Robust Discretization Research Articles

Compactness results for a Dirichlet energy of nonlocal gradient with applications

AbstractWe prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem.

An isogeometric approach to coupled thermomechanics in 3D via hierarchical adaptivity

In the present work, a complete numerical scheme to perform adaptive refinement and coarsening in the context of isogeometric analysis applied to thermomechanics is introduced. In particular, truncated hierarchical B-splines (THB-splines) are considered as a basis for the analysis and suitably graded isogeometric meshes are constructed to guarantee admissible configurations at each refinement and coarsening iteration. The algorithms are developed and implemented in a full three-dimensional framework and applied to solve coupled thermoelastic problems. The proposed numerical framework is verified on two relevant problems with respect to overkill solutions, obtained either with a commercial finite element software or with uniform isogeometric meshes. Different refinement and coarsening strategies are considered and compared. The obtained results demonstrate the ability of the THB-spline-based adaptive scheme to deliver robust discretizations guaranteeing a suitable trade-off between computational accuracy and number of degrees of freedom.

Bregman dynamics, contact transformations and convex optimization

Recent research on accelerated gradient methods of use in optimization has demonstrated that these methods can be derived as discretizations of dynamical systems. This, in turn, has provided a basis for more systematic investigations, especially into the geometric structure of those dynamical systems and their structure-preserving discretizations. In this work, we introduce dynamical systems defined through a contact geometry which are not only naturally suited to the optimization goal but also subsume all previous methods based on geometric dynamical systems. As a consequence, all the deterministic flows used in optimization share an extremely interesting geometric property: they are invariant under contact transformations. In our main result, we exploit this observation to show that the celebrated Bregman Hamiltonian system can always be transformed into an equivalent but separable Hamiltonian by means of a contact transformation. This in turn enables the development of fast and robust discretizations through geometric contact splitting integrators. As an illustration, we propose the Relativistic Bregman algorithm, and show in some paradigmatic examples that it compares favorably with respect to standard optimization algorithms such as classical momentum and Nesterov’s accelerated gradient.

A robust discretization method of factor screening for landslide susceptibility mapping using convolution neural network, random forest, and logistic regression models

ABSTRACT The selection of discretization criteria and interval numbers of landslide-related environmental factors generally fails to quantitatively determine or filter, resulting in uncertainties and limitations in the performance of machine learning (ML) methods for landslide susceptibility mapping (LSM). The aim of this study is to propose a robust discretization criterion (RDC) to quantify and explore the uncertainty and subjectivity of different discretization methods. The RDC consists of two steps: raw classification dataset generation and optimal dataset extraction. To evaluate the robustness of the proposed RDC method, Lushan County of Sichuan Province in China was chosen as the study area to generate the LSM based on three datasets (optimal dataset, original dataset with continuous values, and statistical dataset) using three popular ML methods, namely, convolution neural network, random forest, and logistic regression. The results show that the areas under the receiver operating characteristic curve (AUCs) of the optimal dataset for the abovementioned ML models are 0.963, 0.961, and 0.930 which are higher than those of the original dataset (0.938, 0.947, and 0.900) and statistical dataset (0.948, 0.954, and 0.897). In conclusion, the RDC method can extract the more representative features from environmental factors and outperform the other conventional discretization methods.

A Robust Discrete Scheme based on Staggered Grids for Poroelastic-Elastic Coupled Problems

A Robust Discrete Scheme based on Staggered Grids for Poroelastic-Elastic Coupled Problems

On robust discretization methods for poroelastic problems: Numerical examples and counter-examples

Finite element approximations of poroelastic materials are nowadays used within multiple applications. Due to wide variation of possible material parameters, robustness of the considered discretization is important. Within this contribution robust of discretization schemes, initially developed for Biot’s theory, will be applied within the Theory of Porous Media. Selected numerical test-cases, special attention will be paid to incompressible and impermeable regimes, are conducted.

A new method for controlling an induction motor using a hybrid discretization model predictive field orientated control.

The dynamic performance of the Model Predictive Control (MPC) of an Induction Motor (IM) relies on the accuracy and computational efficiency of the Discretisation Technique (DT). If the discretisation process is inaccurate or slow approximation, the MPC will exhibit high torque ripple and lower load handling capabilities. Traditionally, Euler’s method is used to discretise the MPC, which merely relies on the predictor to yield a fast, but less accurate system approximation. In contrast, Heun’s method uses a combination of predictor and corrector at alternate sampling intervals to improve the discretisation accuracy; however, the controller response becomes slow due to increased computational intensity of the algorithm. In this study, a new Hybrid Discretisation Technique (HDT) for Model Predictive Field Oriented Control (MPFOC) for IM control systems is presented to achieve robust discretisation with improved accuracy. In the proposed approach, Euler’s method is used to discretise the system at the first nine samples, followed by the predictor-corrector at the tenth sampling interval, accomplishing the desired speed and accuracy of discretisation. This newly proposed HDT in MPFOC is verified with Processor-In-Loop (PIL) for a three-phase IM with bi-directional rotation under varying load conditions. The results indicate that the IM torque ripple is reduced by up to 20%, whereas, the load handling capability is increased by up to 10%. Moreover, the controller gives 20% and 23% improvement in rise time and settling time, respectively, under high loading conditions, as compared to traditional Euler and Heun methods.

Very high-order accurate finite volume scheme for the steady-state incompressible Navier–Stokes equations with polygonal meshes on arbitrary curved boundaries

The numerical solution of the incompressible Navier–Stokes equations raises challenging numerical issues on the development of accurate and robust discretisation techniques due to the div-grad duality. Significant advances have been achieved in the context of high-order of convergence methods, but many questions remain unsolved in the quest of practical, accurate, and efficient approaches. In particular, the discretisation of problems with arbitrary curved boundaries has been the subject of intensive research, where the conventional practice consists in employing curved meshes to avoid accuracy deterioration. The present work proposes a novel approach, the reconstruction for off-site data method, which employs polygonal meshes to approximate arbitrary smooth curved boundaries with linear piecewise elements. The Navier–Stokes equations are discretised with a staggered finite volume method, and the numerical fluxes are computed on the polygonal mesh elements. Boundary conditions are taken into account via polynomial reconstructions with specific linear constraints defined for a set of points on the physical boundary. The very-high order of convergence is preserved without relying on the full curve parametrisation, avoiding the limitations of curved mesh approaches, such as sophisticated mesh generation algorithms, cumbersome quadrature rules, and complex non-linear transformations. A comprehensive verification benchmark is provided, with numerical test cases for several fluid flow problems, to demonstrate the capability of the proposed approach to achieve very high-orders of convergence.

A nonsymmetric approach and a quasi-optimal and robust discretization for the Biot’s model

We consider the system of partial differential equations stemming from the time discretization of the two-field formulation of the Biot’s model with the backward Euler scheme. A typical difficulty encountered in the space discretization of this problem is the robustness with respect to various material parameters. We deal with this issue by observing that the problem is uniformly stable, irrespective of all parameters, in a suitable nonsymmetric variational setting. Guided by this result, we design a novel nonconforming discretization, which employs Crouzeix-Raviart and discontinuous elements. We prove that the proposed discretization is quasi-optimal and robust in a parameter-dependent norm and discuss the consequences of this result.

Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

Abstract We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}} . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

Locking-Free and Gradient-Robust $${\varvec{H}}({{\,\mathrm{{\text {div}}}\,}})$$-Conforming HDG Methods for Linear Elasticity

Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new: it assures that dominant gradient fields in the momentum balance do not lead to spurious displacements. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

A Robust Discrete Wavelet Transform Based Adaptive Watermarking Scheme in YCbCr Color Space against Camcorder Recording in Cinema/Movie Theates

A Robust Discrete Wavelet Transform Based Adaptive Watermarking Scheme in YCbCr Color Space against Camcorder Recording in Cinema/Movie Theates

Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model

We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements. We consider the mathematical analysis and the numerical approximation of the problem. More precisely, we carry out firstly the well-posedness analysis of the model. Then, we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation; stability and numerical error estimates are proved.Particular attention is dedicated to the study of the saddle-point structure of the problem, that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi-incompressibility constraint. Our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity, solid displacement and pressure, to obtain a stable and robust discretization based on Taylor–Hood type finite element spaces. Interestingly, we show how this choice depends on the porosity of the mixture, i.e. the volume fraction of the fluid phase.

Quasi-Optimal and Pressure Robust Discretizations of the Stokes Equations by Moment- and Divergence-Preserving Operators

Abstract We approximate the solution of the Stokes equations by a new quasi-optimal and pressure robust discontinuous Galerkin discretization of arbitrary order. This means quasi-optimality of the velocity error independent of the pressure. Moreover, the discretization is well-defined for any load which is admissible for the continuous problem and it also provides classical quasi-optimal estimates for the sum of velocity and pressure errors. The key design principle is a careful discretization of the load involving a linear operator, which maps discontinuous Galerkin test functions onto conforming ones thereby preserving the discrete divergence and certain moment conditions on faces and elements.

A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree

A Robust Discretization of the Reissner-Mindlin Plate with Arbitrary Polynomial Degree

Asymptotically Compatible Schemes for Robust Discretization of Parametrized Problems with Applications to Nonlocal Models

Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are con...

SIGEST

The SIGEST article in this issue, “Asymptotically Compatible Schemes for Robust Discretization of Parametrized Problems with Applications to Nonlocal Models,” by Xiaochuan Tian and Qiang Du, concerns differential equation models that (a) involve a physical parameter, and (b) change their nature when this parameter reaches an asymptotic limit. The work is motivated by nonlocal peridynamic models in continuum mechanics, which reduce to classical differential equation models as the horizon parameter tends to zero. In this setting, a numerical method is said to be asymptotically compatible if it produces approximations that are robust in such a limit. This is an important property which allows us to avoid inaccurate or unphysical numerical solutions from inappropriate choices of model and discretization parameters. The authors develop a general framework that relies on minimal assumptions. Their high-level results, provided in Theorems 2.6 to 2.8, are neatly summarized in Figure 2.1. The results are then applied to finite element discretizations of nonlocal diffusion and peridynamic models. Numerical experiments are also provided to validate the results. The original article appeared in the SIAM Journal on Numerical Analysis (SINUM) in 2014. As indicated by its high citation count, this work is relevant to many circumstances where multiscale or nonlocal effects are present; examples include viscosity limits in nonlinear conservation laws, phase field models with sharp interfaces, and smoothed particle hydrodynamics. The original SINUM article presented an asymptotically compatible framework for self-adjoint problems. In converting their article into SIGEST form, the authors have extended the results to cover both self-adjoint and non-self-adjoint problems with parameter-dependent data. Here, some modifications, and new notation, were required to distinguish between operators and their adjoints, between test and trial functions, and between the associated function spaces. The authors also reorganized the derivation of the main results and streamlined the presentation of numerical experiments. In addition, they have discussed subsequent developments in the field, increasing the size of the bibliography from 40 to 63, and highlighted some current challenges. We also note that the results in the original SINUM article formed part of the 2017 Columbia University Ph.D. dissertation of the first author, Xiaochuan Tian, who was subsequently awarded an Association for Woman in Mathematics Dissertation Prize.

Generalized Scharfetter–Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

Many challenges faced in today's semiconductor devices are related to self-heating phenomena. The optimization of device designs can be assisted by numerical simulations using the non-isothermal drift-diffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the so-called Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known self-heating effects, namely Joule, recombination and Thomson–Peltier heating, without any further (transient) contributions. Moreover, the thermal driving force in the electrical current density expressions can be entirely absorbed in the diffusion coefficient via a generalized Einstein relation. The efficient numerical simulation relies on an accurate and robust discretization technique for the fluxes (finite volume Scharfetter–Gummel method), which allows to cope with the typically stiff solutions of the semiconductor device equations. We derive two non-isothermal generalizations of the Scharfetter–Gummel scheme for degenerate semiconductors (Fermi–Dirac statistics) obeying the Kelvin formula. The approaches differ in the treatment of degeneration effects: The first is based on an approximation of the discrete generalized Einstein relation implying a specifically modified thermal voltage, whereas the second scheme follows the conventionally used approach employing a modified electric field. We present a detailed analysis and comparison of both schemes, indicating a superior performance of the modified thermal voltage scheme.

Image deconvolution for confocal laser scanning microscopy using constrained total variation with a gradient field

As for the confocal laser scanning microscope (CLSM) imaging system, the collected weak fluorescence signals are always distorted by optic blur and severe photon-counting noise, and the deconvolution for CLSM images is a typical ill-posed inverse problem, which is highly sensitive to the measurement noise. To promote the reconstruction quality for characteristics of low intensity and strong noise, we employed the prominent total variation regularization (TV) to enforce the sparsity of a fluorescent image gradient with rich details. However, the well-known reconstruction artifacts (e.g., artificial staircase) emerge with TV prior. To settle this issue, we utilized a robust first-order discretization yielding near-isotropy with a gradient field to depress the reconstruction artifacts. Furthermore, the bound constraint was suited to restrain final reconstruction results from appearing unreasonably explosive. For the proposed optimization minimizer with linear constraint, we take one proximal gradient for approximate estimation of each subproblem under the framework of the inexact alternating direction method of multipliers. Moreover, we incorporated a Nesterov's scheme into the numerical method for acceleration of iteration updating. Compared with other competing methods, both the simulation and practical results demonstrate the effectiveness of our proposed model for CLSM image deconvolution.

Tomographic Image Reconstruction Based on Minimization of Symmetrized Kullback-Leibler Divergence

Iterative reconstruction (IR) algorithms based on the principle of optimization are known for producing better reconstructed images in computed tomography. In this paper, we present an IR algorithm based on minimizing a symmetrized Kullback-Leibler divergence (SKLD) that is called Jeffreys’ J-divergence. The SKLD with iterative steps is guaranteed to decrease in convergence monotonically using a continuous dynamical method for consistent inverse problems. Specifically, we construct an autonomous differential equation for which the proposed iterative formula gives a first-order numerical discretization and demonstrate the stability of a desired solution using Lyapunov’s theorem. We describe a hybrid Euler method combined with additive and multiplicative calculus for constructing an effective and robust discretization method, thereby enabling us to obtain an approximate solution to the differential equation. We performed experiments and found that the IR algorithm derived from the hybrid discretization achieved high performance.