The scaled boundary finite element method is known for its capability in reproducing highly-detailed solution fields. This, however, is only attainable in those cases where analytical solutions exist. Many others invoke the use of numerical methods that only provide the response of boundaries. Hence, no information on the inner-subdomain solution fields can be recovered. As a remedy, we propose a new solution scheme by which the interior fields of subdomains can be recovered.
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The starting point of this paper is a collection of properties of an algorithm that have been distilled from the informal descriptions of what an algorithm is that are given in standard works from the mathematical and computer science literature. Based on that, the notion of a proto-algorithm is introduced. The thought is that algorithms are equivalence classes of proto-algorithms under some equivalence relation. Three equivalence relations are defined. Two of them give bounds between which an appropriate equivalence relation must lie. The third lies in between these two and is likely an appropriate equivalence relation. A sound method is presented to prove, using an imperative process algebra based on ACP, that this equivalence relation holds between two proto-algorithms.
Two approximations of the integral of a class of sinusoidal composite functions, for which an explicit form does not exist, are derived. Numerical experiments show that the proposed approximations yield an error that does not depend on the width of the integration interval. Using such approximations, definite integrals can be computed in almost real-time.
We investigate distributional properties of a class of spectral spatial statistics under irregular sampling of a random field that is defined on $\mathbb{R}^d$, and use this to obtain a test for isotropy. Within this context, edge effects are well-known to create a bias in classical estimators commonly encountered in the analysis of spatial data. This bias increases with dimension $d$ and, for $d>1$, can become non-negligible in the limiting distribution of such statistics to the extent that a nondegenerate distribution does not exist. We provide a general theory for a class of (integrated) spectral statistics that enables to 1) significantly reduce this bias and 2) that ensures that asymptotically Gaussian limits can be derived for $d \le 3$ for appropriately tapered versions of such statistics. We use this to address some crucial gaps in the literature, and demonstrate that tapering with a sufficiently smooth function is necessary to achieve such results. Our findings specifically shed a new light on a recent result in Subba Rao (2018a). Our theory then is used to propose a novel test for isotropy. In contrast to most of the literature, which validates this assumption on a finite number of spatial locations (or a finite number of Fourier frequencies), we develop a test for isotropy on the full spatial domain by means of its characterization in the frequency domain. More precisely, we derive an explicit expression for the minimum $L^2$-distance between the spectral density of the random field and its best approximation by a spectral density of an isotropic process. We prove asymptotic normality of an estimator of this quantity in the mixed increasing domain framework and use this result to derive an asymptotic level $\alpha$-test.
In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation {on convex domains}. Our approach is a simple discontinuous Galerkin method similar to the Crouzeix--Raviart finite element method. As our primary contribution, we show a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart--Thomas finite element space and a discontinuous space. While inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed, our results show that the Stokes element satisfies the inf-sup condition on anisotropic meshes. Furthermore, we also provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions.
The distribution for the minimum of Brownian motion or the Cauchy process is well-known using the reflection principle. Here we consider the problem of finding the sample-by-sample minimum, which we call the online minimum search. We consider the possibility of the golden search method, but we show quantitatively that the bisection method is more efficient. In the bisection method there is a hierarchical parameter, which tunes the depth to which each sub-search is conducted, somewhat similarly to how a depth-first search works to generate a topological ordering on nodes. Finally, we consider the possibility of using harmonic measure, which is a novel idea that has so far been unexplored.
We propose a Lawson-time-splitting extended Fourier pseudospectral (LTSeFP) method for the numerical integration of the Gross-Pitaevskii equation with time-dependent potential that is of low regularity in space. For the spatial discretization of low regularity potential, we use an extended Fourier pseudospectral (eFP) method, i.e., we compute the discrete Fourier transform of the low regularity potential in an extended window. For the temporal discretization, to efficiently implement the eFP method for time-dependent low regularity potential, we combine the standard time-splitting method with a Lawson-type exponential integrator to integrate potential and nonlinearity differently. The LTSeFP method is both accurate and efficient: it achieves first-order convergence in time and optimal-order convergence in space in $L^2$-norm under low regularity potential, while the computational cost is comparable to the standard time-splitting Fourier pseudospectral method. Theoretically, we also prove such convergence orders for a large class of spatially low regularity time-dependent potential. Extensive numerical results are reported to confirm the error estimates and to demonstrate the superiority of our method.
The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, in the presented examples, the proposed UAL method is 1-2 orders of magnitude faster than force-controlled arc-length and monolithic Newton-Raphson solvers.
We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier-Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper [Li et al. {\it CMAME} 2023], and we extend the methodology by improving the analysis to be in the $L^2$ norm instead of a weighted $H^1$ norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration.
Covariance matrices of random vectors contain information that is crucial for modelling. Certain structures and patterns of the covariances (or correlations) may be used to justify parametric models, e.g., autoregressive models. Until now, there have been only few approaches for testing such covariance structures systematically and in a unified way. In the present paper, we propose such a unified testing procedure, and we will exemplify the approach with a large variety of covariance structure models. This includes common structures such as diagonal matrices, Toeplitz matrices, and compound symmetry but also the more involved autoregressive matrices. We propose hypothesis tests for these structures, and we use bootstrap techniques for better small-sample approximation. The structures of the proposed tests invite for adaptations to other covariance patterns by choosing the hypothesis matrix appropriately. We prove their correctness for large sample sizes. The proposed methods require only weak assumptions. With the help of a simulation study, we assess the small sample properties of the tests. We also analyze a real data set to illustrate the application of the procedure.
We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AILFEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.
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