Stable Approximation Research Articles

An equal-order hybridized discontinuous Galerkin method with a small pressure penalty parameter for the Stokes equations

In this paper, an equal-order hybridized discontinuous Galerkin (HDG) method with a small pressure penalty parameter for the Stokes equations is analyzed. When the pressure penalty parameter γ tends to 0 (γ>0), the velocity approximation tends to be H(div)-conforming and exactly divergence-free, and the velocity error bound tends to be pressure-robust. However, taking the value of γ too small will cause the over-stabilization of the pressure. Then, we can provide a post-processing procedure to obtain a stable pressure approximation.

Machine learning for accelerating 2D flood models: Potential and challenges

AbstractTwo‐dimensional hydrodynamic models numerically solve full Shallow Water Equations (SWEs). Despite their high accuracy, these models have long simulation run times and therefore are of limited use for exploratory or real‐time flood predictions. We investigated the possibility of improving flood modelling speed using Machine Learning (ML). We propose a new method that replaces the computationally expensive parts of the hydrodynamic models with simple and efficient data‐driven approximations. Our hypothesis is that by integrating ML with physics‐based numerical methods, we can achieve improved generalization performance: that is, the trained model for one case study can be used in other studies without the need for new training. We tested two ML approaches: for the first, we integrated curve fitting, and, for the second, artificial neural networks (ANN) with a finite volume scheme to solve the local inertial approximation of the SWEs. The data‐driven models approximated the Momentum Equation, which explicitly solved the time derivative of flow rates. Water depths were then updated by applying a water balance equation. We also tested two different training datasets: the simulated dataset, generated from the results of hydrodynamic model, and the random dataset, generated by directly solving the momentum equation on randomly sampled input data. Various combinations of input features, for example, water slope and depth, were explored. The proposed models were trained in a small hypothetical case and tested in a different hypothetical and in two real case studies. Results showed that the curve‐fitting method can be implemented successfully, given sufficient training and input data. The ANN model trained with a random dataset was substantially more accurate than that of the model trained with the simulated dataset. However, it was not successful in the real case studies. The curve‐fitting method resulted in better generalization performance and increased the simulation speed of the local inertial model by 23%. Future research should test the performance of ML in terms of an increase in stable time step size and approximation of the full SWEs.

Non-integrable Stable Approximation by Stein’s Method

We develop Stein’s method for $$\alpha $$ -stable approximation with $$\alpha \in (0,1]$$ , continuing the recent line of research by Xu (Ann Appl Probab 29(1):458–504, 2019) and Chen et al. (J Theor Probab, 2018. https://doi.org/10.1007/s10959-020-01004-1 ) in the case $$\alpha \in (1,2)$$ . The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, the proof strategy is significantly different from that in the integrable case. We rely on fine regularity estimates of the solution to Stein’s equation established in this paper.

Existence and Uniqueness of Weak Solutions for Novel Anisotropic Nonlinear Diffusion Equations Related to Image Analysis

This paper establishes the existence and uniqueness of weak solutions for the initial-boundary value problem of anisotropic nonlinear diffusion partial differential equations related to image processing and analysis. An implicit iterative method combined with a variational approach has been applied to construct approximate solutions for this problem. Then, under some a priori estimates and a monotonicity condition, the existence of unique weak solutions for this problem has been proven. This work has been complemented by a consistent and stable approximation scheme showing its great significance as an image restoration technique.

Appointment Capacity Planning With Overbooking for Outpatient Clinics With Patient No-Shows

Outpatient clinics typically face a critical issue in patient adherence to their clinic appointments and suffer a huge backlog of patients waiting to be seen. The goal of this study is to enhance care provider productivity and patient access through effective clinic appointment scheduling. To achieve this goal, we introduce discrete-time bulk-service queues to model the backlog dynamics accounting for the no-show behavior of patients and consider different overbooking strategies to reduce backlogs at a minimum risk of working overtime. Numerically stable solution procedures and approximation methods are introduced to significantly reduce the computational burden, making the queueing models a practical tool for outpatient appointment template design. The insights obtained from this work will support capacity planning decision-making and are critical to improving the operational efficiency of outpatient clinics.

Goal-oriented adaptivity for a conforming residual minimization method in a dual discontinuous Galerkin norm

We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal-oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection–diffusion–reaction problems.

Performance of FSPAOR iteration for solving one-dimensional space-fractional diffusion equation

This paper considers the numerical solution of a one-dimensional space-fractional diffusion equation. To obtain the solution, we use an unconditionally stable implicit finite difference approximation with the Caputo’s space-fractional operator. We study on improving the convergence rate of the solution while solving the generated linear system through the approximation equation iteratively. In our study, we apply the preconditioning technique to construct a preconditioned linear system which eventually derives into a Full-Sweep Preconditioned AOR. From the presented results, we show that the proposed Full-Sweep Preconditioned AOR iterative method has superiority in efficiency compared to the basic Full-Sweep Preconditioned SOR and Full-Sweep Preconditioned Gauss-Seidel iterative methods.

Non-Iterative Solution Methods for Cauchy Problems for Laplace and Helmholtz Equation in Annulus Domain

This note is concerned with two new methods for the solution of a Cauchy problem. The first method is based on homotopy-perturbation approach which leads to solving a series of well-posed boundary value problems. No regularization is needed in this method. Laplace and Helmholtz equations are considered in an annular region. It is also proved that the homotopy solution for the Laplace operator converges to the actual exact solution. The second method is also non-iterative. It is based on the application of the Green’s second identity which leads to a moment problem for the unknown boundary condition. Tikhonov regularization is used to obtain a stable and close approximation of the missing boundary condition. A number of examples are used to study the applicability of the methods with the presence of noise.

Robust and resource efficient identification of shallow neural networks by fewest samples

Abstract We address the structure identification and the uniform approximation of sums of ridge functions $f(x)=\sum _{i=1}^m g_i(\langle a_i,x\rangle )$ on ${\mathbb{R}}^d$, representing a general form of a shallow feed-forward neural network, from a small number of query samples. Higher order differentiation, as used in our constructive approximations, of sums of ridge functions or of their compositions, as in deeper neural network, yields a natural connection between neural network weight identification and tensor product decomposition identification. In the case of the shallowest feed-forward neural network, second-order differentiation and tensors of order two (i.e., matrices) suffice as we prove in this paper. We use two sampling schemes to perform approximate differentiation—active sampling, where the sampling points are universal, actively and randomly designed, and passive sampling, where sampling points were preselected at random from a distribution with known density. Based on multiple gathered approximated first- and second-order differentials, our general approximation strategy is developed as a sequence of algorithms to perform individual sub-tasks. We first perform an active subspace search by approximating the span of the weight vectors $a_1,\dots ,a_m$. Then we use a straightforward substitution, which reduces the dimensionality of the problem from $d$ to $m$. The core of the construction is then the stable and efficient approximation of weights expressed in terms of rank-$1$ matrices $a_i \otimes a_i$, realized by formulating their individual identification as a suitable nonlinear program. We prove the successful identification by this program of weight vectors being close to orthonormal and we also show how we can constructively reduce to this case by a whitening procedure, without loss of any generality. We finally discuss the implementation and the performance of the proposed algorithmic pipeline with extensive numerical experiments, which illustrate and confirm the theoretical results.

Fast and stable approximation of laminar and turbulent flows in channels by Darcy’s Law

In the design of flow geometries consisting of channels such as high-temperature gas heat exchangers, a great number of design parameters can be chosen, why solving the turbulent Reynolds Averaged Navier-Stokes (RANS) equations by CFD coupled with other physics can become computational demanding. Therefore, we here propose a method for a significant reduction of computational resources and consequent high speed. This is done by using the less computational demanding Darcy’s Law (DL) to approximate the laminar and turbulent flows in channels with circular and arbitrary cross-sections. To obtain the right velocity profile, an artificial permeability variation across the cross-section of the channel is determined. This is obtained based on the analogy of the DL and Darcy–Weisbach equation (DW). Results demonstrate that the DL approximations predict velocities and pressures distributions from the laminar and turbulent flows in the channels with circular and arbitrary cross-sections very well. At the same time, the models with DL approximations reduce the runtime up to ~40 times as compared to RANS, and improve the stability and convergence of the model. Lastly, a cross-flow heat exchanger is studied as an application of the DL approximations.

Hybridized weak Galerkin finite element methods for Brinkman equations

<p style='text-indent:20px;'>This paper presents a hybridized weak Galerkin (HWG) finite element method for solving the Brinkman equations. Mathematically, Brinkman equations can model the Stokes and Darcy flows in a unified framework so as to describe the fluid motion in porous media with fractures. Numerical schemes for Brinkman equations, therefore, must be designed to tackle Stokes and Darcy flows at the same time. We demonstrate that HWG is capable of providing very accurate and stable numerical approximations for both Darcy and Stokes. The main features of HWG is that it approximates the differential operators by their weak forms as distributions and it introduces the Lagrange multipliers to relax certain constraints. We establish the optimal order error estimates for HWG solutions of Brinkman equations. We also present a Schur complement formulation of HWG, which reduces the systems' computational complexity significantly. A number of numerical experiments are provided to confirm the theoretical developments.</p>

On the Energy Stable Approximation of Hamiltonian and Gradient Systems

Abstract A general framework for the numerical approximation of evolution problems is presented that allows to preserve an underlying dissipative Hamiltonian or gradient structure exactly. The approach relies on rewriting the evolution problem in a particular form that complies with the underlying geometric structure. The Galerkin approximation of a corresponding variational formulation in space then automatically preserves this structure which allows to deduce important properties for appropriate discretization schemes including projection based model order reduction. We further show that the underlying structure is preserved also under time discretization by a Petrov–Galerkin approach. The presented framework is rather general and allows the numerical approximation of a wide range of applications, including nonlinear partial differential equations and port-Hamiltonian systems. Some examples will be discussed for illustration of our theoretical results, and connections to other discretization approaches will be highlighted.

Goal-oriented error estimation for the automatic variationally stable FE method for convection-dominated diffusion problems

We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS-FE) method (Calo et al., 2020) for scalar-valued convection–diffusion problems. The AVS-FE method is a Petrov–Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C0 or Raviart–Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov–Galerkin (DPG) method by Demkowicz and Gopalakrishnan (Demkowicz and Gopalakrishnan, 2010; Carstensen et al., 2014; Demkowicz and Gopalakrishnan, 2011a; Demkowicz and Gopalakrishnan, 2011b; Demkowicz and Gopalakrishnan, 2012), leading to unconditionally stable FE approximations. Remarkably, by using C0 or Raviart–Thomas trial spaces, the optimal discontinuous test functions can be computed in a completely decoupled element-by-element fashion.To establish the error estimators we present two approaches: (i) following the classical approach of Becker and Rannacher (Becker and Rannacher, 2001), i.e., the dual solution is sought in the (broken) test space, and (ii) introducing an alternative approach in which we seek C0, or Raviart–Thomas, AVS-FE approximations of the dual solution by using the underlying strong form of the dual boundary value problem (BVP). Various numerical verifications for 2D convection-dominated diffusion BVPs show that the estimates of the approximation error by the new alternative method are highly accurate, while the classical approach leads to error estimates of poor quality. Lastly, we present an algorithm for h-adaptive processes based on control of the numerical approximation error via the new alternative approach. Numerical verifications show that the estimator maintains high accuracy as the error converges to zero.

Foundations for high-order, conservative cut-cell methods: Stable discretizations on degenerate meshes

Cut-cell methods for unsteady flow problems can greatly simplify the grid generation process and allow for high-fidelity simulations on complex geometries. However, cut-cell methods have been limited to low orders of accuracy. This is driven, largely, by the variety of procedures typically introduced to evaluate derivatives in a stable manner near the highly irregular embedded geometry. In the present work, a completely new approach, termed TEMO (truncation error matching and optimization), is taken to solve this problem. The approach is based on two simple and intuitive design principles. These principles directly allow for the construction of stable 8th order approximations to elliptic and parabolic problems. In addition, when combined with the non-linear optimization process of Ref. [6], these principles allow for stable and conservative 4th order approximations to hyperbolic problems without the addition of numerical dissipation. To the best of the authors' knowledge, these are the highest orders ever achieved for a cut-cell discretization by a significant margin. This is done for both explicit and compact finite differences and is accomplished without any geometric transformations or artificial stabilization procedures.

On the evaluation of highly oscillatory integrals with high frequency

We study accurate and stable approximations for highly oscillatory integrals (HOIs) which are demanding in computational engineering, especially for high frequency. In literature, the Levin method with radial basis functions (RBFs) is considered for better accuracy instead of stability of the algorithm. In this work, we demonstrate an accurate and stable Levin method for a class of RBFs. In addition, hybrid functions are coupled with Levin method to compute HOIs in case of stationary point, which returns a stable splitting procedure. Furthermore, error analysis of Levin method in the context of Gaussian RBFs is obtained. Although our error analysis is not significantly improved, however, we present a stable algorithm for evaluation of HOIs. The methods are implemented for both stationary and without stationary point for low and high frequencies. In each case, we address accuracy and stability of the algorithm which is reflected in numerical experiments. The obtained results are compared with well-known methods in the literature. Also, the numerical examples show that the proposed methods are highly accurate even for high frequency.

A neural network based shock detection and localization approach for discontinuous Galerkin methods

The stable and accurate approximation of discontinuities such as shocks on a finite computational mesh is a challenging task. Detection of shocks or strong discontinuities in the flow solution is typically achieved through a priori troubled cell indicators, which guide the subsequent action of an appropriate shock capturing mechanism. Arriving at a stable and accurate solution often requires empirically based parameter tuning and adjustments of the indicator settings to the discretization and solution at hand. In this work, we propose to separate the task of shock detection and shock capturing more strongly and aim to develop a shock indicator that is robust, accurate, requires minimal user input and is suitable for high order element-based methods like discontinuous Galerkin and flux reconstruction methods. The novel indicator is learned from analytical data through a supervised learning strategy; its input is given by the high order solution field, its output is an element-local map of the shock position. We use state of the art methods from edge detection in image analysis based on deep convolutional multiscale networks and deep supervision to train the indicators. The resulting networks are then used as black box indicators, showing their robustness and accuracy on well established canonical testcases. All simulations are run ab initio using the developed indicators, showing that they provide also stability during the strongly transient phases. In particular for high order schemes with large cells and considerable inner-cell resolution capabilities, we demonstrate how the additional accurate prediction of the position of the shock front can be exploited to guide inner-element shock capturing strategies.

Boundary variation diminishing algorithm for high‐order local polynomial‐based schemes

SummaryA novel approach for selecting appropriate reconstructions is implemented to the hyperbolic conservation laws in the high‐order local polynomial‐based framework, for example, the discontinuous Galerkin (DG) and flux reconstruction (FR) schemes. The high‐order polynomial approximation generally fails to correctly capture a strong discontinuity inside a cell due to the Gibbs phenomenon, which is replaced by more stable approximation on the basis of a troubled‐cell indicator such as that used with the total variation bounded (TVB) limiter. This paper examines the applicability of a new algorithm, so‐called boundary variation diminishing (BVD) reconstruction, to the weighted essentially nonoscillatory methodology in the FR framework including the nodal type DG method. The BVD reconstruction adaptively chooses a proper approximation for the solution function so as to minimize the jump between values at the left and right side of cell boundaries. The results of the BVD algorithm are comparable to those with the conventional TVB limiter in terms of oscillation suppression and numerical dissipation in one‐dimensional linear advection and nonlinear system equations, while the TVB limiter performs better in the case with strong discontinuities (the blast wave problem in the Euler equations). Overall, since the present BVD algorithm does not need any ad hoc constant such as the TVB parameter, it could be more reliable than the conventional TVB limiter that is often used in the DG and FR communities for shocks and other discontinuities. The proposed method would lead to a parameter‐free robust algorithm for the local polynomial‐based high‐order schemes.

Image Morphing in Deep Feature Spaces: Theory and Applications

This paper combines image metamorphosis with deep features. To this end, images are considered as maps into a high-dimensional feature space and a structure-sensitive, anisotropic flow regularization is incorporated in the metamorphosis model proposed by Miller and Younes (Int J Comput Vis 41(1):61–84, 2001) and Trouvé and Younes (Found Comput Math 5(2):173–198, 2005). For this model, a variational time discretization of the Riemannian path energy is presented and the existence of discrete geodesic paths minimizing this energy is demonstrated. Furthermore, convergence of discrete geodesic paths to geodesic paths in the time continuous model is investigated. The spatial discretization is based on a finite difference approximation in image space and a stable spline approximation in deformation space; the fully discrete model is optimized using the iPALM algorithm. Numerical experiments indicate that the incorporation of semantic deep features is superior to intensity-based approaches.

Non-intrusive model reduction of large-scale, nonlinear dynamical systems using deep learning

Projection-based model reduction has become a popular approach to reduce the cost associated with integrating large-scale dynamical systems so they can be used in many-query settings such as optimization and uncertainty quantification. For nonlinear systems, significant cost reduction is only possible with an additional layer of approximation to reduce the computational bottleneck of evaluating the projected nonlinear terms. Prevailing methods to approximate the nonlinear terms are code intrusive, potentially requiring years of development time to integrate into an existing codebase, and have been known to lack parametric robustness.This work develops a non-intrusive method to efficiently and accurately approximate the expensive nonlinear terms that arise in reduced nonlinear dynamical system using deep neural networks. The neural network is trained using only the simulation data used to construct the reduced basis and evaluations of the nonlinear terms at these snapshots. Once trained, the neural network-based reduced-order model only requires forward and backward propagation through the network to evaluate the nonlinear term and its derivative, which are used to integrate the reduced dynamical system at a new parameter configuration. We provide two numerical experiments – the dynamical systems result from the semi-discretization of parametrized, nonlinear, hyperbolic partial differential equations – that show, in addition to non-intrusivity, the proposed approach provides more stable and accurate approximations to each dynamical system across a large number of training and testing points than the popular empirical interpolation method.

Pricing American Options with a Non-Constant Penalty Parameter

As the American early exercise results in a free boundary problem, in this article we add a penalty term to obtain a partial differential equation, and we also focus on an improved definition of the penalty term for American options. We replace the constant penalty parameter with a time-dependent function. The novelty and advantage of our approach consists in introducing a bounded, time-dependent penalty function, enabling us to construct an efficient, stable, and adaptive numerical approximation scheme, while in contrast, the existing standard approach to the penalisation of the American put option-free boundary problem involves a constant penalty parameter. To gain insight into the accuracy of our proposed extension, we compare the solution of the extension to standard reference solutions from the literature. This illustrates the improvement of using a penalty function instead of a penalising constant.