Theoretical Convergence Rates Research Articles

Velocity-vorticity-pressure formulation for the Oseen problem with variable viscosity

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babu\vska-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation.

Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model

The Cahn-Hilliard phase-field model of two-phase incompressible flows, namely the Cahn-Hilliard-Navier-Stokes (CH-NS) problem represents the fundamental building blocks of hydrodynamic phase-field models for multiphase fluid flow dynamics. Since finding solution (numerically and theoretically) of the CH-NS system is non-trivial (owing to the coupling between the Navier-Stokes equation and the Cahn-Hilliard equation), in this paper, we propose and analyze a numerical scheme with the following properties: (1) first-order in time, (2) nonlinear, (3) fully coupled and (4) energy stable; for solving CH-NS system, in the framework of weak Galerkin (WG) method. More precisely, we employ the WG method, which uses discontinuous functions to construct the approximation space, and the first-order backward Euler (implicit) method for space and time discretizations, respectively. We first recall the corresponding variational formulation, and then summarize the main WG method ingredients that are required for our discrete analysis. In particular, in order to define the weak discrete bilinear (and trilinear) form, whose continuous version involves classical differential operators, we propose two well-known alternatives for gradient and divergence operators onto a suitable polynomial subspace. Next, we show that the weak global discrete bilinear form satisfies the hypotheses required by the Lax-Milgram lemma. In this way, we derive the associated a priori error estimates for phase field variable, chemical potential, velocity and further the pressure in L2 norm. Finally, several numerical results confirming the theoretical rates of convergence and illustrating the good performance of the method for simulation influence of surface tension, small density variations, and the driven cavity are presented.

Numerical analysis of a porous–elastic model for convection enhanced drug delivery

Convection enhanced drug delivery (CED) is a technique used to make therapeutic agents reach, through a catheter, sites of difficult access. The name of this technique comes from the convective flow originated by a pressure gradient induced at the tip of the catheter. This flow enhances passive diffusion and allows a more efficient spread of the agents by the target site. CED is particularly useful in the treatment of diseases that affect the central nervous system, where the blood–brain barrier prevents the diffusion of most therapeutic agents from the cerebral blood vessels to the brain interstitial space.In this work we deal with the numerical analysis of a coupled system of partial differential equations that can be used to simulate CED in an elastic medium like brain tissue. The model variables are the fluid velocity, the pressure, the tissue deformation, and the agents concentration. We prove the stability of the coupled problem and from the numerical point of view we propose a fully discrete piecewise linear finite element method (FEM). The convergence analysis shows that the method has second order convergence for the pressure, displacement, and concentration. Numerical experiments illustrating the theoretical convergence rates and the behavior of the system are also given.

Some aspects on the computational implementation of diverse terms arising in mixed virtual element formulations

In the present paper, we describe the computational implementation of some integral terms that arise from mixed virtual element methods (mixed-VEM) in two-dimensional pseudostress-velocity formulations. The implementation presented here considers any polynomial degree k ≥ 0 in a natural way by building several local matrices of small size through the matrix multiplication and the Kronecker product. In particular, we apply the foregoing mentioned matrices to the Navier-Stokes equations with Dirichlet boundary conditions, whose mixed-VEM formulation was originally proposed and analyzed in a recent work using virtual element subspaces for H(div) and H1, simultaneously. In addition, an algorithm is proposed for the assembly of the associated global linear system for Newton’s iteration. Finally, we present a numerical example in order to illustrate the performance of the mixed-VEM scheme and confirm the expected theoretical convergence rates.

Federated Block Coordinate Descent Scheme for Learning Global and Personalized Models

In federated learning, models are learned from users’ data that are held private in their edge devices, by aggregating them in the service provider’s “cloud” to obtain a global model. Such global model is of great commercial value in, e.g., improving the customers’ experience. In this paper we focus on two possible areas of improvement of the state of the art. First, we take the difference between user habits into account and propose a quadratic penalty-based formulation, for efficient learning of the global model that allows to personalize local models. Second, we address the latency issue associated with the heterogeneous training time on edge devices, by exploiting a hierarchical structure modeling communication not only between the cloud and edge devices, but also within the cloud. Specifically, we devise a tailored block coordinate descent-based computation scheme, accompanied with communication protocols for both the synchronous and asynchronous cloud settings. We characterize the theoretical convergence rate of the algorithm, and provide a variant that performs empirically better. We also prove that the asynchronous protocol, inspired by multi-agent consensus technique, has the potential for large gains in latency compared to a synchronous setting when the edge-device updates are intermittent. Finally, experimental results are provided that corroborate not only the theory, but also show that the system leads to faster convergence for personalized models on the edge devices, compared to the state of the art.

A generalized finite element method for problems with sign-changing coefficients

Problems with sign-changing coefficients occur, for instance, in the study of transmission problems with metamaterials. In this work, we present and analyze a generalized finite element method in the spirit of the localized orthogonal decomposition, that is especially efficient when the negative and positive materials exhibit multiscale features. We derive optimal linear convergence in the energy norm independently of the potentially low regularity of the exact solution. Numerical experiments illustrate the theoretical convergence rates and show the applicability of the method for a large class of sign-changing diffusion problems.

A stochastic collocation approach for parabolic PDEs with random domain deformations

In this article we analyze the linear parabolic partial differential equation with a stochastic domain deformation. In particular, we concentrate on the problem of numerically approximating the statistical moments of a given Quantity of Interest (QoI). The geometry is assumed to be random. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. The stochastic moments of the QoI are computed by employing a collocation method in conjunction with an isotropic Smolyak sparse grid. Theoretical sub-exponential convergence rates as a function to the number of collocation interpolation knots are derived. Numerical experiments are performed and they confirm the theoretical error estimates.

Spectral extended finite element method for band structure calculations in phononic crystals

This talk introduces the extended finite element method (X-FEM) on structured higher-order (spectral) finite element meshes for computing the band structure of one- and two-dimensional phononic composites. The X-FEM enables the modeling of holes and inclusions in geometry through the framework of partition-of-unity enrichment. This reduces the burden on mesh generation, since meshes do not need to conform to geometric features in the periodic domain. Further, with iterative design processes such as phononic shape optimization, the need for remeshing is eliminated. To obtain theoretical rates of convergence with higher-order extended finite elements, careful consideration of curved geometry representation and numerical integration is required. In two dimensions, we adopt rational Bezier representation of conic sections or optimized cubic Hermite functions to model implicit boundaries described by level sets. Efficient computation of weak form integrals is realized through the homogeneous numerical integration scheme—a method that uses Euler’s homogeneous function theorem and Stokes’s theorem to reduce integration to the boundary of the domain. Ghost penalty stabilization is used to improve matrix conditioning on finite elements that are cut by a hole. Band structure calculations on perforated materials as well as on two-phase phononic crystals are presented that affirm the sound accuracy and optimal convergence of the method on structured, higher-order spectral finite element meshes.

An improved banded estimation for large covariance matrix

The modified Cholesky decomposition (MCD) is a powerful and efficient tool for the large covariance matrix estimation, which guarantees the positive definite property of the estimated matrix. However, when implementing the MCD, it requires a pre-knowledge of the variable ordering, which is often unknown before analysis or does not exist for some real data. In this work, we propose a positive definite Cholesky-based estimate for the large banded covariance matrix by recovering the variable ordering before applying the MCD technique. The asymptotically theoretical convergence rate is established under some regularity conditions. The merits of the proposed model is illustrated by simulation study and applications to two gene expression data sets.

Ensemble sparse estimation of covariance structure for exploring genetic disease data

High-dimensional data often occur nowadays in various areas, such as genetic and microarray data. The covariance matrix is of fundamental importance in analyzing the relationship between multivariate variables. A powerful tool for estimating a covariance matrix is the modified Cholesky decomposition, which allows for unconstrained estimation and guarantees the positive definiteness of the estimate. However, it requires a pre-specified ordering of variables before analysis, which is often not available in the real data. Hence, an ensemble Cholesky-based sparse estimation is proposed for a high-dimensional covariance matrix by adopting the model averaging idea. The asymptotically theoretical convergence rate is established under some regularity conditions. The merits of the proposed model are illustrated by the numerical study and two genetic disease data.

Variational multiscale virtual element method for the convection-dominated diffusion problem

In this paper a new stabilized virtual element method for the convection-dominated diffusion problem is proposed. By taking advantage of robustness of variational multiscale method and flexibility of virtual element method, the proposed method is proven to be very easy to implement. The adding stabilized term in the variational formulation can be calculated directly using the stability portion of the virtual element method. There is no user-defined parameter. And the priori error estimate can be easily obtained as there is no change in the consistency portion in the virtual element method. Numerical experiments show good agreement with the theoretical convergence rate and validate the stabilization effect.

Improving accuracy in the Leray model for incompressible nonisothermal flows via adaptive deconvolution‐based nonlinear filtering

This paper considers a Leray regularization model of incompressible, nonisothermal fluid flows which uses nonlinear filtering based on indicator functions and introduces an efficient numerical method for solving it. The proposed method uses a multistep, second‐order temporal discretization with a finite element (FE) spatial discretization in such a way that the resulting algorithm is linear at each time level and decouples the evolution equations from the velocity filter step. Since the indicator function chosen in this model is mathematically based on approximation theory, the proposed numerical algorithm can be analyzed robustly, i.e., the stability and convergence of the method is provable. A series of numerical tests are carried out to verify the theoretical convergence rates and to compare the algorithm with direct numerical simulation and the usual Leray‐α model of the flow problem.

Curve Based Approximation of Measures on Manifolds by Discrepancy Minimization

The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve’s length and Lipschitz constant. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on mathbb R^3 and the Grassmannian of all 2-dimensional linear subspaces of {mathbb {R}}^4. Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.

A high-order moment limiter for the discontinuous Galerkin method on triangular meshes

We propose a moment limiter of arbitrary high order for the discontinuous Galerkin method on unstructured triangular meshes. The limiter works by hierarchically limiting solution coefficients (moments) by comparing them to reconstructed directional derivatives along specific directions. Limiting along these directions is performed using one-dimensional minmod slope limiters. The stencil used to reconstruct directional derivatives consists of only eight mesh elements and can be computed in the pre-processing stage. Due to a low number of operations involved in limiting, the limiter takes only a fraction of the total computing time for the modal discontinuous Galerkin method. We present numerical examples showing that limited solutions retain the theoretical rate of convergence and are robust in the presence of discontinuities.

Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

We present a novel approach aimed at high-performance uncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loève expansion and compute statistics of high-dimensional quantities-of-interest, such as the cardiac activation potential. Our methodology relies on a close integration of multilevel Monte Carlo methods, parallel iterative solvers, and a space-time discretization. This combination allows for space-time adaptivity, time-changing domains, and to take advantage of past samples to initialize the space-time solution. The resulting sequence of problems is distributed using a multilevel parallelization strategy, allocating batches of samples having different sizes to a different number of processors. We assess the performance of the proposed framework by showing in detail its application to the solution of nonlinear equations arising from cardiac electrophysiology. Specifically, we study the effect of spatially-correlated perturbations of the heart fibers' conductivities on the mean and variance of the resulting activation map. As shown by the experiments, the theoretical rates of convergence of multilevel Monte Carlo are achieved. Moreover, the total computational work for a prescribed accuracy is reduced by an order of magnitude with respect to standard Monte Carlo methods.

Well-posedness and discrete analysis for advection-diffusion-reaction in poroelastic media

ABSTRACT We analyse a PDE system modelling poromechanical processes (formulated in mixed form using the solid deformation, fluid pressure, and total pressure) interacting with diffusing and reacting solutes in the medium. We investigate the well-posedness of the nonlinear set of equations using fixed-point theory, Fredholm's alternative, a priori estimates, and compactness arguments. We also propose a mixed finite element method and demonstrate the stability of the scheme. Error estimates are derived in suitable norms, and numerical experiments are conducted to illustrate the mechano-chemical coupling and to verify the theoretical rates of convergence.

Poisson reduced-rank models with sparse loadings

High-dimensional Poisson reduced-rank models have been considered for statistical inference on low-dimensional locations of the individuals based on the observations of high-dimensional count vectors. In this study, we assume sparsity on a so-called loading matrix to enhance its interpretability. The sparsity assumption leads to the use of $$L_1$$ penalty, for the estimation of the loading. We provide novel computational and theoretical analyses for the corresponding penalized Poisson maximum likelihood estimation. We establish theoretical convergence rates for the parameters under weak-dependence conditions; this implies consistency even in large-dimensional problems. To implement the proposed method involving several computational issues, including nonconvex log-likelihoods, $$L_1$$ penalty, and orthogonal constraints, we developed an iterative algorithm. Further, we propose a Bayesian-Information-Criteria-based penalty parameter selection, which works well in the implementation. Some numerical evidence is provided by conducting real-data-based simulation analyses and the proposed method is illustrated with the analysis of German party manifesto data.

Simulation of McKean–Vlasov SDEs with super-linear growth

Abstract We present two fully probabilistic Euler schemes, one explicit and one implicit, for the simulation of McKean–Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth and random initial condition. We provide a pathwise propagation of chaos result and show strong convergence for both schemes on the consequent particle system. The explicit scheme attains the standard $1/2$ rate in stepsize. From a technical point of view, we successfully use stopping times to prove the convergence of the implicit method; although we avoid them altogether for the explicit one. The combination of particle interactions and random initial condition makes the proofs technically more involved. Numerical tests recover the theoretical convergence rates and illustrate a computational complexity advantage of the explicit over the implicit scheme. Comparative analysis is carried out on a stylized non-Lipschitz MV-SDE and a mean-field model for FitzHugh–Nagumo neurons. We provide numerical tests illustrating particle corruption effect where one single particle diverging can ‘corrupt’ the whole particle system. Moreover, the more particles in the system the more likely this divergence is to occur.

Banach spaces-based analysis of a fully-mixed finite element method for the steady-state model of fluidized beds

In this paper we propose and analyze a fully-mixed finite element method for the steady-state model of fluidized beds. This numerical technique, which arises from the use of a dual-mixed approach in each phase, is motivated by a methodology previously applied to the stationary Navier–Stokes equations and related models. More precisely, we modify the stress tensors of the fluid and solid phases by defining pseudostresses as phasic stresses that include shear, pressure, and convective effects. Next, we eliminate the pressures from the equations and derive constitutive relations depending only on the aforementioned pseudostresses and the velocities of the fluid and the particles. In this way, these variables, together with the skew-symmetric parts of the velocity gradients, also named vorticities, become the only unknowns of our variational formulation. As usual, the latter is obtained by testing against suitable functions, and then integrating and integrating by parts, respectively, the equilibrium and the constitutive equations. The particle pressure, a known function of the concentration, is given as a datum, and the fluid pressure is computed afterwards via a postprocessing formula. The continuous setting, lying in a Banach spaces framework rather than in a Hilbertian one, is rewritten as an equivalent fixed-point equation, and hence the well-posedness analysis is carried out by combining the Babusˇka–Brezzi theory, the Banach–Necˇas–Babusˇka Theorem, and the classical Banach fixed-point Theorem. Thus, existence of a unique solution in a closed ball is guaranteed for sufficiently small data. In turn, the associated Galerkin scheme is introduced and analyzed analogously, so that, under suitable assumptions on generic finite element subspaces, and for sufficiently small data as well, the Brouwer and Banach fixed-point Theorems allow to conclude existence and uniqueness of solution, respectively. Specific finite element subspaces satisfying the required hypotheses are described, and optimal a priori error estimates are derived. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence, are reported.

A fast and high-order numerical method for nonlinear fractional-order differential equations with non-singular kernel

Efficient and fast explicit methods are proposed to solve nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed methods produce the second-order for linear interpolation and the third-order accuracy for quadratic interpolation, respectively. The convergence analysis is proved by using discrete Gronwall's inequality. Furthermore, applying the recurrence relation of the memory term, it reduces CPU time executed the proposed methods. The proposed fast algorithm requires approximately O(N) arithmetic operations while O(N2) is required in case of the regular predictor-corrector schemes, where N is the total number of the time step. The following numerical examples demonstrate the accuracy of the proposed methods as well as the efficiency: nonlinear fractional differential equations, time-fraction sub-diffusion, and time-fractional advection-diffusion equation. Numerical experiments also verify the theoretical convergence rates.