Optimal Convergence Properties Research Articles

Advancing nonlinear problem solutions: Newton-like iterative techniques for varied convergence orders using homotopy perturbation

Abstract Iterative methods are crucial in numerical analysis for approximating solutions to nonlinear problems encountered in various fields, such as biomathematics, thermodynamics, chemical engineering, and fluid mechanics. This paper introduces innovative iterative methods of convergence orders three and six to eight, developed using the homotopy perturbation technique (HPT). We address the limitations of traditional derivative-based methods by incorporating divided differences, resulting in derivative-free schemes with optimal convergence properties. We present four newly designed algorithms (HPM1, HPM2, HPM3, and HPM4) and provide a comprehensive convergence analysis. Numerical simulations and basins of attraction demonstrate the superior performance of the proposed methods compared to existing methods of the same order. The proposed methods exhibit faster convergence and larger basins of attraction, making them highly effective for solving nonlinear equations. Our new numerical methods rely on approximations and iterative processes, with each small step bringing us closer to the exact solution. Exploring these numerical methods encourages us to apply mathematics to solve challenging problems in physics, engineering, finance, and more.

Mixed finite element analysis for a modified Poisson–Fermi interface problem accounting for electrostatic correlations

The Bazant–Storey–Kornyshev (BSK) theory [1–3] recently developed an important continuum framework to expound the nonlocal dielectric permittivity of ionic liquids due to electrostatic correlations, leading to a fourth-order modified Poisson–Fermi equation to model the electrostatic potential field in the solvent (e.g., the electrolyte), while the standard second-order Poisson–Fermi equation is still valid in modeling the electrostatic potential field in the solute. Thus an interface problem is formed between the fourth-order and second-order Poisson–Fermi equations through the interface of solvent and solute with jump coefficients, which has exerted tremendous impacts on applications of electrokinetics, electrochemistry, biophysics, and etc. In this paper, a type of mixed finite element method is developed to solve the proposed interface problem once for all variables: the electrostatic potential, electric field, electric displacement field, electrostatic stress as well as interactional force in the electrolyte in an accurate fashion, and its optimal convergence properties are analyzed for all variables in their respective norms. Numerical experiments are carried out through self-defined mathematical examples to validate all attained theoretical results. Furthermore, as a part of modeling verification for the presented interface problem that models the BSK theory, a practically physical example is investigated to validate the necessity of introducing the fourth-order modified Poisson–Fermi equation to describe the electrostatic correlation effects due to charge reversal phenomenon.

Structure-preserving finite element methods for computing dynamics of rotating Bose-Einstein condensate

This work is concerned with the construction and analysis of structure-preserving Galerkin methods for computing the dynamics of rotating Bose-Einstein condensate (BEC) based on the Gross-Pitaevskii equation with angular momentum rotation. Due to the presence of the rotation term, constructing finite element methods (FEMs) that preserve both mass and energy remains an unresolved issue, particularly in the context of nonconforming FEMs. Furthermore, in comparison to existing works, we provide a comprehensive convergence analysis, offering a thorough demonstration of the methods’ optimal and high-order convergence properties. Finally, extensive numerical results are presented to check the theoretical analysis of the structure-preserving numerical method for rotating BEC, and the quantized vortex lattice’s behavior is scrutinized through a series of numerical tests.

Automatic variationally stable analysis for finite element computations: Transient convection-diffusion problems

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time “slices” in an attempt to reduce the computational cost of the space-time discretizations.We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6], the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serves as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive strategies.

A comparison between marginal likelihood and data augmented MCMC algorithms for Gaussian hidden Markov models

We provide a comparison between marginal likelihood and data augmented Markov chain Monte Carlo (MCMC) algorithms for Bayesian estimation of hidden Markov models. In particular, we focus on a specification of these models based on a multivariate Gaussian distribution for the response variables. We evaluate the performance of both methods on the basis of simulated samples and an empirical application, where we compare these approaches in terms of standard measures of estimation and time efficiency. The results show that the augmented algorithm is preferable, obtaining optimal convergence properties and producing larger values of the effective sample size. Instead of using variable dimensional techniques for model selection, we suggest a method based on the so-called parallel sampling for this aim.

EXtended Hybridizable Discontinuous Galerkin (X-HDG) method for linear convection-diffusion equations on unfitted domains

In this work, we propose a novel strategy for the numerical solution of linear convection diffusion equation (CDE) over unfitted domains. In the proposed numerical scheme, strategies from high order Hybridized Discontinuous Galerkin method and eXtended Finite Element method are combined with the level set definition of the boundaries. The proposed scheme and hence, is named as eXtended Hybridizable Discontinuous Galerkin (XHDG) method. In this regard, the Hybridizable Discontinuous Galerkin (HDG) method is eXtended to the unfitted domains; i.e., the computational mesh does not need to fit to the domain boundary; instead, the boundary is defined by a level set function and cuts through the background mesh arbitrarily. The original unknown structure of HDG and its hybrid nature ensuring the local conservation of fluxes is kept, while developing a modified bilinear form for the elements cut by the boundary. At every cut element, an auxiliary nodal trace variable on the boundary is introduced, which is eliminated afterwards while imposing the boundary conditions. Both stationary and time dependent CDEs are studied over a range of flow regimes from diffusion to convection dominated; using high order (p≤4) XHDG through benchmark numerical examples over arbitrary unfitted domains. Results proved that XHDG inherits optimal (p+1) and super (p+2) convergence properties of HDG while removing the fitting mesh restriction.

Energy-stable finite element method for a class of nonlinear fourth-order parabolic equations

In this paper, an energy-stable finite element method with the Crank–Nicolson type of temporal discretization scheme is developed and analyzed for a class of nonlinear fourth-order parabolic equations (including the Swift–Hohenberg (SH) equation and the extended Fisher–Kolmogorov (EFK) equation). In addition to the energy stability properties, the optimal spatial convergence properties in both L∞(L2)- and L∞(H1)-norm and the second-order temporal approximation rate are also obtained for numerical solutions approximating to the real solution and its Laplacian for the developed energy-stable finite element method in both semi- and fully discrete schemes. Numerical experiments are carried out to validate all attained theoretical results.

Simulating hyperelasticity and fractional viscoelasticity in the human heart

Biomechanics plays an important role in the diagnosis and treatment of pathological conditions of the heart. Computational models are paving the way for personalized therapeutic treatment but they rely on accurate constitutive equations for predicting their biomechanical behavior. Even so, viscoelasticity remains under-explored in computational modeling despite experimental observations. To facilitate the viscoelastic modeling of cardiovascular soft tissues, we previously developed a fractional viscoelastic modeling approach, which extends existing hyperelastic models. This has comparable computational costs to the conventional hyperelastic model and only requires two additional material parameters for the viscoelastic response. This approach was demonstrated to be able to accurately capture the viscoelastic response of the human myocardium. However, the numerical properties of this fractional viscoelastic approach have not yet been examined. In this work, we present its implementation in Finite Element Analysis, examine its numerical properties in uniaxial extension and 2D inflation test examples, and examine its physiological implication in a computational model of an idealized left ventricle in a fully idealized circulatory system. Optimal convergence properties were observed and the importance of viscoelasticity during passive filling, ventricular motion, and regional fiber strain and stresses were explained.

Fluid-Fluid Interaction Problems at High Reynolds Numbers: Reducing the Modeling Error with LES-C

We consider a fluid-fluid interaction problem, where two flows (with high Reynolds numbers for one or both of these flows) are coupled through a joint interface. A nonlinear coupling equation, known as the rigid lid condition, creates an extra level of difficulty, typical for atmosphere-ocean problems. We propose a novel turbulence model, NS- -C, from the recently introduced family of LES-C (large eddy simulation with correction) models. Combining it with the so-called geometric averaging (GA) partitioning method, we obtain the NS- -C-GA model that is shown to possess several key properties. First, the preexisting solvers for the subdomains can be used, which is critical, e.g., for atmosphere-ocean applications. Second, the LES-C turbulence models use defect correction to efficiently reduce the modeling error of the corresponding LES models; we demonstrate numerically that the NS- -C model outperforms its LES counterpart, the NS- model. It has also been shown recently that it is favorable for an LES model to have the nonfiltered velocity in the interface terms. The NS- -C-GA model possesses this important property; we also show it to be stable and have optimal convergence properties.

A priori error analysis for an isogeometric discontinuous Galerkin approximation for convection problems on surfaces

In this paper, we identify and study the new isogeometric analysis penalty discontinuous Galerkin (DG) methods of convection problems on implicitly defined surfaces with optimal convergence properties. Like all other known discontinuous Galerkin methods on flat space or Euclidean space using polynomials of degree k≥0 for the unknown, the orders of convergence in L2 norm and DG norm are k+1 and k+12, respectively, which shows the resulting methods on surfaces can be implemented as efficiently as they are for the case of flat space or Euclidean space. The theoretical results are illustrated by two numerical experiments.

An energy-conserving finite element method for nonlinear fourth-order wave equations

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.

Multistep collocation methods for integral-algebraic equations with non-vanishing delays

In this article, we study the piecewise multistep collocation method for a class of functional integral equations with non-vanishing delays. Based on the notions of the tractability index and the ν-smoothing property of a Volterra integral operator, our numerical analysis and optimal convergence properties are investigated. Here, multistep collocation method which depends on the numerical solution in a fixed number of previous time steps is described by the constructive technique and dividing the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. Numerical experiments confirm the theoretical expectations.

Finite element approximations to a fourth-order modified Poisson-Fermi equation for electrostatic correlations in concentrated electrolytes

In this paper, both the standard finite element method (FEM) and the mixed FEM are developed for the fourth-order modified Poisson-Fermi equation resulted from the Bazant-Storey-Kornyshev (BSK) theory to account for electrostatic correlations in concentrated electrolytes. Optimal convergence properties are obtained for both FEMs in their respective norms, additionally, the mixed FEM can produce one-order higher approximation accuracy for the electric field in L2 norm than that of the standard FEM, resulting in more accurate approximations to the electrostatic stress and the force of interaction in concentrated electrolytes. All attained theoretical results are validated by numerical experiments. Furthermore, a practical example is studied to validate the necessity of introducing the fourth-order modified Poisson-Fermi equation to describe the electrostatic potential field due to correlation effects by comparing with the classical Poisson equation that accounts for the classical mean-field Poisson-Boltzmann (PB) theory, illustrating that numerical trends obtained from the modified model are all consistent with experimental results under the circumstance of electrostatic correlations in concentrated electrolytes, which however cannot be predicted by the classical model.

NS-[formula omitted] model for fluid–fluid interaction problems at high Reynolds numbers

We consider a fluid–fluid interaction problem, where two flows are coupled through a nonlinear rigid lid condition, and one or both of these flows are at high Reynolds numbers. A method is proposed, that combines the NS-ω turbulence model with a partitioning method—so that the resulting model is efficiently decoupled, allows for the usage of preexisting solvers (e.g., air and water domain solvers), and is capable of resolving the flows at high Reynolds numbers. The model is shown to be unconditionally stable and have optimal convergence properties. The model also allows for the non-filtered velocity in the interface terms; it has been shown recently that filtering the velocity in the interface terms could corrupt the quality of the model’s solution. To the best of the authors’ knowledge, this is the first model with non-filtered velocity on the interface, that lends itself to full numerical analysis.

Bayesian group testing with dilution effects.

A Bayesian framework for group testing under dilution effects has been developed, using lattice-based models. This work has particular relevance given the pressing public health need to enhance testing capacity for coronavirus disease 2019 and future pandemics, and the need for wide-scale and repeated testing for surveillance under constantly varying conditions. The proposed Bayesian approach allows for dilution effects in group testing and for general test response distributions beyond just binary outcomes. It is shown that even under strong dilution effects, an intuitive group testing selection rule that relies on the model order structure, referred to as the Bayesian halving algorithm, has attractive optimal convergence properties. Analogous look-ahead rules that can reduce the number of stages in classification by selecting several pooled tests at a time are proposed and evaluated as well. Group testing is demonstrated to provide great savings over individual testing in the number of tests needed, even for moderately high prevalence levels. However, there is a trade-off with higher number of testing stages, and increased variability. A web-based calculator is introduced to assist in weighing these factors and to guide decisions on when and how to pool under various conditions. High-performance distributed computing methods have also been implemented for considering larger pool sizes, when savings from group testing can be even more dramatic.

Cut finite element methods and ghost stabilization techniques for space‐time discretizations of the Navier–Stokes equations

AbstractWe propose and analyze computationally a new fictitious domain method, based on higher order space‐time finite element discretizations, for the simulation of the nonstationary, incompressible Navier–Stokes equations on evolving domains. The physical domain is embedded into a fixed computational mesh such that arbitrary intersections of the moving domain's boundaries with the background mesh occur. The potential of such cut finite element techniques for higher order space‐time finite element methods has rarely been studied in the literature so far and deserves further elucidation. The key ingredients of the approach are the weak formulation of Dirichlet boundary conditions by Nitsche's method, the flexible and efficient integration over all types of intersections of cells by moving boundaries and the spatial extension of the discrete physical quantities to the entire computational background mesh including fictitious (ghost) subdomains of fluid flow. Thereby, an expensive remeshing and adaptation of the sparse matrix data structure are avoided and the computations are accelerated. To prevent spurious oscillations caused by irregular intersections of mesh cells, a penalization, defining also implicitly the extension to ghost domains, is added. These techniques are embedded in an arbitrary order, discontinuous Galerkin discretization of the time variable and an inf‐sup stable discretization of the spatial variables. The parallel implementation of the matrix assembly is described. The optimal order convergence properties of the algorithm are illustrated in a numerical experiment for an evolving domain. The well‐known 2d benchmark of flow around a cylinder as well as flow around moving obstacles with arising cut cells and fictitious domains are considered further.

A Basic AC Power Flow Based on the Bus Admittance Matrix Incorporating Loads and Generators Including Slack Bus

This paper presents an algorithm for solving the AC basic power flow based on some enrichments provided in the bus admittance matrix methods findable in the literature. In particular, the interpretation of the slack bus generator as a current source rather than a voltage one and its inclusion inside an “all-inclusive” admittance matrix allows obtaining strong performances of the algorithm. In fact, this method gives both a well conditioning of the admittance matrix and the reduction of matrix partitioning for each iteration. As a result, a greater precision of the solution, a shorter execution time compared to classical commercial methods, a decreasing number of iterations and optimal convergence properties are obtained. Eventually, in order to show the efficiency of the method, real and fictious networks are tested, by comparing its results and performances with robust open source/commercial software packages that use well-known methods ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i.e.</i> , Newton-Raphson and Fast Decoupled Load Flow methods).

CRPS learning

Combination and aggregation techniques can significantly improve forecast accuracy. This also holds for probabilistic forecasting methods where predictive distributions are combined. There are several time-varying and adaptive weighting schemes such as Bayesian model averaging (BMA). However, the quality of different forecasts may vary not only over time but also within the distribution. For example, some distribution forecasts may be more accurate in the center of the distributions, while others are better at predicting the tails. Therefore, we introduce a new weighting method that considers the differences in performance over time and within the distribution. We discuss pointwise combination algorithms based on aggregation across quantiles that optimize with respect to the continuous ranked probability score (CRPS). After analyzing the theoretical properties of pointwise CRPS learning, we discuss B- and P-Spline-based estimation techniques for batch and online learning, based on quantile regression and prediction with expert advice. We prove that the proposed fully adaptive Bernstein online aggregation (BOA) method for pointwise CRPS online learning has optimal convergence properties. They are confirmed in simulations and a probabilistic forecasting study for European emission allowance (EUA) prices.

An overlapping local projection stabilization for Galerkin approximations of Stokes and Darcy flow problems

An a priori analysis for a generalized local projection stabilized finite element approximation of the Stokes, and the Darcy flow equations are presented in this paper. A first-order conforming P1c finite element space is used to approximate both the velocity and pressure. It is shown that the stabilized discrete bilinear form satisfies the inf-sup condition in the generalized local projection norm. Moreover, a priori error estimates are established in a mesh-dependent norm as well as in the L2-norm for the velocity and pressure. The optimal and quasi-optimal convergence properties are derived for the Stokes and the Darcy flow problems. Finally, the derived estimates are numerically validated with appropriate examples.

Optimally convergent mixed finite element methods for the stochastic Stokes equations

Abstract We propose some new mixed finite element methods for the time-dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known (Langa, J. A., Real, J. &amp; Simon, J. (2003) Existence and regularity of the pressure for the stochastic Navier--Stokes equations. Appl. Math. Optim., 48, 195--210) that the pressure solution has low regularity, which manifests in suboptimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations; see Feng X., &amp; Qiu, H. (Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise. arXiv:1905.03289v2 [math.NA]). We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods and that this conceptual idea may be used to retool numerical methods that are well known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptual usefulness of the proposed numerical approach.