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This paper extends interpolatory model reduction to systems with (up to) quadratic-bilinear dynamics and quadratic-bilinear outputs. These systems are referred to as QB-QB systems and arise in a number of applications, including fluid dynamics, optimal control, and uncertainty quantification. In the interpolatory approach, the reduced order models (ROMs) are based on a Petrov-Galerkin projection, and the projection matrices are constructed so that transfer function components of the ROM interpolate the corresponding transfer function components of the original system. To extend the approach to systems with QB outputs, this paper derives system transfer functions and sufficient conditions on the projection matrices that guarantee the aforementioned interpolation properties. Alternatively, if the system has linear dynamics and quadratic outputs, one can introduce auxiliary state variables to transform it into a system with QB dynamics and linear outputs to which known interpolatory model reduction can be applied. This transformation approach is compared with the proposed extension that directly treats quadratic outputs. The comparison shows that transformation hides the problem structure. Numerical examples illustrate that keeping the original QB-QB structure leads to ROMs with better approximation properties. [ABSTRACT FROM AUTHOR]

In this paper, a class of 3D elliptic equations is solved by using the combination of the finite difference method in one direction and nonconforming finite element methods in the other two directions. A finite-difference (FD) discretization based on P 1 -element in the z-direction and a finite-element (FE) discretization based on P 1 NC -nonconforming element in the (x, y)-plane are used to convert the 3D equation into a series of 2D ones. This paper analyzes the convergence of P 1 NC -nonconforming finite element methods in the 2D elliptic equation and the error estimation of the H 1 -norm of the DFE method. Finally, in this paper, the DFE method is tested on the 3D elliptic equation with the FD method based on the P 1 element in the z-direction and the FE method based on the Crouzeix-Raviart element, the P 1 linear element, the Park-Sheen element, and the Q 1 bilinear element, respectively, in the (x, y)-plane. [ABSTRACT FROM AUTHOR]

The author’s recent research papers, “Cumulative deviation of a subpopulation from the full population” and “A graphical method of cumulative differences between two subpopulations” (both published in volume 8 of Springer’s open-access Journal of Big Data during 2021), propose graphical methods and summary statistics, without extensively calibrating formal significance tests. The summary metrics and methods can measure the calibration of probabilistic predictions and can assess differences in responses between a subpopulation and the full population while controlling for a covariate or score via conditioning on it. These recently published papers construct significance tests based on the scalar summary statistics, but only sketch how to calibrate the attained significance levels (also known as “P-values”) for the tests. The present article reviews and synthesizes work spanning many decades in order to detail how to calibrate the P-values. The present paper presents computationally efficient, easily implemented numerical methods for evaluating properly calibrated P-values, together with rigorous mathematical proofs guaranteeing their accuracy, and illustrates and validates the methods with open-source software and numerical examples. [ABSTRACT FROM AUTHOR]

Manifold models in data analysis and signal processing have become more prominent in recent years. In this paper, we will look at one of the main tasks of modern signal processing, namely, at analog-to-digital (A/D) conversion in connection with a simple manifold model — the circle. We will focus on Sigma-Delta modulation which is a popular method for A/D conversion of bandlimited signals that employs coarse quantization coupled with oversampling. Classical Sigma-Delta schemes provide mismatches and large errors at the initialization point if the signal to be converted is defined on the circle. In this paper, our goal is to get around these problems for Sigma-Delta schemes. Our results show how to design an update for the first and the second order schemes based on the reconstruction error analysis such that for the updated scheme the reconstruction error is improved. [ABSTRACT FROM AUTHOR]

In d dimensions, accurately approximating an arbitrary function oscillating with frequency ≲ k requires ∼ k d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k) suffers from the pollution effect if, as k → ∞ , the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the h-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth h and keeping the polynomial degree p fixed) suffers from the pollution effect, the hp-FEM (where accuracy is increased by decreasing the meshwidth h and increasing the polynomial degree p) does not suffer from the pollution effect. The heart of the proof of this result is a PDE result splitting the solution of the Helmholtz equation into “high” and “low” frequency components. This result for the constant-coefficient Helmholtz equation in full space (i.e. in ℝ d ) was originally proved in Melenk and Sauter (Math. Comp79(272), 1871–1914, 2010). In this paper, we prove this result using only integration by parts and elementary properties of the Fourier transform. The proof in this paper is motivated by the recent proof in Lafontaine et al. (Comp. Math. Appl.113, 59–69, 2022) of this splitting for the variable-coefficient Helmholtz equation in full space use the more-sophisticated tools of semiclassical pseudodifferential operators. [ABSTRACT FROM AUTHOR]

In this paper, we propose an unconditionally stable and L 2 optimal quadratic finite volume (FV) scheme for solving the two-dimensional anisotropic elliptic equation on triangular meshes. In quadratic FV schemes, the construction of the dual partition is closely related to the L 2 error estimate. While many dual partitions over triangular meshes have been investigated in the literature, only the one proposed by Wang and Li (SIAM J. Numer. Anal. 54:2729–2749, 2016) has been proven to achieve optimal L 2 norm convergence rate. This paper introduces a novel approach for constructing the dual partition using multiblock control volumes, which is also shown to optimally converge in the L 2 norm ( O (h 3) ). Furthermore, we present a new mapping from the trial space to the test space, which enables us to demonstrate that the inf-sup condition of the scheme holds independently of the minimal angle of the underlying mesh. To the best of our knowledge, this is the first unconditionally stable quadratic FV scheme over triangular meshes that achieves optimal L 2 norm convergence rate. We provide numerical experiments to validate our findings. [ABSTRACT FROM AUTHOR]

The article on Riemann derivatives by P. L. Butzer and W. Kozakiewicz of 1954 was the basis to generalizations of the classical scalar-valued derivatives to Taylor, Peano, and Riemann derivatives in the setting of semigroup theory. The present paper gives an overview of the 1954 article, describes its influence, and integrates it into the literature on related problems. It also describes the state of the mathematics department at McGill University where the article was written. [ABSTRACT FROM AUTHOR]

The paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on Ω (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on Ω . Moreover, we prove with an overwhelming probability that O (μ (Ω) (log μ (Ω)) 3) many random points uniformly distributed over Ω yield a stable set of sampling for functions concentrated on Ω . [ABSTRACT FROM AUTHOR]

In this paper, acceleration of gradient methods for convex optimization problems with weak levels of convexity and smoothness is considered. Starting from the universal fast gradient method which was designed to be an optimal method for weakly smooth problems whose gradients are Hölder continuous, its momentum is modified appropriately so that it can also accommodate uniformly convex and weakly smooth problems. Different from the existing works, fast gradient methods proposed in this paper do not use the restarting technique but use momentums that are suitably designed to reflect both the uniform convexity and weak smoothness information of the target energy function. Both theoretical and numerical results that support the superiority of the proposed methods are presented. [ABSTRACT FROM AUTHOR]

In this paper we present a construction of interpolatory Hermite multiwavelets for functions that take values in nonlinear geometries such as Riemannian manifolds or Lie groups. We rely on the strong connection between wavelets and subdivision schemes to define a prediction-correction approach based on Hermite subdivision schemes that operate on manifold-valued data. The main result concerns the decay of the wavelet coefficients: We show that our manifold-valued construction essentially admits the same coefficient decay as linear Hermite wavelets, which also generalizes results on manifold-valued scalar wavelets. [ABSTRACT FROM AUTHOR]

We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average — a notion introduced in this paper. We provide an example of such an average and show, via an illustrative interpolating subdivision scheme, how the limits of the subdivision scheme inherit geometric properties of the average. Finally, we prove the convergence of this subdivision scheme, whose limit interpolates the geometric Hermite data and approximates the sampled curve. We conclude the paper with various numerical examples that elucidate the advantages of our approach. [ABSTRACT FROM AUTHOR]

Numerical analysts and scientists working in applications often observe that once they improve their techniques to get a better accuracy, some instability of the evaluation creeps in through the back door. This paper shows for a large class of numerical methods that such a Trade-off Principle between error and evaluation stability is unavoidable. It is an instance of a no free lunch theorem. Here, evaluation is the mathematical map that takes input data to output data. This is independent from the numerical routine that calculates the output. Therefore, evaluation stability is different from computational stability. The setting is confined to recovery of functions from data, but it includes solving differential equations by writing such methods as a recovery of functions under constraints imposed by differential operators and boundary values. The trade-off principle bounds the product of two terms from below. The first is related to errors, and the second turns out to be related to evaluation instability. Under certain conditions satisfied for splines and kernel-based interpolation, both can be minimized. Then the lower bound is attained, and the error term is the inverse of the instability term. As a byproduct, it is shown that Kansa’s Unsymmetric Collocation Method sacrifices accuracy for improved evaluation stability, when compared to symmetric collocation. [ABSTRACT FROM AUTHOR]

The paper presents a versatile library of quasi-analytic complex-valued wavelet packets (qWPs) which originate from polynomial splines of arbitrary orders. Discrete Fourier transforms (DFT) of qWPs are located in either positive or negative half-band of the frequency domain. Consequently, the DFTs of 2D qWPs, which are derived by the tensor products of 1D qWPs, occupy one of quadrants of 2D frequency domain. Such a structure of the DFT spectra of the 2D qWPs results in the directionality of their real parts. Due to the fact that the spectra of qWPs are well localized in the frequency domain, the shapes of real parts of the qWPs are close to windowed cosine waves oscillating in a variety of different directions with a variety of frequencies. For example, a set of the fourth-level qWPs comprises 314 different directions and 256 different frequencies. The above properties combined with a fast transform implementation make directional qWPs a strong tool for the application to a variety of image processing tasks such as restoration of degraded images and extraction of characteristic features from images to use them in deep learning. A few illustrations of successful application of the designed qWPs to image denoising, inpainting, and classification are given in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Scaling Fractional Asymptotical Regularization (S-FAR) method for solving linear ill-posed operator equations in Hilbert spaces, inspired by the work of (2019 Fract. Calc. Appl. Anal. 22(3) 699-721). Our method is incorporated into the general framework of linear regularization and demonstrates that, under both Hölder and logarithmic source conditions, the S-FAR with fractional orders in the range (1, 2] offers accelerated convergence compared to comparable order optimal regularization methods. Additionally, we introduce a de-biasing strategy that significantly outperforms previous approaches, alongside a thresholding technique for achieving sparse solutions, which greatly enhances the accuracy of approximations. A variety of numerical examples, including one- and two-dimensional model problems, are provided to validate the accuracy and acceleration benefits of the FAR method. [ABSTRACT FROM AUTHOR]

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 is then solved using the MINI-element method and a simple DtD iteration strategy, resulting into a sequence of unique and geometrically (h- independent) convergent solutions. The quasi-optimal error estimate is proved for the iterative solution at the end of the iteration process. Numerical experiments are presented to demonstrate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel unfitted finite element method to solve the quad-curl interface problem. We adapt Nitsche’s method for curl 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 and demonstrate an optimal error bound in the discrete energy norm. We also analyze the stiffness matrix’s condition number. Our numerical tests back up our theory on convergence rates and condition numbers. [ABSTRACT FROM AUTHOR]

We revisit the moving least squares (MLS) approximation scheme on the sphere S d - 1 ⊂ R d , where d > 1 . It is well known that using the spherical harmonics up to degree L ∈ N as ansatz space yields for functions in C L + 1 (S d - 1) the approximation order O h L + 1 , where h denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to L, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as h → 0 . Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space. [ABSTRACT FROM AUTHOR]

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 of vehicle. By means of a machine learning model based on an LSTM recursive neural network, we extrapolate two important pieces of information: (1) if congestion is appearing under the sensor, and (2) the total amount of vehicles which is going to pass under the sensor in the next future (30 min). These pieces of information are then used to improve the accuracy of an LWR-based first-order multi-class model describing the dynamics of traffic flow between sensors. The first piece of information is used to invert the (concave) fundamental diagram, thus recovering the density of vehicles from the flux data, and then inject directly the density datum in the model. This allows one to better approximate the dynamics between sensors, especially if an accident/bottleneck happens in a not monitored stretch of the road. The second piece of information is used instead as boundary conditions for the equations underlying the traffic model, to better predict the total amount of vehicles on the road at any future time. Some examples motivated by real scenarios will be discussed. Real data are provided by the Italian motorway company Autovie Venete S.p.A. [ABSTRACT FROM AUTHOR]

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 non-monomial basis, we combine the tropical scaling with the block Kronecker linearization. We then establish upper bounds for the backward error of an approximate eigenpair of the QEP relative to the backward error of an approximate eigenpair of the block Kronecker linearization with and without tropical scaling. Moreover, we get bounds for the normwise condition number of an eigenvalue of the QEP relative to that of the block Kronecker linearization. Our investigation is accompanied by adequate numerical experiments to justify our theoretical findings. [ABSTRACT FROM AUTHOR]

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 with an error close to the best tree approximation error for a given cardinality. We support our findings by numerical experiments demonstrating that this approach can be used to achieve inverse-exponential convergence rates. [ABSTRACT FROM AUTHOR]

In this paper, we present the fundamentals of a hierarchical algorithm for computing the N-dimensional integral ϕ (a , b ; A) = ∫ a b H (x) f (x | A) d x representing the expectation of a function H(X) where f(x|A) is the truncated multi-variate normal (TMVN) distribution with zero mean, x is the vector of integration variables for the N-dimensional random vector X, A is the inverse of the covariance matrix Σ, and a and b are constant vectors. The algorithm assumes that H(x) is “low-rank” and is designed for properly clustered X so that the matrix A has “low-rank” blocks and “low-dimensional” features. We demonstrate the divide-and-conquer idea when A is a symmetric positive definite tridiagonal matrix and present the necessary building blocks and rigorous potential theory–based algorithm analysis when A is given by the exponential covariance model. The algorithm overall complexity is O(N) for N-dimensional problems, with a prefactor determined by the rank of the off-diagonal matrix blocks and number of effective variables. Very high accuracy results for N as large as 2048 are obtained on a desktop computer with 16G memory using the fast Fourier transform (FFT) and non-uniform FFT to validate the analysis. The current paper focuses on the ideas using the simple yet representative examples where the off-diagonal matrix blocks are rank 1 and the number of effective variables is bounded by 2, to allow concise notations and easier explanation. In a subsequent paper, we discuss the generalization of current scheme using the sparse grid technique for higher rank problems and demonstrate how all the moments of kth order or less (a total of O(Nk) integrals) can be computed using O(Nk) operations for k ≥ 2 and O (N log N) operations for k = 1. [ABSTRACT FROM AUTHOR]

This paper presents the coupling method of direct discontinuous Garlekin (DDG) and natural boundary element (NBE) for exterior interface problem. The coupled DDG method is based on the Dirichlet transmission condition on a circular artificial boundary. The feature allows one to use continuous basis functions on the artificial boundary for the NBE method but apply discontinuous basis functions in interior domain bounded by this boundary for the DDG method and it is compatible with the spirit of DG methods. Based on trace, inverse estimate and optimal polynomial estimate for straight elements and curved elements caused by curved interface and circular artificial boundary, error and stability analysis allowed are presented. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

This paper extends a conventional, general framework for online adaptive estimation problems for systems governed by unknown or uncertain nonlinear ordinary differential equations. The central feature of the theory introduced in this paper represents the unknown function as a member of a reproducing kernel Hilbert space (RKHS) and defines a distributed parameter system (DPS) that governs state estimates and estimates of the unknown function. Under the assumption that full state measurements are available, this paper (1) derives sufficient conditions for the existence and stability of the infinite dimensional online estimation problem, (2) derives existence and stability of finite dimensional approximations of the infinite dimensional approximations, and (3) determines sufficient conditions for the convergence of finite dimensional approximations to the infinite dimensional online estimates. A new condition for persistency of excitation in a RKHS in terms of its evaluation functionals is introduced in the paper that enables proof of convergence of the finite dimensional approximations of the unknown function in the RKHS. This paper studies two particular choices of the RKHS, those that are generated by exponential functions and those that are generated by multiscale kernels defined from a multiresolution analysis. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi–Raugel-like H (div) -conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. 42 (2022) 3711–3734]. Therein, the lowest-order H (div) -conforming Raviart–Thomas space ( RT 0 ) was added to the classical conforming P 1 × P 0 pair to meet the inf-sup condition, while preserving the divergence constraint and some important features of conforming methods. Due to the inf-sup stability of the P 1 ⊕ RT 0 × P 0 pair, a locking-free elasticity discretization with respect to the Lamé constant λ can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneous displacement boundary problem, that is, the discrete H 1 -norm of the displacement is O (λ - 1) when the external body force is a gradient field. We also consider the mixed displacement and stress boundary problem, whose P 1 ⊕ RT 0 discretization should be carefully designed due to a consistency error arising from the RT 0 part. We propose both symmetric and nonsymmetric schemes to approximate the mixed boundary case. The optimal error estimates are derived for the energy norm and/or L 2 -norm. Numerical experiments demonstrate the accuracy and robustness of our schemes. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set E. Starting from a frame (x n) n = 1 ∞ and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of (x n) n ∈ E c so that the perfect reconstruction can be obtained from the preserved frame coefficients. The work is motivated by methods using the canonical dual frame of (x n) n = 1 ∞ , which however do not extend automatically to the case when the canonical dual is replaced with another dual frame. The differences between the cases when the starting dual frame is the canonical dual and when it is not the canonical dual are investigated. We also give several ways of computing a dual of the reduced frame, among which we are the most interested in the iterative procedure for computing this dual frame. [ABSTRACT FROM AUTHOR]

The elastic transmission eigenvalue problem, arising from the inverse scattering theory, plays a critical role in the qualitative reconstruction methods for elastic media. This paper proposes and analyzes an a posteriori error estimator of the finite element method for solving the elastic transmission eigenvalue problem with different elastic tensors and different mass densities in R d (d = 2 , 3) . An adaptive algorithm based on the a posteriori error estimators is designed. Numerical results are provided to illustrate the efficiency of our adaptive algorithm. [ABSTRACT FROM AUTHOR]

This paper introduces a novel model for image fusion that is based on a fractional-order osmosis approach. The model incorporates a definition of osmosis energy that takes into account nonlocal pixel relationships using fractional derivatives and contrast change. The proposed model was subjected to theoretical and experimental investigation. The semigroup theory was used to demonstrate the existence and uniqueness of the evolution equation solution. Additionally, the model was validated and tested using numerical experiments and compared to local image fusion methods. The findings demonstrate that the proposed model outperforms the competitive local image fusion models. [ABSTRACT FROM AUTHOR]

In this paper, we study some techniques for solving numerically magnetostatic systems. We consider fairly general assumptions on the magnetic permeability tensor. It is elliptic, but can be nonhermitian. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of T-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on an augmented variational formulation obtained with the help of the weighted regularization method. [ABSTRACT FROM AUTHOR]

In various applications, the adoption of optimal energy matching pursuit with dictionary elements is common. When the dictionary elements are indexed by parameters within a bounded region, exhaustion-type algorithms can be employed. This article aims to investigate a process that converts the optimal parameter selection in unbounded regions to a bounded and closed (compact) sub-domain. Such a process provides accessibility for energy matching pursuit in a wide range of applications. The paper initially focuses on the open unit disc and the upper-half complex plane, introducing adaptive Fourier decomposition as the underlying methodology. It then extends this concept to general Hilbert spaces with a dictionary and Bochner spaces for random signals. Computational examples are included to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical symplectic approximation of the charged-particle dynamics (CPD) with a homogeneous magnetic field and its extension to a non-homogeneous magnetic field. By utilizing continuous-stage methods and exponential integrators, a general class of symplectic methods is formulated for CPD under a homogeneous magnetic field. Based on the derived symplectic conditions, two practical symplectic methods up to order four are constructed where the error estimates show that the proposed second order scheme has a uniform accuracy in the position w.r.t. the strength of the magnetic field. Moreover, the symplectic methods are extended to CPD under a non-homogeneous magnetic field and three algorithms are formulated. Rigorous error estimates are investigated for the proposed methods and one method is proved to have a uniform accuracy in the position w.r.t. the strength of the magnetic field. Numerical experiments are provided for CPD under homogeneous and non-homogeneous magnetic fields, and the numerical results support the theoretical analysis and demonstrate the remarkable numerical behavior of our methods. [ABSTRACT FROM AUTHOR]

Numerical simulations with rigid particles, drops, or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper, we present quadrature error estimates for layer potentials evaluated near surfaces of genus 0, parametrized using a polar and an azimuthal angle, discretized by a combination of the Gauss-Legendre and the trapezoidal quadrature rules. The error estimates involve no unknown coefficients, but complex-valued roots of a specified distance function. The evaluation of the error estimates in general requires a one-dimensional local root-finding procedure, but for specific geometries, we obtain analytical results. Based on these explicit solutions, we derive simplified error estimates for layer potentials evaluated near spheres; these simple formulas depend only on the distance from the surface, the radius of the sphere, and the number of discretization points. The usefulness of these error estimates is illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze the iterative two-grid methods for the discontinuous Galerkin (DG) discretization of semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that is just like the classical iterative two-grid methods for nonsymmetric or indefinite linear elliptic PDEs, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Secondly, we designed another iterative two-grid method, which replace the semilinear term by using the corresponding first-order Taylor expansion. Specifically, we need to construct a suitable initial value, which can be sorted out from an auxiliary variational problem, for the second iterative method. We also provide the error estimates for the second iterative algorithm and present numerical experiments to illustrate the theoretical result. [ABSTRACT FROM AUTHOR]

This paper is devoted to the backward problem of determining the initial value and initial velocity simultaneously in a time-fractional wave equation, with the aid of extra measurement data at two fixed times. Uniqueness results are obtained by using the analyticity and the asymptotics of the Mittag-Leffler functions provided that the two fixed measurement times are sufficiently close. Since this problem is ill-posed, we propose a quasi-reversibility method whose regularization parameters are given by the a priori parameter choice rule. Finally, several one- and two-dimensional numerical examples are presented to show the accuracy and efficiency of the proposed regularization method. [ABSTRACT FROM AUTHOR]

In this paper, we propose an image multi-frame Super Resolution (SR) method based on fractional-time Caputo derivative combined with Weickert-type diffusion process idea. We provide the existence and uniqueness results with a detailed discretization using the finite difference scheme. Our approach is based on anisotropic diffusion behavior with coherence enhancing diffusion tensor together with the fractional-time derivative to benefit from its memory effect potential and to control the smoothing process near strong edges and flat regions while avoiding tiny corners destruction. The experimental results confirm the effectiveness of fractional-time derivative and the robustness of the proposed Partial Differential Equation (PDE) compared with some competitive super-resolution methods. [ABSTRACT FROM AUTHOR]

In a recent work (J. Sci. Comput. 85 (2020), no. 3), the author generalized the Chebyshev-Davidson method appeared in standard eigenvalue problems to symmetric generalized eigenvalue problems. The theoretical derivation indicates that the Chebyshev-Davidson method for symmetric generalized eigenvalue problems only admits local convergence; thus, in this paper, we adopt a flexible strategy to improve the global convergence and to save number of iteration steps. Moreover, the deflation technique used for computing several eigenpairs in the proposed Chebyshev-Davidson method cannot be implemented in parallel; therefore, we construct a flexible block Chebyshev-Davidson method for computing several eigenpairs of symmetric generalized eigenvalue problems. The block implementation is important in scientific computing since it allows parallelism and efficient use of local memory. Numerical experiments are carried out to show great superiority and robustness over some state-of-the-art iteration methods. [ABSTRACT FROM AUTHOR]

In this paper, we present a new methodology to develop arbitrary high-order structure-preserving methods for solving the quantum Zakharov system. The key ingredients of our method are as follows: (i) the original Hamiltonian energy is reformulated into a quadratic form by introducing a new quadratic auxiliary variable; (ii) based on the energy variational principle, the original system is then rewritten into a new equivalent system which inherits the mass conservation law and a quadratic energy; and (iii) the resulting system is discretized by symplectic Runge–Kutta method in time combining with the Fourier pseudo-spectral method in space. The proposed method achieves arbitrary high-order accurate in time in a periodic domain and can exactly preserve the discrete mass and original Hamiltonian energy. Moreover, an efficient iterative solver is presented to solve the resulting discrete nonlinear equations. Finally, ample numerical examples are presented to demonstrate the theoretical claims and illustrate the efficiency of our methods. [ABSTRACT FROM AUTHOR]

The synchrosqueezing transform (SST) was developed recently to separate the components of non-stationary multicomponent signals. The continuous wavelet transform-based SST (WSST) reassigns the scale variable of the continuous wavelet transform of a signal to the frequency variable and sharpens the time-frequency representation. The WSST with a time-varying parameter, called the adaptive WSST, was introduced very recently in the paper “Adaptive synchrosqueezing transform with a time-varying parameter for non-stationary signal separation.” The well-separated conditions of non-stationary multicomponent signals with the adaptive WSST and a method to select the time-varying parameter were proposed in that paper. In addition, simulation experiments in that paper show that the adaptive WSST is very promising in estimating the instantaneous frequency of a multicomponent signal, and in accurate component recovery. However, the theoremretical analysis of the adaptive WSST has not been studied. In this paper, we carry out such analysis and obtain error bounds for component recovery with the adaptive WSST and the 2nd-order adaptive WSST. These results provide a mathematical guarantee to non-stationary multicomponent signal separation with the adaptive WSST. [ABSTRACT FROM AUTHOR]