Robust goal-oriented error estimation based on the constitutive relation error for stochastic problems

Domain decomposition based finite element verification in linear framework

PurposeThis paper aims to deal with the verification of local quantities of interest obtained through linear elastic finite element analysis. A technique is presented for determining the most accurate error estimation. This technique enables one to address industrial‐size problems while keeping computing costs reasonable.Design/methodology/approachThe concept of error in constitutive relation is used to assess the quality of the finite element solution. The key issue is the construction of admissible fields. The objective is to show that it is possible to build admissible fields using a new method. These fields are obtained by using a high‐quality construction over a limited zone while the construction is less refined and less expensive elsewhere.FindingsNumerical tests are presented in order to illustrate a very satisfying presented methodology. It shows clearly how to take advantage of the method to treat large examples. They clearly show the interest of this new method to treat large examples.Originality/valueThe paper demonstrates clearly that verification of large finite element problem must have dedicated methods in order to be applicable.

In recent publications, the author and his coworkers have shown robust approximation error estimates for B-splines of maximum smoothness and have proposed multigrid methods based on them. These methods allow to solve the linear system arising from the discretization of a partial differential equation in Isogeometric Analysis in a single-patch setting with convergence rates that are provably robust both in the grid size and the spline degree. In real-world problems, the computational domain cannot be nicely represented by just one B-spline patch. In computer aided design, such domains are typically represented as a union of multiple patches. In this paper, we extend — for two-dimensional domains — the approximation error estimates and the multigrid solver to this multi-patch case.

High-resolution airborne synthetic aperture radar (SAR) systems are very sensible to deviations of the aircraft from the reference track. In high-resolution imagery, the improvement of range resolution increases the difficulty of implementing range cell migration correction (RCMC), while a wider synthetic aperture increases the cumulative time of motion errors which will affect the image quality. To enable accurate motion compensation in image processing, a high-precision navigation system is needed. However, in many cases, due to the limit of accuracy of such systems, motion errors are hard to be compensated correctly, causing mainly the resolution decrease in final image. Moreover, in large swath mode, the range-dependent phase errors are difficult to be compensated by using the conventional autofocus algorithm only. In this paper, we propose a robust motion error estimation method based on raw SAR data. To apply this estimation method, we first estimate the double phase gradients in subaperture. Second, a filtering method based on curve fitting was proposed to reduce the phase estimation errors caused by low signal-to-clutter ratio (SCR). Finally, we propose a weighted total least square method to calculate the motion errors using the filtered phase gradients. Because the proposed algorithm is nonparametric, it can estimate high-order motion errors. This is very important for the airborne SAR, particularly the light aircraft SAR platform, due to their more complicated movement in air turbulence. The versatility that the proposed method can be used in any imaging algorithms is another advantage. The processing of large number of raw SAR data shows that the algorithm is as robust and practical as phase gradient autofocus and can generate better focused images.

We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties, where the vector x is known to satisfy a weighted norm constraint. We first assume that the model is known exactly and seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible values of x. We show that for an arbitrary choice of weighting, the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved very efficiently. We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke, with a specific choice of shrinkage factor that explicitly takes the prior information into account. Next, we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst-case MSE across all possible values of x and all possible values of the model matrix. As we show, the robust minimax MSE estimator can also be formulated as a solution to an SDP. Finally, we demonstrate through several examples that the minimax MSE estimator can significantly increase the performance over the conventional least-squares estimator, and when the model matrix is subject to uncertainties, the robust minimax MSE estimator can lead to a considerable improvement in performance over the minimax MSE estimator.

We introduce a goal-oriented strategy for multiscale computations performed using the Multiscale Finite Element Method (MsFEM). In a previous work, we have shown how to use, in the MsFEM framework, the concept of Constitutive Relation Error (CRE) to obtain a guaranteed and fully computable a posteriori error estimate in the energy norm (as well as error indicators on various error sources). Here, the CRE concept is coupled with the solution of an adjoint problem to control the error and drive an adaptive procedure with respect to a given output of interest. Furthermore, a local and non-intrusive enrichment technique is proposed to enhance the accuracy of error bounds. The overall strategy, which is fully automatic and robust, enables to reach an appropriate trade-off between certified reliability and computational cost in the MsFEM context. The performances of the proposed method are investigated on several illustrative numerical test cases. In particular, the error estimation is observed to be very accurate, yielding a very efficient adaptive procedure.

A central problem in learning is selection of an appropriate model. This is typically done by estimating the unknown generalization errors of a set of models to be selected from and then choosing the model with minimal generalization error estimate. In this article, we discuss the problem of model selection and generalization error estimation in the context of kernel regression models, e.g., kernel ridge regression, kernel subset regression or Gaussian process regression. Previously, a non-asymptotic generalization error estimator called the subspace information criterion (SIC) was proposed, that could be successfully applied to finite dimensional subspace models. SIC is an unbiased estimator of the generalization error for the finite sample case under the conditions that the learning target function belongs to a specified reproducing kernel Hilbert space (RKHS) H and the reproducing kernels centered on training sample points span the whole space H. These conditions hold only if dim H < l, where l < ∞ is the number of training examples. Therefore, SIC could be applied only to finite dimensional RKHSs. In this paper, we extend the range of applicability of SIC, and show that even if the reproducing kernels centered on training sample points do not span the whole space H, SIC is an unbiased estimator of an essential part of the generalization error. Our extension allows the use of any RKHSs including infinite dimensional ones, i.e., richer function classes commonly used in Gaussian processes, support vector machines or boosting. We further show that when the kernel matrix is invertible, SIC can be expressed in a much simpler form, making its computation highly efficient. In computer simulations on ridge parameter selection with real and artificial data sets, SIC is compared favorably with other standard model selection techniques for instance leave-one-out cross-validation or an empirical Bayesian method.

In this paper, we propose joint angle-delay subspace based channel estimation in single cell for broadband massive multiple-input and multiple-output (MIMO) systems employing orthogonal frequency division multiplexing (OFDM) modulation. Based on a parametric channel model, we present a new concept of the joint angle-delay subspace which can be tracked by the low-complexity low-rank adaptive filtering (LORAF) algorithm. Then, we investigate an interference-free transmission condition that the joint angle-delay subspaces of the users reusing the same pilots are non-overlapping. Since the channel statistics are usually unknown, we develop a robust minimum mean square error (MMSE) estimator under the worst precondition of pilot decontamination, considering that the joint angle-delay subspaces of the interfering users fully overlap. Furthermore, motivated by the interference-free transmission criteria, we present a novel low-complexity greedy pilot scheduling algorithm to avoid the problem of initial value sensitivity. Simulation results show that the joint angle-delay subspace can be estimated effectively, and the proposed pilot reuse scheme combined with robust MMSE channel estimation offers significant performance gains.

Error estimates for a class of partial functional differential equation with small dissipation

Accurate masses and radii for normal stars derived from observations of detached eclipsing binary stars are of fundamental importance for testing stellar models and may be useful for calibrating free parameters in these model if the masses and radii are sufficiently precise and accurate. We aim to measure precise masses and radii for the stars in the bright eclipsing binary AI Phe, and to quantify the level of systematic error in these estimates. We use several different methods to model the Transiting Exoplanet Survey Satellite (TESS) light curve of AI Phe combined with spectroscopic orbits from multiple sources to estimate precisely the stellar masses and radii together with robust error estimates. We find that the agreement between different methods for the light-curve analysis is very good but some methods underestimate the errors on the model parameters. The semi-amplitudes of the spectroscopic orbits derived from spectra obtained with modern échelle spectrographs are consistent to within 0.1 per cent. The masses of the stars in AI Phe are $M_1 = 1.1938 \pm 0.0008\, \rm M_{\odot }$ and $M_2 = 1.2438 \pm 0.0008\, \rm M_{\odot }$, and the radii are $R_1 = 1.8050 \pm 0.0022\, \rm R_{\odot }$ and $R_2 = 2.9332 \pm 0.0023\, \rm R_{\odot }$. We conclude that it is possible to measure accurate masses and radii for stars in bright eclipsing binary stars to a precision of 0.2 per cent or better using photometry from TESS and spectroscopy obtained with modern échelle spectrographs. We provide recommendations for publishing masses and radii of eclipsing binary stars at this level of precision.

We present a parallel goal‐oriented adaptive finite element algorithm that can be used to rapidly compute highly accurate solutions for 2.5D controlled‐source electromagnetic (CSEM) and 2D magnetotelluric (MT) modeling problems. The use of an unstructured triangular modeling grid allows for efficient use of mesh nodes for discretizing arbitrarily complex domains. A goal‐oriented error estimator based on the dual residual weighting method computed through hierarchical bases provides robust error estimation that is used to guide an iterative adaptive mesh refinement process. Our formulation of the error estimator considers the relative error in the strike aligned fields and their spatial gradients, and therefore results in a more efficient use of mesh nodes than previous error estimators based on absolute field errors. This algorithm has been parallelized over frequencies, transmitters, receivers and wave‐numbers, enabling it to achieve accurate solutions in run‐times of seconds to tens of seconds for realistic models and data parameters when run on cluster computers of up to a thousand processors. Application of this new algorithm to a complex model that includes strong seafloor topography variations and multiple thin stacked reservoirs demonstrates the performance and scalability on a large cluster computer.

We consider transient convection–diffusion equations with a velocity vector field with multiscale character and rough data. We assume that the velocity field has two scales, a coarse scale with slow spatial variation, which is responsible for advective transport and a fine scale with small amplitude that contributes to the mixing. For this problem we consider the estimation of filtered error quantities for solutions computed using a finite element method with symmetric stabilization. A posteriori error estimates and a priori error estimates are derived using the multiscale decomposition of the advective velocity to improve stability. All estimates are independent both of the Péclet number and of the regularity of the exact solution.

In this paper the performance of back-propagation algorithm is studied in the presence of noise in the training data. A general method of using robust error estimators with back-propagation algorithm is explained and a family of so called robust M-estimates is investigated in detail.

Convergence analysis of a fully-discrete FEM for singularly perturbed two-parameter parabolic PDE

Binge drinking and risky sexual behavior among HIV-negative and unknown HIV status men who have sex with men, 20 US cities