We introduce three new stopping criteria that balance algebraic and discretization errors for the conjugate gradient algorithm applied to high-order finite element discretizations of Poisson problems. The current state of the art stopping criteria compare a posteriori estimates of discretization error against estimates of the algebraic error. Firstly, we propose a new error indicator derived from a recovery-based error estimator that is less computationally expensive and more reliable. Secondly, we introduce a new stopping criterion that suggests stopping when the norm of the linear residual is less than a small fraction of an error indicator derived directly from the residual. This indicator shares the same mesh size and polynomial degree scaling as the norm of the residual, resulting in a robust criterion regardless of the mesh size, the polynomial degree, and the shape regularity of the mesh. Thirdly, in solving Poisson problems with highly variable piecewise constant coefficients, we introduce a subdomain-based criterion that recommends stopping when the norm of the linear residual restricted to each subdomain is smaller than the corresponding indicator also restricted to that subdomain. Numerical experiments, including tests with anisotropic meshes and highly variable piecewise constant coefficients, demonstrate that the proposed criteria efficiently avoid both premature termination and over-solving.
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The goal of this work is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional Boussinesq-Abbott system. The problem can be reduced to a transmission problem for this Boussinesq system, in which the transmission conditions between the components of the domain at the left and at the right of the object are determined through the resolution of coupled forced ODEs in time satisfied by the vertical displacement of the object and the average discharge in the portion of the fluid located under the object. We propose a new extended formulation in which these ODEs are complemented by two other forced ODEs satisfied by the trace of the surface elevation at the contact points. The interest of this new extended formulation is that the forcing terms are easy to compute numerically and that the surface elevation at the contact points is furnished for free. Based on this formulation, we propose a second order scheme that involves a generalization of the MacCormack scheme with nonlocal flux and a source term, which is coupled to a second order Heun scheme for the ODEs. In order to validate this scheme, several explicit solutions for this wave-structure interaction problem are derived and can serve as benchmark for future codes. As a byproduct, our method provides a second order scheme for the generation of waves at the entrance of the numerical domain for the Boussinesq-Abbott system.
In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.
We consider the problem of inference for projection parameters in linear regression with increasing dimensions. This problem has been studied under a variety of assumptions in the literature. The classical asymptotic normality result for the least squares estimator of the projection parameter only holds when the dimension $d$ of the covariates is of smaller order than $n^{1/2}$, where $n$ is the sample size. Traditional sandwich estimator-based Wald intervals are asymptotically valid in this regime. In this work, we propose a bias correction for the least squares estimator and prove the asymptotic normality of the resulting debiased estimator as long as $d = o(n^{2/3})$, with an explicit bound on the rate of convergence to normality. We leverage recent methods of statistical inference that do not require an estimator of the variance to perform asymptotically valid statistical inference. We provide a discussion of how our techniques can be generalized to increase the allowable range of $d$ even further.
This paper introduces an extension of the well-known Morley element for the biharmonic equation, extending its application from triangular elements to general polytopal elements using the weak Galerkin finite element methods. By leveraging the Schur complement of the weak Galerkin method, this extension not only preserves the same degrees of freedom as the Morley element on triangular elements but also expands its applicability to general polytopal elements. The numerical scheme is devised by locally constructing weak tangential derivatives and weak second-order partial derivatives. Error estimates for the numerical approximation are established in both the energy norm and the $L^2$ norm. A series of numerical experiments are conducted to validate the theoretical developments.
In this paper, we propose a Dimension-Reduced Second-Order Method (DRSOM) for convex and nonconvex (unconstrained) optimization. Under a trust-region-like framework, our method preserves the convergence of the second-order method while using only curvature information in a few directions. Consequently, the computational overhead of our method remains comparable to the first-order such as the gradient descent method. Theoretically, we show that the method has a local quadratic convergence and a global convergence rate of $O(\epsilon^{-3/2})$ to satisfy the first-order and second-order conditions if the subspace satisfies a commonly adopted approximated Hessian assumption. We further show that this assumption can be removed if we perform a corrector step using a Krylov-like method periodically at the end stage of the algorithm. The applicability and performance of DRSOM are exhibited by various computational experiments, including $L_2 - L_p$ minimization, CUTEst problems, and sensor network localization.
SCoTLASS is the first sparse principal component analysis (SPCA) model which imposes extra l1 norm constraints on the measured variables to obtain sparse loadings. Due to the the difficulty of finding projections on the intersection of an l1 ball/sphere and an l2 ball/sphere, early approaches to solving the SCoTLASS problems were focused on penalty function methods or conditional gradient methods. In this paper, we re-examine the SCoTLASS problems, denoted by SPCA-P1, SPCA-P2 or SPCA-P3 when using the intersection of an l1 ball and an l2 ball, an l1 sphere and an l2 sphere, or an l1 ball and an l2 sphere as constrained set, respectively. We prove the equivalence of the solutions to SPCA-P1 and SPCA-P3, and the solutions to SPCA-P2 and SPCA-P3 are the same in most case. Then by employing the projection method onto the intersection of an l1 ball/sphere and an l2 ball/sphere, we design a gradient projection method (GPSPCA for short) and an approximate Newton algorithm (ANSPCA for short) for SPCA-P1, SPCA-P2 and SPCA-P3 problems, and prove the global convergence of the proposed GPSPCA and ANSPCA algorithms. Finally, we conduct several numerical experiments in MATLAB environment to evaluate the performance of our proposed GPSPCA and ANSPCA algorithms. Simulation results confirm the assertions that the solutions to SPCA-P1 and SPCA-P3 are the same, and the solutions to SPCA-P2 and SPCA-P3 are the same in most case, and show that ANSPCA is faster than GPSPCA for large-scale data. Furthermore, GPSPCA and ANSPCA perform well as a whole comparing with the typical SPCA methods: the l0-constrained GPBB algorithm, the l1-constrained BCD-SPCAl1 algorithm, the l1-penalized ConGradU and Gpowerl1 algorithms, and can be used for large-scale computation.
We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a new weak formulation for the problem, in which the interface and its contact line are evolved simultaneously. By using piecewise linear elements in space and backward Euler in time, we then discretize the weak formulation to obtain a fully discretized parametric finite element approximation. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is extended for solving the sharp interface model of solid-state dewetting with anisotropic surface energies in the Riemmanian metric form. Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.
In this paper we formulate and analyze a space-time finite element method for the numerical simulation of rotating electric machines where the finite element mesh is fixed in space-time domain. Based on the Babu\v{s}ka--Ne\v{c}as theory we prove unique solvability both for the continuous variational formulation and for a standard Galerkin finite element discretization in the space-time domain. This approach allows for an adaptive resolution of the solution both in space and time, but it requires the solution of the overall system of algebraic equations. While the use of parallel solution algorithms seems to be mandatory, this also allows for a parallelization simultaneously in space and time. This approach is used for the eddy current approximation of the Maxwell equations which results in an elliptic-parabolic interface problem. Numerical results for linear and nonlinear constitutive material relations confirm the applicability, efficiency and accuracy of the proposed approach.
We propose and analyze unfitted finite element approximations for the two-phase incompressible Navier--Stokes flow in an axisymmetric setting. The discretized schemes are based on an Eulerian weak formulation for the Navier--Stokes equation in the 2d-meridian halfplane, together with a parametric formulation for the generating curve of the evolving interface. We use the lowest order Taylor--Hood and piecewise linear elements for discretizing the Navier--Stokes formulation in the bulk and the moving interface, respectively. We discuss a variety of schemes, amongst which is a linear scheme that enjoys an equidistribution property on the discrete interface and good volume conservation. An alternative scheme can be shown to be unconditionally stable and to conserve the volume of the two phases exactly. Numerical results are presented to show the robustness and accuracy of the introduced methods for simulating both rising bubble and oscillating droplet experiments.
This paper presents the Residual QPAS Subspace (ResQPASS) method that solves large-scale linear least-squares problems with bound constraints on the variables. The problem is solved by creating a series of small projected problems with increasing size. We project on the basis spanned by the residuals. Each projected problem is solved by the QPAS method that is warm-started with the working set and the solution of the previous problem. The method coincides with conjugate gradients (CG) applied to the normal equations when none of the constraints is active. When only a few constraints are active the method converges, after a few initial iterations, as the CG method. Our analysis links the convergence to Krylov subspaces. We also present an efficient implementation where the matrix factorizations using QR are updated over the inner iterations and Cholesky over the outer iterations.
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