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Abstract: Abstract We propose a methodology to solve high-order PDE boundary value problems with generalised periodicity, in the framework of the \(\mathcal {C}^0\) interior penalty method. The method is developed for the analysis of flexoelectricity-based metamaterial unit cells, formalising the corresponding problem statement and weak form, and giving details on the implementation of the local and macro conditions for generalised periodicity. Numerical examples demonstrate the high-order convergence of the method and its applicability in realistic problem settings. PubDate: 2022-05-18

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Abstract: Abstract The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications. As a matter of fact we also show at the end an application in the framework of plasma physics. PubDate: 2022-05-17

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Abstract: Abstract In this paper, an effective finite element method with shifted fractional powers bases is developed for fractional convection diffusion equations involving a Riemann–Liouville derivative of order \(\alpha \in (3/2,2)\) . A Petrov-Galerkin variational formulation is constructed on the domain \(\tilde{H}^{\alpha -1}(\Omega )\times \tilde{H}^{1}(\Omega )\) , based on which the finite element approximation scheme is developed by employing shifted fractional power functions and continuous piecewise polynomials of degree up to \(m~(m\in \mathbb {N}^+)\) for trial and test finite element spaces, respectively. The approximation property of trial finite element space and \(\inf \) - \(\sup \) condition for discrete variational form are derived, which enables us to derive the error estimates in \(L^2(\Omega )\) and \(H^{\alpha -1}(\Omega )\) norms. Numerical examples are included to verify the theoretical findings and demonstrate an actual convergence rate of order \(\alpha -1+m\) , where m equals to 1 or 2. PubDate: 2022-05-17

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Abstract: Abstract We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the \(h\) - and \(p\) -versions of the scheme, under a threshold condition on the approximability properties of the discrete spaces. Amongst others, an essential tool in the analysis is a novel discontinuous-to-continuous reconstruction operator on tetrahedral meshes with curved faces. PubDate: 2022-05-16

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Abstract: Abstract In this paper, motivated by some recent progress in the development of Partial Newton-Correction Method for finding multiple solutions to nonlinear PDEs and by a closer observation to the implementation of the Local Min-Orthogonal (LMO) method developed in 2004, first a new L- \(\bot \) selection expression is introduced, then the (strict) separation condition and the continuity condition used in the mathematical framework of the LMO method are successively improved or weakened so that they are not only closer to the real algorithm’s implementation but also able to improve the relevant analysis. A new step-size rule and a new local characterization on saddle points are then established, based on which an improved LMO method is developed. The results in the paper can be further used to improve analysis of relevant numerical methods for solving multiple-solution problems in nonlinear PDEs. Finally, two numerical examples are carried out to illustrate the effectiveness of the new method and some new numerical findings are also presented. PubDate: 2022-05-16

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Abstract: A new algorithm is developed to jointly recover a temporal sequence of images from noisy and under-sampled Fourier data. Specifically we consider the case where each data set is missing vital information that prevents its (individual) accurate recovery. Our new method is designed to restore the missing information in each individual image by “borrowing” it from the other images in the sequence. As a result, all of the individual reconstructions yield improved accuracy. The use of high resolution Fourier edge detection methods is essential to our algorithm. In particular, edge information is obtained directly from the Fourier data which leads to an accurate coupling term between data sets. Moreover, data loss is largely avoided as coarse reconstructions are not required to process inter- and intra-image information. Numerical examples are provided to demonstrate the accuracy, efficiency and robustness of our new method. PubDate: 2022-05-11

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Abstract: In tensor computations, tensor–vector multiplication is one of the main computational costs. We recently studied algorithms with wider applicability and more computational potential for computing the largest eigenpair of a weakly irreducible nonnegative mth-order tensor \({\mathscr {A}}\) , called higher-order Noda iteration (HONI). This method is an eigenvalue solver which uses an inner-outer scheme. The outer iteration is the update of the approximate eigenpair(s), while in the inner iteration a multilinear system has to be solved, often iteratively. For the inner iteration, we also provide a Newton-type method to solve multilinear systems, and prove that the algorithm converges to the unique solution of multilinear systems and the convergence rate is quadratic. HONI has superior performance in terms of fast convergence and positivity preserving property, and its main advantage is to use simple recursive relations to compute the approximate eigenvalue, which means that no additional tensor–vector multiplication is required. Moreover, we devise a practical relaxation criterion based on our theoretical results to improve the efficiency and practicality of HONI, called inexact HONI, and further explain the relationship between HONI and Newton–Noda iteration. Numerical experiments are provided to support the theoretical results. PubDate: 2022-05-10

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Abstract: The invariant distribution, which is characterized by the stationary Fokker-Planck equation, is an important object in the study of randomly perturbed dynamical systems. Traditional numerical methods for computing the invariant distribution based on the Fokker-Planck equation, such as finite difference or finite element methods, are limited to low-dimensional systems due to the curse of dimensionality. In this work, we propose a deep learning based method to compute the generalized potential, i.e. the negative logarithm of the invariant distribution multiplied by the noise. The idea of the method is to learn a decomposition of the force field, as specified by the Fokker-Planck equation, from the trajectory data. The potential component of the decomposition gives the generalized potential. The method can deal with high-dimensional systems, possibly with partially known dynamics. Using the generalized potential also allows us to deal with systems at low temperatures, where the invariant distribution becomes singular around the metastable states. These advantages make it an efficient method to analyze invariant distributions for practical dynamical systems. The effectiveness of the proposed method is demonstrated by numerical examples. PubDate: 2022-05-10

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Abstract: We investigate multiscale finite element methods for an elliptic distributed optimal control problem with rough coefficients. They are based on the (local) orthogonal decomposition methodology of Målqvist and Peterseim. PubDate: 2022-05-07

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Abstract: Abstract In this paper, the optimization problem of supervised distance preserving projection (SDPP) for data dimensionality reduction is considered, which is equivalent to a rank constrained least squares semidefinite programming (RCLSSDP). Due to the combinatorial nature of rank function, the rank constrained optimization problems are NP-hard in most cases. In order to overcome the difficulties caused by rank constraint, a difference-of-convex (DC) regularization strategy is employed, then RCLSSDP is transferred into a DC programming. For solving the corresponding DC problem, an inexact proximal DC algorithm with sieving strategy (s-iPDCA) is proposed, whose subproblems are solved by an accelerated block coordinate descent method. The global convergence of the sequence generated by s-iPDCA is proved. To illustrate the efficiency of the proposed algorithm for solving RCLSSDP, s-iPDCA is compared with classical proximal DC algorithm, proximal gradient method, proximal gradient-DC algorithm and proximal DC algorithm with extrapolation by performing dimensionality reduction experiment on COIL-20 database. From the computation time and the quality of solution, the numerical results demonstrate that s-iPDCA outperforms other methods. Moreover, dimensionality reduction experiments for face recognition on ORL and YaleB databases demonstrate that rank constrained kernel SDPP is efficient and competitive when comparing with kernel semidefinite SDPP and kernel principal component analysis in terms of recognition accuracy. PubDate: 2022-04-30

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Abstract: Abstract We study the global convergence of the gradient descent method of the minimization of strictly convex functionals on an open and bounded set of a Hilbert space. Such results are unknown for this type of sets, unlike the case of the entire Hilbert space. Then, we use our result to establish a general framework to numerically solve boundary value problems for quasi-linear partial differential equations with noisy Cauchy data. The procedure involves the use of Carleman weight functions to convexify a cost functional arising from the given boundary value problem and thus to ensure the convergence of the gradient descent method above. We prove the global convergence of the method as the noise tends to 0. The convergence rate is Lipschitz. Next, we apply this method to solve a highly nonlinear and severely ill-posed coefficient inverse problem, which is the so-called back scattering inverse problem. This problem has many real-world applications. Numerical examples are presented. PubDate: 2022-04-29

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Abstract: Abstract We propose a cell-centered Lagrangian scheme for solving the three dimensional ideal magnetohydrodynamics (MHD) equations on unstructured meshes. The physical conservation laws are compatibly discretized on the unstructured meshes to satisfy the geometric conservation law (GCL). By introducing a generalized Lagrange multiplier, the magnetic divergence constraint is coupled with the conservation laws hence the magnetic divergence errors can dissipate and transport to the domain boundaries. Invoking the Galilean invariance, magnetic flux conservation and the thermodynamic consistency, the nodal approximate Riemann solver is derived and the corresponding first order finite volume scheme is then constructed. The piecewise linear spatial reconstruction and two step predictor corrector time integration are then adopted to increase the accuracy of the scheme. Various numerical tests are presented to assert the robustness and accuracy of our scheme. PubDate: 2022-04-29

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Abstract: Abstract We derive a priori error estimates of the Godunov method for the multidimensional compressible Euler system of gas dynamics. To this end we apply the relative energy principle and estimate the distance between the numerical solution and the strong solution. This yields also the estimates of the \(L^2\) -norms of the errors in density, momentum and entropy. Under the assumption, that the numerical density is uniformly bounded from below by a positive constant and that the energy is uniformly bounded from above and stays positive, we obtain a convergence rate of 1/2 for the relative energy in the \(L^1\) -norm, that is to say, a convergence rate of 1/4 for the \(L^2\) -error of the numerical solution. Further, under the assumption—the total variation of the numerical solution is uniformly bounded, we obtain the first order convergence rate for the relative energy in the \(L^1\) -norm, consequently, the numerical solution converges in the \(L^2\) -norm with the convergence rate of 1/2. The numerical results presented are consistent with our theoretical analysis. PubDate: 2022-04-26

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Abstract: Abstract Concerning the satisfaction of free-stream preservation which reduces errors in the numerical evaluation of grid metrics, the nonlinear central schemes based on WENO interpolations are preferable because of their easy achievement of preservation even on distorted grids, whereas the preservation is hardly achieved by canonical WENO schemes due to their nonlinear upwind nature. However, the latter are typically robust in terms of capturing shocks on uniform or smooth grids, whereas the former are sometimes liable to numerical instability. Aiming at above predicaments, an upwind-biased approach in the presence of flux splitting is established by integrating two seemingly distinct techniques, WENO interpolation and reconstruction, and meanwhile free-stream preservation and increased robustness are acquired. Specifically, the proposed recipe consists of variable and corresponding flux at one midpoint as well as fluxes at sets of nodes as that in WENO; WENO interpolation is cast to derive the former and WENO reconstruction is implemented using the latter to incarnate their contribution; results by two nonlinearities are hybridized carefully and the target order can be achieved. The developed schemes include third-, fifth-, and seventh-order nonlinear methods. By the way, a WENO implementation with the free-stream preservation is obtained as a special case of the proposed method. Numerical examples are used to validate the third- and fifth-order schemes. The achievement of free-stream preservation property is testified (including the aforementioned WENO implementations). 1-D problems by Euler equations indicate the capability of proposed schemes to resolve shock discontinuities and their good resolution. In 2-D situation, vortex preservation and double Mach reflection problems on uniform and randomized grids are chosen, and the computation on the latter grids has been known to accomplish only if free-stream preservation is satisfied. The proposed schemes produce well results in both cases. Comparative studies demonstrate the accuracy and increased robustness of the newly developed schemes for solving flow problems under non-smooth grids. PubDate: 2022-04-26

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Abstract: Abstract The parallel Schwarz method (PSM) is an overlapping domain decomposition (DD) method to solve partial differential equations (PDEs). Similarly to classical nonoverlapping DD methods, the PSM admits a substructured formulation, that is, it can be formulated as an iteration acting on variables defined exclusively on the interfaces of the overlapping decomposition. In this manuscript, spectral coarse spaces are considered to improve the convergence and robustness of the substructured PSM. In this framework, the coarse space functions are defined exclusively on the interfaces. This is in contrast to classical two-level volume methods, where the coarse functions are defined in volume, though with local support. The approach presented in this work has several advantages. First, it allows one to use some of the well-known efficient coarse spaces proposed in the literature, and facilitates the numerical construction of efficient coarse spaces. Second, the computational work is comparable or lower than standard volume two-level methods. Third, it opens new interesting perspectives as the analysis of the new two-level substructured method requires the development of a new convergence analysis of general two-level iterative methods. The new analysis casts light on the optimality of coarse spaces: given a fixed dimension m, the spectral coarse space made by the first m dominant eigenvectors is not necessarily the minimizer of the asymptotic convergence factor. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework. PubDate: 2022-04-16

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Abstract: Abstract A class of time–space tempered fractional diffusion equations is considered in this paper. The solution of these problems generally have a weak singularity near the initial time \(t = 0\) . To solve the time–space tempered fractional diffusion equations, a fully discrete local discontinuous Galerkin (LDG) method is proposed. The basic idea is to apply LDG method in the space on uniform meshes and a finite difference method in the time on graded meshes to deal with the weak singularity at initial time \(t = 0\) . The discrete fractional Grönwall inequality is used to analyze the stability and convergence of the method. Numerical results show that the proposed method for time–space tempered fractional diffusion equation is accurate and reliable. PubDate: 2022-04-16

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Abstract: Abstract A continuous Galerkin method based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard “radial” algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales. PubDate: 2022-04-15

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Abstract: Abstract This paper presents a lowest-order immersed Raviart–Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element (IFE) is constructed by modifying the traditional Raviart–Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal finite element error estimates are proved rigorously with the constant independent of the interface location relative to the mesh. Some numerical examples are provided to validate the theoretical analysis. PubDate: 2022-04-12

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Abstract: Abstract The harmonic balance method has emerged as an efficient and accurate approach for computing periodic, as well as almost periodic, solutions to nonlinear ordinary differential equations. The accuracy of the harmonic balance method can however be negatively impacted by aliasing. Aliasing occurs because Fourier coefficients of nonlinear terms in the governing equations are approximated by a discrete Fourier transform (DFT). Understanding how aliasing occurs when the DFT is applied is therefore essential in improving the accuracy of the harmonic balance method. In this work, a new operator that describe the fold-back, i.e. aliasing, of unresolved frequencies onto the resolved ones is developed. The norm of this operator is then used as a metric for investigating how the time sampling should be performed to minimize aliasing. It is found that a time sampling which minimizes the condition number of the DFT matrix is the best choice in this regard, both for single and multiple frequency problems. These findings are also verified for the Duffing oscillator. Finally, a strategy for oversampling multiple frequency harmonic balance computations is developed and tested. PubDate: 2022-04-11