right matrices U and V, you can convert most elements of the matrix to zero and make it diagonal with non-negative elements. The novelty and relevance of scientific solutions lies in the feasibility of using a singular decomposition of the matrix to obtain linear equations of the least squares method, which can be used to solve incorrect geodetic problems. The purpose of scientific research is to obtain a stable solution of parametric equations of corrections to the results of measurements in incorrect geodetic problems. Based on the performed research on the application of the singular decomposition method in solving incorrect geodetic problems, we can summarize the following. A singular expansion of a real matrix is any factorization of a matrix with orthogonal columns , an orthogonal matrix and a diagonal matrix , the elements of which are called singular numbers of the matrix , and the columns of matrices and left and right singular vectors. If the matrix has a full rank, then its solution will be unique and stable, which can be obtained by different methods. But the method of singular decomposition, in contrast to other methods, makes it possible to solve problems with incomplete rank. Research shows that the method of solving normal equations by sequential exclusion of unknowns (Gaussian method), which is quite common in geodesy, does not provide stable solutions for poorly conditioned or incorrect geodetic problems. Therefore, in the case of unstable systems of equations, it is proposed to use the method of singular matrix decomposition, which in computational mathematics is called SVD. The SVD singular decomposition method makes it possible to obtain stable solutions of both stable and unstable problems by nature. This possibility to solve incorrect geodetic problems is associated with the application of some limit τ, the choice of which can be made by the relative errors of the matrix of coefficients of parametric equations of corrections and the vector of results of geodetic measurements . Moreover, the solution of the system of normal equations obtained by the SVD method will have the shortest length. Thus, applying the apparatus of the singular decomposition of the matrix of coefficients of parametric equations of corrections to the results of geodetic measurements, we obtained new formulas for estimating the accuracy of the least squares method in solving incorrect geodetic problems. The derived formulas have a compact form and make it possible to easily calculate the elements and estimates of accuracy, almost ignoring the complex procedure of rotation of the matrix of coefficients of normal equations.

GRACE/GRACE-FO mascons and satellite laser ranging: Updates from GSFC

<p>We present a summary of our recent work on time-variable gravity estimates derived from GRACE/GRACE-FO and satellite laser ranging (SLR). We show the latest results of our monthly mascon solution, with special attention paid to the selection and impact of the damping parameter applied to the regularization matrices. Additionally, we present a new method of regularized mascon estimation from spherical harmonics. Provided that the full normal equations for these coefficients are available, this method allows for a mathematically equivalent mascon estimate to those determined from a single iteration solution from Level-1B observations, but at a fraction of the computational cost. For this we use the ITSG-Grace2018 solution and full normal equations to produce a new mascon solution of comparable quality to current mascon products, and we are in the process of using this rapid approach to perform a trade study of various regularization strategies. Lastly, we present low degree SLR solutions that form the basis of the C20 and C30 solutions provided in Technical Note 14, and present preliminary results of large SLR-derived mascons from 1993 – Present.</p>

Progress towards a rigorous error propagation for total least-squares estimates

After several attempts at a formal derivation of the dispersion matrix for Total Least-Squares (TLS) estimates within an Errors-In-Variables (EIV) Model, here a refined approach is presented that makes rigorous use of the nonlinear normal equations, though assuming a Kronecker product structure for both observational dispersion matrices at this point. In this way, iterative linearization of a model (that can be established as being equivalent to the original EIV-Model) is avoided, which might be preferred since such techniques are based on the last iteration step only and, therefore, produce dispersion matrices for the estimated parameters that are generally too optimistic. Here, the error propagation is based on the (linearized total differential of the) exact nonlinear normal equations, which should lead to more trustworthy measures of precision.

Structure of the system of normal equations in the construction and equalization of analytical phototriangulation by the ligament method using the coplanarity condition

Рассматривается построение и уравнивание аналитической пространственной фототриангуляции, ос- нованной на совместном использованием условия коллинеарности и условия компланарности векторов. Получена структура системы нормальных уравнений. Предлагаемое решение позволяет выполнять по- строение фототриангуляции с одновременным определением (уточнением) элементов внешнего ори- ентирования и внутреннего ориентирования съёмочной камеры. Для назначения начальных значений весов уравнений, составленных с использованием условия компланарности, для некоторой точки ис- следуемого объекта (местности) предлагается использовать ошибки определения пространственных координат точки, рассчитанные методом прямой фотограмметрической засечки для произвольного случая съёмки.