Local Error Estimates Research Articles

Sample-efficient deep learning for accelerating photonic inverse design

Data-driven techniques like deep learning (DL) are currently being explored for inverse design problems in photonics (especially nanophotonics) to deal with the vast search space of materials and nanostructures. Many challenges need to be overcome to fully realize the potential of this approach; current workflows are specific to predefined shapes and require large upfront investments in dataset creation and model hyperparameter search. We report an improved workflow for DL based acceleration of evolutionary optimizations for scenarios where past simulation data is nonexistent or highly inadequate and demonstrate its utility considering the example problem of multilayered thin-film optics design. For obtaining sample-efficiency in surrogate training, novel training loss functions that emphasize a model’s ability to predict a structurally similar spectral response rather than minimizing local approximation error are proposed. The workflow is of interest to extend the ambit of DL based optics design to complicated structures whose spectra are computationally expensive to calculate.

Accuracy of Difference Schemes in Electromagnetic Applications: A Trefftz Analysis

The article examines local approximation errors of finite-difference schemes in electromagnetic analysis. Despite a long history of the subject, several accuracy-related issues have been overlooked and/or remain controversial. For example, conflicting claims have been made in the literature about the order of Yee-like schemes in the vicinity of slanted or curved material interfaces. Two novel practical methods for comparison of local accuracy of difference schemes are proposed: one makes use of Trefftz test matrices, and the other one relies on a new measure of approximation accuracy in scheme-exact Trefftz subspaces. One particular conclusion is that a loss of accuracy for Yee-like schemes at slanted material boundaries is unavoidable.

A Hierarchical Error Estimator for the MSFEM for the Eddy Current Problem in 3-D

The aim of this article is to develop an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> hierarchical error estimator for the multiscale finite element method (MSFEM). The MSFEM is used to solve the linear eddy current problem in a stack of iron sheets. It allows for a calculation of the solution without having to resolve each sheet in the finite element mesh. A hierarchical local error estimator for nodal elements and edge elements is adapted to the multiscale setting. The estimator allows for adaptive p-refinement on the coarse multiscale mesh. Numerical examples show an increased order of convergence compared to uniform refinement. The proposed error estimator increases the efficiency of the MSFEM, requiring even fewer degrees of freedom. It is the first error estimator presented for the 3-D MSFEM for the eddy current problem.

Poisson-Gaussian Noise Reduction for X-Ray Images Based on Local Linear Minimum Mean Square Error Shrinkage in Nonsubsampled Contourlet Transform Domain

Noise reduction is important for X-ray images because it can reduce radiation exposure to patients. X-ray image noise has a Poisson-Gaussian distribution, and recently, noise analysis and removal in multiscale transformations have been widely implemented. The nonsubsampled contourlet transform (NSCT) is a multiscale transformation suitable for medical images that separates the scale and direction. This study proposes a Poisson-Gaussian noise-removal method using NSCT shrinkage that is based on the characteristics of Poisson-Gaussian noise in NSCT domain. It has the structure of a block-matching 3D filtering algorithm in the form of basic estimation and noise removal process; however, the main processes are modified to consider Poisson-Gaussian noise characteristics. In the basic estimation process, an NSCT shrinkage method that is suitable for Poisson-Gaussian noise characteristics is developed by optimizing the local linear minimum mean square error estimator in the NSCT domain. In the denoising step, the noise term of the Wiener filter is determined using the result of the NSCT shrinkage, and finally, the denoised image is obtained. The proposed method is applied to simulated and real X-ray images and is compared with other state-of-the-art Poisson-Gaussian noise removal methods; it exhibits excellent results in both quantitative and qualitative aspects.

Adaptive time stepping for commutator free Lie group integrators

Lie group integrators are steadily gaining popularity in many application areas. Until recently, there has been little work on Lie group integrators with adaptive time stepping. The most popular method for automatic step size selection seems in general to be that of embedded pairs of methods. For schemes that can be interpreted via local parameterisations, error estimation and step size selection can be determined in the parameter space. However, not all Lie group integrators fall into this class. Commutator-free Lie group integrators or Crouch-Grossman methods do not have such an interpretation. The problem can be resolved by using non-commutative order conditions. We develop new examples of embedded pairs of commutator-free Lie group integrators of orders 3(2) and 4(3), based on earlier work by Curry, C and Owren, B. (2019). Variable step size commutator-free Lie group integrators. Numer. Algorithms, 82(4), 1359-1376 where optimal pairs in terms of flow calculations and vector field evaluations per step were derived for orders 3(2) and 4(3). In particular we introduce a non-optimal embedded pair of orders 4(3) that actually behave better than the optimal ones for a popular test case, the heavy top.

Pointwise error estimates for 𝐶⁰ interior penalty approximation of biharmonic problems

The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C 0 C^0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green’s function estimates for the continuous problem which has independent interest.

An adaptive sampling method for Kriging surrogate model with multiple outputs

The sample distribution has a vital influence on the quality of a Kriging surrogate model, which may further influence the required cost or convergence of the surrogate model-based design and optimization problems. Adaptive sampling methods utilize the information from existing samples to reasonably allocate the sequential samples, which can generally build a more accurate Kriging surrogate model under the same computational budget. However, most of the existing adaptive sampling methods for the Kriging surrogate model are only available for single-output problems, and there are few studies on problems with multiple responses. In this paper, an adaptive sampling method based on Delaunay triangulation and technique for order preference by similarity to ideal solution (TOPSIS) is proposed for Kriging surrogate model with multiple outputs (mKMDT). In the proposed mKMDT, Delaunay triangulation is used to partition the design space into multiple triangle regions, whose area denotes the dispersion of the sample points. The prediction error at each triangle’s centroid represents the local approximation error. Specifically, three different strategies are developed when allocating weights to the area and the prediction error of each triangle with the entropy method and the TOPSIS method. The performance of the proposed method is illustrated through numerical examples with different numbers of outputs and a collision problem between the missile and the adapter. Results show that the proposed method can construct an accuracy surrogate model with few samples, which is useful for practical engineering design problems with multiple outputs.

Data assimilation finite element method for the linearized Navier–Stokes equations in the low Reynolds regime

In this paper, we are interested in designing and analyzing a finite element data assimilation method for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We propose a weakly consistent stabilized finite element method which reconstructs the whole fluid flow from noisy velocity measurements in a subset of the computational domain. Using the stability of the continuous problem in the form of a three balls inequality, we derive quantitative local error estimates for the velocity. Numerical simulations illustrate these convergence properties and we finally apply our method to the flow reconstruction in a blood vessel.

Anisotropic mesh adaptation for continuous finite element discretization through mesh optimization via error sampling and synthesis

Anisotropic output-based mesh adaptivity is a powerful technique for controlling the output error of finite element simulations, particularly when used in conjunction with higher-order discretization. The Mesh Optimization via Error Sampling and Synthesis (MOESS) algorithm makes use of the continuous mesh model which encodes local mesh sizing and anisotropy in a Riemannian metric field, and was developed for Discontinuous Galerkin (DG) discretization. In this paper, we outline an extension of the MOESS algorithm for discretizations which are defined by basis functions that are continuous across element boundaries. Error models are defined in terms of local error estimates defined at vertices, and local solutions are computed on local patches defined in terms of the edges of the mesh. The resulting algorithm displays reduced error for the same number of degrees of freedom compared to the MOESS algorithm for DG discretization, as illustrated by some numerical examples for L2 projection and linear advection diffusion.

Regularity for the stationary Navier–Stokes equations over bumpy boundaries and a local wall law

We investigate regularity estimates for the stationary Navier–Stokes equations above a highly oscillating Lipschitz boundary with the no-slip boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the boundary layer thickness. We also obtain an improved \(C^{1,\mu }\) estimate and identify the building blocks of the regularity theory, dubbed ‘Navier polynomials’. In the case when some structure is assumed on the oscillations of the boundary, for instance periodicity, these estimates can be seen as local error estimates. Although we handle the regularity of the nonlinear stationary Navier–Stokes equations, our results do not require any smallness assumption on the solutions.

Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data

In this paper we derive local L2 error estimates for penalized least-squares approximation on the d-dimensional unit sphere Sd⊆Rd+1, given noisy, scattered, local data representing an underlying function from a Sobolev space of order s>d∕2 defined on a non-empty connected open set Ω⊆Sd with Lipschitz-continuous boundary. The quadratic regularization functional has two terms, one measuring the squared pointwise ℓ2-discrepancy from the local data, the other containing the squared native space norm of a radial basis function (RBF), multiplied by a regularization parameter. The RBF is chosen so that its native space is equivalent to the (global) Sobolev space of order s on Sd. While both the data and the approximated function are local, we minimize the quadratic functional over all functions in the native space of the RBF, and obtain as exact minimizer a (global) radial basis function approximation. By choosing the RBF to be a Wendland function the resulting linear system has a sparse matrix which is easily computed. We consider three different strategies for choosing the smoothing parameter, namely Morozov’s discrepancy principle and two a priori strategies, and derive L2(Ω) error estimates for each strategy. As auxiliary tools for proving the local L2 error estimates we develop both a local L2 sampling inequality and a suitable Sobolev extension theorem. The paper concludes with numerical experiments.

Local error estimation and step size control in adaptive linear multistep methods

In a k-step adaptive linear multistep methods the coefficients depend on the k − 1 most recent step size ratios. In a similar way, both the actual and the estimated local error will depend on these step ratios. The classical error model has been the asymptotic model, chp+ 1y(p+ 1)(t), based on the constant step size analysis, where all past step sizes simultaneously go to zero. This does not reflect actual computations with multistep methods, where the step size control selects the next step, based on error information from previously accepted steps and the recent step size history. In variable step size implementations the error model must therefore be dynamic and include past step ratios, even in the asymptotic regime. In this paper we derive dynamic asymptotic models of the local error and its estimator, and show how to use dynamically compensated step size controllers that keep the asymptotic local error near a prescribed tolerance tol. The new error models enable the use of controllers with enhanced stability, producing more regular step size sequences. Numerical examples illustrate the impact of dynamically compensated control, and that the proper choice of error estimator affects efficiency.

Goal Oriented Time Adaptivity Using Local Error Estimates

We consider initial value problems (IVPs) where we are interested in a quantity of interest (QoI) that is the integral in time of a functional of the solution. For these, we analyze goal oriented time adaptive methods that use only local error estimates. A local error estimate and timestep controller for step-wise contributions to the QoI are derived. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical tests verify these results. We compare performance with classical local error based time-adaptivity and a posteriori based adaptivity using the dual-weighted residual (DWR) method. For dissipative problems, local error based methods show better performance than DWR and the goal oriented method shows good results in most examples, with significant speedups in some cases.

Embedded Exponentially-Fitted Explicit Runge-Kutta-Nyström Methods for Solving Periodic Problems

In this work, a pair of embedded explicit exponentially-fitted Runge–Kutta–Nyström methods is formulated for solving special second-order ordinary differential equations (ODEs) with periodic solutions. A variable step-size technique is used for the derivation of the 5(3) embedded pair, which provides a cheap local error estimation. The numerical results obtained signify that the new adapted method is more efficient and accurate compared with the existing methods.

Robustness and efficiency of an implicit time-adaptive discontinuous Galerkin solver for unsteady flows

High-fidelity fluid dynamics simulations of unsteady flows are nowadays of great interest for many industrial fields. This class of simulations, as they are characterized by a wide range of temporal scales, requires robust, accurate and efficient long time integration strategies. These features can be achieved by an appropriate coupling of high-order time integration schemes and time-step adaptation algorithms. The adaptation algorithms are typically based on a local error estimator, which exploits the local truncation error of the time integration scheme and of its lower order embedded scheme.In literature few information are available to assess the benefits in terms of robustness, accuracy, and efficiency provided by the coupling between temporal schemes and adaptation strategies for unsteady CFD simulations. The aim of this work is to reduce this gap, presenting a numerical investigation of the performance for different adaptive time-step strategies, based on implicit Rosenbrock-type temporal schemes, in a high-order discontinuous Galerkin solver.The performance of the considered time integration strategies for the autonomous ODE system resulting from the DG space discretization of the Navier–Stokes equations is assessed for several test cases of increasing stiffness and difficulty, identifying the best scheme and algorithm: (i) the 2D laminar flow around a circular cylinder and around a tandem of cylinders at ReD=100; (ii) the 2D viscous flow through a porous media, modelled as an array of cylinders, at ReD=2100 and ReD=10,000; (iii) the 3D turbulent flow through a 4-wheels rudimentary landing gear (RLG) at ReD=1×106.

Verified Solution to Optimal Control Problems of Elastic Rod Motion Based on the Ritz Method

To model vibrations in flexible structures, a generalized variational formulation of PDE control problems is considered in the frame of the method of integro-differential relations. This approach allows us to estimate a posteriori the quality of finite-dimensional approximations and, as a result, either to refine or coarsen them if necessary. Such estimates also make it possible to correct the related control signals. The procedures for solving optimization problems in dynamics of linear elasticity have been proposed based on the Ritz method and FEM. An original FEM solver for mechanical systems withvarying distributed parameters is described. The resulting control law is regularized via a quadratic cost functional including the discrepancy of the constitutive equations. The verification of optimized control for elastic rod motion involves local and integral error estimates proposed.

An adaptive stabilized method for advection–diffusion–reaction equation

We introduce and analyze a Residual Local Projection (RELP) adaptive finite element scheme for the advection–reaction–diffusion equation. This scheme is based on Franca et al. (2009) and combined with a residual a posteriori error estimator (see Verfürth, 1998). The a posteriori estimator is proved to be consistent, reliable, and efficient. More specifically, we proved global upper and local lower error estimates with constants, which are independent of the physical parameters and the mesh-size. Several numerical experiments illustrate the effectiveness of this method.

Estimating local protein model quality: prospects for molecular replacement.

Model quality assessment programs estimate the quality of protein models and can be used to estimate local error in protein models. ProQ3D is the most recent and most accurate version of our software. Here, it is demonstrated that it is possible to use local error estimates to substantially increase the quality of the models for molecular replacement (MR). Adjusting the B factors using ProQ3D improved the log-likelihood gain (LLG) score by over 50% on average, resulting in significantly more successful models in MR compared with not using error estimates. On a data set of 431 homology models to address difficult MR targets, models with error estimates from ProQ3D received an LLG of >50 for almost half of the models 209/431 (48.5%), compared with 175/431 (40.6%) for the previous version, ProQ2, and only 74/431 (17.2%) for models with no error estimates, clearly demonstrating the added value of using error estimates to enable MR for more targets. ProQ3D is available from http://proq3.bioinfo.se/ both as a server and as a standalone download.

A Comparison of Error Estimators for Method of Moments

Local error estimators are investigated for use with numerical solutions of the electric field integral equation. Three-dimensional test targets include a sphere, disk, NASA almond, and a Lockheed Martin Expedite aircraft model. Visual plots and correlation coefficients are used to assess the accuracy of the estimators. It is shown that the inexpensive discontinuity estimators are usually as accurate as the residual method.

Effective Algorithms for Computing Global and Local Posterior Error Estimates of Solutions to Linear Ill-Posed Problems

Effective Algorithms for Computing Global and Local Posterior Error Estimates of Solutions to Linear Ill-Posed Problems