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Abstract: Abstract This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the \(L^2\) -norm, the Galerkin projection onto this generalized finite element space even yields \(\varepsilon \) -independent error bounds, \(\varepsilon \) being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate \(\varepsilon \) -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers. PubDate: 2024-08-05
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Abstract: Abstract Considering positive \(\textbf{H}(\varvec{curl},\Omega )\) problems, we propose a substructured version of the Hiptmair-Xu preconditioner based on a new formula that expresses the inverse of Schur systems in terms of the inverse matrix of the global volume problem. PubDate: 2024-07-15
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Abstract: Abstract The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the \(\delta \) -weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution. PubDate: 2024-07-15
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Abstract: Abstract Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments. PubDate: 2024-07-13
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Abstract: Abstract We prove weak convergence of order one for a class of exponential based integrators for SDEs with non-globally Lipschitz drift. Our analysis covers tamed versions of Geometric Brownian Motion (GBM) based methods as well as the standard exponential schemes. The numerical performance of both the GBM and exponential tamed methods through four different multi-level Monte Carlo techniques are compared. We observe that for linear noise the standard exponential tamed method requires severe restrictions on the step size unlike the GBM tamed method. PubDate: 2024-07-11
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Abstract: Abstract The objective of this work is to develop a conforming virtual element method for viscoelastic wave equations with variable coefficients on polygonal meshes. For problems where the coefficients are variable, the standard virtual element discrete forms do not work efficiently and require modification. For the optimal convergence estimate of the semi-discrete approximation in the \(L^{2}\) norm, a special projection operator is used. In the fully discrete scheme, the implicit second-order Newmark method is employed to approximate the temporal derivatives. Numerical experiments are presented to support the theoretical results. The proposed numerical algorithm can be applied to various problems arising in the engineering and medical fields. PubDate: 2024-07-05
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Abstract: Abstract This paper suggests an hp-discontinuous Galerkin approach for the fractional integro-differential equations with weakly singular kernels. The key idea behind our method is to first convert the fractional integro-differential equations into the second kind of Volterra integral equations, and then solve the equivalent integral equations using the hp-discontinuous Galerkin method. We establish prior error bounds in the \(L^{2}\) -norm that is entirely explicit about the local mesh sizes, local polynomial degrees, and local regularities of the exact solutions. The use of geometrically refined meshes and linearly increasing approximation orders demonstrates, in particular, that exponential convergence is achievable for solutions with endpoint singularities. Numerical results indicate the usefulness of the proposed method. PubDate: 2024-07-04
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Abstract: Abstract In this paper, we consider large-scale ranking problems where one is given a set of (possibly non-redundant) pairwise comparisons and the underlying ranking explained by those comparisons is desired. We show that stochastic gradient descent approaches can be leveraged to offer convergence to a solution that reveals the underlying ranking while requiring low-memory operations. We introduce several variations of this approach that offer a tradeoff in speed and convergence when the pairwise comparisons are noisy (i.e., some comparisons do not respect the underlying ranking). We prove theoretical results for convergence almost surely and study several regimes including those with full observations, partial observations, and noisy observations. Our empirical results give insights into the number of observations required as well as how much noise in those measurements can be tolerated. PubDate: 2024-06-21
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Abstract: Abstract In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston–Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods. PubDate: 2024-06-17 DOI: 10.1007/s10543-024-01028-7
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Abstract: Abstract We study in this paper the monotonicity properties of the numerical solutions to Volterra integral equations with nonincreasing completely positive kernels on nonuniform meshes. There is a duality between the complete positivity and the properties of the complementary kernel being nonnegative and nonincreasing. Based on this, we propose the “complementary monotonicity” to describe the nonincreasing completely positive kernels, and the “right complementary monotone” (R-CMM) kernels as the analogue for nonuniform meshes. We then establish the monotonicity properties of the numerical solutions inherited from the continuous equation if the discretization has the R-CMM property. Such a property seems weaker than log-convexity and there is no restriction on the step size ratio of the discretization for the R-CMM property to hold. PubDate: 2024-06-14 DOI: 10.1007/s10543-024-01027-8
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Abstract: Abstract This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier representing boundary traction and the stabilised formulation is based on simultaneously stabilising both the pressure and the traction. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory. PubDate: 2024-06-13 DOI: 10.1007/s10543-024-01025-w
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Abstract: Abstract Verification methods compute intervals which contain the solution of a given problem with mathematical rigour. In order to compare the quality of intervals some measure is desirable. We identify some anticipated properties and propose a method avoiding drawbacks of previous definitions. PubDate: 2024-05-13 DOI: 10.1007/s10543-024-01020-1
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Abstract: Abstract In this paper, we design a multilevel local defect-correction method to solve the non-selfadjoint Steklov eigenvalue problems. Since the computation work needed for solving the non-selfadjoint Steklov eigenvalue problems increases exponentially as the scale of the problems increase, the main idea of our algorithm is to avoid solving large-scale equations especially large-scale Steklov eigenvalue problems directly. Firstly, we transform the non-selfadjoint Steklov eigenvalue problem into some symmetric boundary value problems defined in a multilevel finite element space sequence, and some small-scale non-selfadjoint Steklov eigenvalue problems defined in a low-dimensional auxiliary subspace. Next, the local defect-correction method is used to solve the symmetric boundary value problems, then the difficulty of solving these symmetric boundary value problems is further reduced by decomposing these large-scale problems into a series of small-scale subproblems. Overall, our algorithm can obtain the optimal error estimates with linear computational complexity, and the conclusions are proved by strict theoretical analysis which are different from the developed conclusions for equations with the Dirichlet boundary conditions. PubDate: 2024-04-09 DOI: 10.1007/s10543-024-01022-z
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Abstract: Abstract We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes. PubDate: 2024-04-07 DOI: 10.1007/s10543-024-01021-0
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Abstract: Abstract Dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov–Poisson equations is based on separation of space and velocity variables, as proposed in several recent works. The standard approach for the time integration in the DLRA model uses a splitting of the tangent space projector for the low-rank manifold according to the separated variables. It can also be modified to allow for rank-adaptivity. A less studied aspect is the incorporation of boundary conditions in the DLRA model. In this work, a variational formulation of the projector splitting is proposed which allows to handle inflow boundary conditions on spatial domains with piecewise linear boundary. Numerical experiments demonstrate the principle feasibility of this approach. PubDate: 2024-04-07 DOI: 10.1007/s10543-024-01019-8
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Abstract: Abstract In recent years, there has been an increasing interest in utilizing deep learning-based techniques to predict solutions to various partial differential equations. In this study, we investigate the identification of an unknown flux function and diffusion coefficient in a one-dimensional convection-diffusion equation. The diffusion function is allowed to vanish on intervals implying that solutions generally possess low regularity, i.e., are discontinuous. Therefore, solutions must be interpreted in the sense of entropy solutions which combine a weak formulation with an additional constraint (entropy condition). We explore a methodology that utilizes symbolic neural networks (S-Nets) in combination with an entropy-consistent discrete numerical scheme (ECDNS). Different types of observation data are explored. Extensive experiments in this paper demonstrate that the proposed method is a robust tool to identify the unknown flux and diffusion function. The flux and diffusion functions are restricted to be rational functions. PubDate: 2024-03-30 DOI: 10.1007/s10543-024-01018-9
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Abstract: Abstract In this article, we derive accurate Horner methods in real and complex floating-point arithmetic. In particular, we show that these methods are as accurate as if computed in k-fold precision and then rounded into the working precision. When k is two, our methods are comparable or faster than the existing compensated Horner routines. When compared to multi-precision software, such as MPFR and MPC, our methods are significantly faster, up to k equal to eight, that is, up to 489 bits in the significand. PubDate: 2024-03-27 DOI: 10.1007/s10543-024-01017-w
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Abstract: Abstract The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the sphere can be used to create a numerically feasible, stable finite difference method based on radial basis functions (an RBF-FD-like method). For certain classes of PDEs this approach leads to rigorous convergence estimates for stencils which grow moderately with increasing discretization fineness. PubDate: 2024-03-15 DOI: 10.1007/s10543-024-01016-x
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Abstract: Abstract We examine some bounded perturbations resilient iterative methods for addressing (constrained) consistent linear systems of equations and (constrained) least squares problems. We introduce multiple frameworks rooted in the operator of the Landweber iteration, adapting the operators to facilitate the minimization of absolute errors or residuals. We demonstrate that our operator-based methods exhibit comparable speed to powerful methods like CGLS, and we establish that the computational cost of our methods is nearly equal to that of CGLS. Furthermore, our methods possess the capability to handle constraints (e.g. non-negativity) and control the semi-convergence phenomenon. In addition, we provide convergence analysis of the methods when the current iterations are perturbed by summable vectors. This allows us to utilize these iterative methods for the superiorization methodology. We showcase their performance using examples drawn from tomographic imaging and compare them with CGLS, superiorized conjugate gradient (S-CG), and the non-negative flexible CGLS (NN-FCGLS) methods. PubDate: 2024-03-05 DOI: 10.1007/s10543-024-01015-y
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Abstract: Abstract The two-step backward difference formula (BDF) method on variable grids for parabolic equations with self-adjoint elliptic part is considered. Standard stability estimates for adjacent time-step ratios \(r_j:=k_j/k_{j-1}\leqslant 1.8685\) and 1.9104, respectively, have been proved by Becker (BIT 38:644–662, 1998) and Emmrich (J Appl Math Comput 19:33–55, 2005) by the energy technique with a single multiplier. Even slightly improving the ratio is cumbersome. In this paper, we present a novel technique to examine the positive definiteness of banded matrices that are neither Toeplitz nor weakly diagonally dominant; this result can be viewed as a variant of the Grenander–Szegő theorem. Then, utilizing the energy technique with two multipliers, we establish stability for adjacent time-step ratios up to 1.9398. PubDate: 2024-03-01 DOI: 10.1007/s10543-024-01007-y
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