In this paper, we introduce a novel unfitted finite element method to solve the quad-curl interface problem. We adapt Nitsche’s method for \({\operatorname {curl}}{\operatorname {curl}}\)-conforming elements and double the degrees of freedom on interface elements. To ensure stability, we incorporate ghost penalty terms and a discrete divergence-free term. We establish the well-posedness of our method

This paper studies the finite element method for solving the exterior Stokes problem in two dimensions. A nonlocal boundary condition is defined using a nonsingular-kernel Dirichlet-to-Dirichlet (DtD) mapping, which maps the Dirichlet data on an interior circle to the Dirichlet data on another circular artificial boundary based on the Poisson integral formula of the Stokes problem. The truncated problem

In this article, we present a numerical method for the Stokes flow of an Oldroyd-B fluid. The viscoelastic stress evolves according to a constitutive law formulated in terms of the upper convected time derivative. A finite difference method is used to discretise along fluid trajectories to approximate the advection and deformation terms of the upper convected derivative in a simple, cheap and cohesive

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional in the sense that an a posteriori computable quantity needs to be small enough—which can be ensured by mesh refinement—and optimal in the sense that the error estimator decays with the same order as the error under mesh

This paper studies a time filtered finite element method for the unsteady inductionless magnetohydrodynamic (MHD) equations. The method uses the semi-implicit backward Euler scheme with a time filter in time and adopts the standard inf-sup stable fluid pairs to discretize the velocity and pressure, and the inf-sup stable face-volume elements for solving the current density and electric potential in

We study Bayesian methods for large-scale linear inverse problems, focusing on the challenging task of hyperparameter estimation. Typical hierarchical Bayesian formulations that follow a Markov Chain Monte Carlo approach are possible for small problems but are not computationally feasible for problems with a very large number of unknown inverse parameters. In this work, we describe an empirical Bayes

We recover the initial status of an evolution system governed by linearized shallow-water wave equations in a 2-dimensional bounded domain by data assimilation technique, with the aim of determining the initial wave height from the measurement of wave distribution in an interior domain. Since we specify only one component of the solution to the governed system and the observation is only measured in

In this paper, we develop new methods to join machine learning techniques and macroscopic differential models, aimed at estimate and forecast vehicular traffic. This is done to complement respective advantages of data-driven and model-driven approaches. We consider here a dataset with flux and velocity data of vehicles moving on a highway, collected by fixed sensors and classified by lane and by class

In this paper, we consider the quadratic eigenvalue problem (QEP) expressed in various commonly used bases, including Taylor, Newton, and Lagrange bases. We propose to investigate the backward errors of the computed eigenpairs and condition numbers of eigenvalues for QEP solved by a class of block Kronecker linearizations. To improve the backward error and condition number of the QEP expressed in a

In two dimensions, we propose and analyse an iterative a posteriori error indicator for the discontinuous Galerkin finite element approximations of the Stokes equations under boundary conditions of friction type. Two sources of error are identified here, namely; the discretisation error and the linearization error. Under a smallness assumption on data, we prove that the devised error estimator is reliable

For h-FEM discretisations of the Helmholtz equation with wavenumber k, we obtain k-explicit analogues of the classic local FEM error bounds of Nitsche and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9), Demlow et al.(Math. Comput. 80(273), 1–9 2011), showing that these bounds hold with constants independent of k, provided one works in Sobolev norms weighted with k in the natural way

For the iterative decoupling of elliptic–parabolic problems such as poroelasticity, we introduce time discretization schemes up to order five based on the backward differentiation formulae. Its analysis combines techniques known from fixed-point iterations with the convergence analysis of the temporal discretization. As the main result, we show that the convergence depends on the interplay between

This paper is concerned with near-optimal approximation of a given univariate function with elements of a polynomially enriched wavelet frame, a so-called quarklet frame. Inspired by hp-approximation techniques of Binev, we use the underlying tree structure of the frame elements to derive an adaptive algorithm that, under standard assumptions concerning the local errors, can be used to create approximations

We investigate Sobolev regularity of bivariate functions obtained in Isogeometric Analysis when using geometry maps that are degenerate in the sense that the first partial derivatives vanish at isolated points. In particular, we show how the known \(C^1\)-conditions for D-patches have to be tightened to guarantee square integrability of second partial derivatives, as required when computing finite

We revisit the moving least squares (MLS) approximation scheme on the sphere \(\mathbb S^{d-1} \subset {\mathbb R}^d\), where \(d>1\). It is well known that using the spherical harmonics up to degree \(L \in {\mathbb N}\) as ansatz space yields for functions in \(\mathcal {C}^{L+1}(\mathbb S^{d-1})\) the approximation order \(\mathcal {O}\left( h^{L+1} \right) \), where h denotes the fill distance

In this work, we propose and analyze a new stabilized virtual element method for the coupled Stokes-Darcy problem with Beavers-Joseph-Saffman interface condition on polygonal meshes. We derive two variants of local projection stabilization methods for the coupled Stokes-Darcy problem. The significance of local projection-based stabilization terms is that they provide reasonable control of the pressure

With local pressure-residual stabilizations as an augmentation to the classical Galerkin/least-squares (GLS) stabilized method, a new locally evaluated residual-based stabilized finite element method is proposed for a type of Stokes equations from the incompressible flows. We focus on the study of a type of nonstandard boundary conditions involving the mixed tangential velocity and pressure Dirichlet

The perturbation of the QR decompostion is analyzed from the probalistic point of view. The perturbation error is approximated by a first-order perturbation expansion with high probability where the perturbation is assumed to be random. Different from the previous normwise perturbation bounds using the Frobenius norm, our techniques are used to develop the spectral norm, as well as the entry-wise perturbation

We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry

We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from

Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical

Standard interpolatory subdivision schemes and their underlying interpolating refinable functions are of interest in CAGD, numerical PDEs, and approximation theory. Generalizing these notions, we introduce and study \(n_s\)-step interpolatory \(\textsf{M}\)-subdivision schemes and their interpolating \(\textsf{M}\)-refinable functions with \(n_s\in \mathbb {N}\cup \{\infty \}\) and a dilation factor

Tensor wheel (TW) decomposition combines the popular tensor ring and fully connected tensor network decompositions and has achieved excellent performance in tensor completion problem. A standard method to compute this decomposition is the alternating least squares (ALS). However, it usually suffers from slow convergence and numerical instability. In this work, the fast and robust SVD-based algorithms

In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulting from the Legendre dual-Petrov-Galerkin (LDPG) method for the mth-order initial value problem (IVP): \(u^{(m)}(t)=\sigma u(t),\, t\in (-1,1)\) with constant \(\sigma \not =0\) and usual initial conditions at t\(=-1,\) are associated with the generalised Bessel polynomials (GBPs). In particular

In this short note we show that Hilbert complexes are strongly related to what we shall call annihilating sets of skew-selfadjoint operators. This provides for a new perspective on the classical topic of Hilbert complexes viewed as families of commuting normal operators.

This paper focuses on computing the eigenvalues of the generalized collocation matrix of the rational Said–Ball basis, also called as the quasi-rational Said–Ball–Vandermonde (q-RSBV) matrix, with high relative accuracy. To achieve this, we propose explicit expressions for the minors of the q-RSBV matrix and develop a high-precision algorithm to compute these parameters. Additionally, we present perturbation

This paper analyses the nonconforming Morley type virtual element method to approximate a regular solution to the von Kármán equations that describes bending of very thin elastic plates. Local existence and uniqueness of a discrete solution to the non-linear problem is discussed. A priori error estimate in the energy norm is established under minimal regularity assumptions on the exact solution. Error

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a

This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with

Let H be a separable Hilbert space and let \(\{x_{n}\}\) be a sequence in H that does not contain any zero elements. We say that \(\{x_{n}\}\) is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence \({\bigl \{\frac{x_n}{\Vert x_n\Vert }\bigr \}}\) is a Bessel sequence or a frame for H, respectively. In this paper, several necessary and sufficient conditions for sequences

In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by

We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians

This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential.

We focus on the control of unknown partial differential equations (PDEs). The system dynamics is unknown, but we assume we are able to observe its evolution for a given control input, as typical in a reinforcement learning framework. We propose an algorithm based on the idea to control and identify on the fly the unknown system configuration. In this work, the control is based on the state-dependent

Based on local Gauss integral technique and backtracking technique, this paper presents and studies three kinds of two-grid stabilized finite element algorithms for the stationary Navier-Stokes equations. The proposed methods consist of deducing a coarse solution on the nonlinear system, updating the solution on a fine mesh via three different methods, and solving a linear correction problem on the

In exterior calculus on smooth manifolds, the exterior derivative and wedge products are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and an antisymmetrized cup-like product provides a

The discrete element method (DEM) is a numerical technique widely used to simulate granular materials. The temporal evolution of these simulations is often performed using a Verlet-type algorithm, because of its second order and its desirable property of better energy conservation. However, when dissipative forces are considered in the model, such as the nonlinear Kuwabara-Kono model, the Verlet method

Given samples of a real or complex-valued function on a set of distinct nodes, the traditional linear Chebyshev approximation is to compute the minimax approximation on a prescribed linear functional space. Lawson’s iteration is a classical and well-known method for the task. However, Lawson’s iteration converges only linearly and in many cases, the convergence is very slow. In this paper, relying

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative

Krylov subspace methods for solving linear systems of equations involving skew-symmetric matrices have gained recent attention. Numerical equivalences among Krylov subspace methods for nonsingular skew-symmetric linear systems have been given in Greif et al. [SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1071–1087]. In this work, we extend the results of Greif et al. to singular skew-symmetric linear

In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in \(\mathbb {R}^{2}\). The main idea is to smooth the parametrization of the curve by iteratively filtering the Fourier or Chebyshev coefficients of both the derivative of the arc-length function and the tangential angle of the curve and applying smooth

Multi-dimensional images can be viewed as tensors and have often embedded a low-rankness property that can be evaluated by tensor low-rank measures. In this paper, we first introduce a tensor low-rank and sparsity measure and then propose low-rank and sparsity models for tensor completion, tensor robust principal component analysis, and tensor denoising. The resulting tensor recovery models are further

Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction

We present a scheme for finding all roots of an analytic function in a square domain in the complex plane. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on the real line, by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” Our extension of the classical approach is based

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising, e.g., as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for deep neural operator and generalized polynomial chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up

We develop a sparse spectral method for a class of fractional differential equations, posed on \(\mathbb {R}\), in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on

Regularized pairwise ranking with Gaussian kernels is one of the cutting-edge learning algorithms. Despite a wide range of applications, a rigorous theoretical demonstration still lacks to support the performance of such ranking estimators. This work aims to fill this gap by developing novel oracle inequalities for regularized pairwise ranking. With the help of these oracle inequalities, we derive

Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable and isotropic elastic solid, which is immersed in a homogeneous compressible air/fluid. By the Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the model is formulated as a boundary value problem of acoustic-elastic interaction. Based on a duality argument technique

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the

In the triangular/tetrahedral spectral finite elements, we apply a bilinear/trilinear transformation to map a reference square/cube to a triangle/tetrahedron, which consequently maps the \(\varvec{Q_k}\) polynomial space on the reference element to a finite element space of rational/algebraic functions on the triangle/tetrahedron. We prove that the resulting finite element space, even under this singular

An adaptive time-stepping scheme is developed for the Zakharov-Rubenchik system to resolve the multiple time scales accurately and to improve the computational efficiency during long-time simulations. The Crank-Nicolson formula and the Fourier pseudo-spectral method are respectively utilized for the temporal and spatial approximations. The proposed numerical method is proved to preserve the mass and

Multilevel Stein variational gradient descent is a method for particle-based variational inference that leverages hierarchies of surrogate target distributions with varying costs and fidelity to computationally speed up inference. The contribution of this work is twofold. First, an extension of a previous cost complexity analysis is presented that applies even when the exponential convergence rate

We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter