• Abstract

We propose a new variational approach for recovering a general source profile in an elliptic system, using measurement data from the interior of the physical domain. The solution of the ill-posed inverse source problem is achieved by solving only one well-posed direct elliptic problem, resulting in the same computational cost as the one for the direct problem, and hence making the whole solution process of the inverse problem much less expensive than most existing methods. The resulting approximate solution is shown to be stable with respect to the change of the noise in the observation data, and a desired error estimate is also established in terms of the mesh size and the noise level in observation data. Numerical experiments are presented to confirm the theoretical predictions.

• Abstract

In this paper we present a novel approach for solving linear elliptic PDEs in regular convex polygons. The proposed algorithm relies on the so-called unified transform, or Fokas method. The basic step of this method involves the formulation of an equation coupling the finite Fourier transforms of the given boundary data and of the unknown boundary values, which is called the global relation. Herewith, a numerical scheme is proposed which computes the solution in the interior of a regular convex polygon using only the associated global relation. In particular, an adaptive complex collocation method is presented in order to solve numerically the global relation, using discrete boundary data. Additionally, the solution of a given PDE is computed in the entire computational domain, using a spatial-stepping scheme in conjunction with an adaptive complex collocation method. Moreover, a polynomial interpolation scheme is used near the center of the domain, and this increases the accuracy of the proposed method. We provide numerical results illustrating the applicability of the method as well as a comparison to a finite element formulation.

• Abstract

This paper studies  image decomposition models which involve functional related to total variation and Euler's elastica energy. Such kind of variational models with first order and higher order derivatives have been widely used in image processing to accomplish advanced tasks. However, these non-linear partial differential equations usually take high computational cost by the gradient descent method.  In this paper, we propose a proximal alternating direction method of multipliers (ADMM) for total variation (TV) based Vese-Osher's decomposition model [L. A. Vese and S. J. Osher, J. Sci.  Comput., 19.1 (2003), pp. 553-572]  and its extension with Euler's elastica regularization. We demonstrate that efficient and effective solutions to these minimization problems can be obtained by proximal based numerical algorithms. In numerical experiments, we present numerous results on image decomposition and image denoising, which conforms significant improvement of the proposed models over standard models.

• Abstract

Selective image segmentation is the task of extracting one object of interest among many others in an image based on minimal user input. Two-phase segmentation models cannot guarantee to locate this object, while multiphase models are more likely to classify this object with another features in the image. Several selective models were proposed recently and they would find local minimizers (sensitive to initialization) because non-convex minimization functionals are involved. Recently, Spencer-Chen (CMS 2015) has successfully proposed a convex selective variational image segmentation model (named CDSS), allowing a global minimizer to be found independently of initialization. However, their algorithm is sensitive to  the regularization parameter $\mu$ and the area parameter $\theta $ due to nonlinearity in the functional and additionally it is only effective for images of moderate size. In order to process images of large size associated with high resolution, urgent need exists in developing fast iterative solvers. In this paper, a stabilized variant of CDSS model through primal-dual formulation is proposed and an optimization based multilevel algorithm for  the new model is introduced. Numerical results show that  the new model   is less sensitive to parameter $\mu$ and $\theta$ compared to the original CDSS model and the multilevel algorithm produces quality segmentation in optimal computational time.

• Abstract

This paper considers the semi-algebraic split feasibility problem (SASFP), i.e., the split feasibility problem defined by polynomials. It is more than a special case of the split feasibility problem (SFP) or the multiple-sets split feasibility problem (MSFP), since the solution set could be nonconvex or empty. We first establish the semi-definite relaxation for the SASFP, then discuss on the relationship of feasibility between the SASFP and its SDP relaxation, especially focus on infeasibility. Finally, some numerical experiments for different cases are implemented, and the corresponding results are reported.

• Abstract

The $p$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil's exponential sum theorem presents an upper bound of the exponential sum over the $p$-set. Based on the result, one shows that the $p$-set performs well in numerical integration, in compressed sensing as well as in uncertainty quantification. However, $p$-set is somewhat rigid since the cardinality of the $p$-set is a prime $p$ and the set only depends on the prime number $p$. The purpose of this paper is to present generalizations of $p$-sets, say $\mathcal{P}_{d,p}^{a,\epsilon}$, which is more flexible. Particularly, when a prime number $p$ is given, we have many different choices of the new $p$-sets. Under the assumption that Goldbach conjecture holds, for any even number $m$, we present a point set, say ${\mathcal L}_{p,q}$, with cardinality $m-1$ by combining two different new $p$-sets, which overcomes a major bottleneck of the $p$-set. We also present the upper bounds of the exponential sums over  $\mathcal{P}_{d,p}^{a,\epsilon}$ and ${\mathcal L}_{p,q}$, which imply these sets have many potential applications.

• Abstract

Piecewise constant images, which are sampled from piecewise constant functions, are an important kind of  images data. Typical examples include QR codes (Quick Response codes), logos and text images, which are widely used in both general commercial and automotive industry use. In this paper, we consider the problem of removing Gaussian noise from this kind of images. A novel method based on selective averaging and outlier removal is proposed. The selective averaging updates the intensity value at each pixel by averaging pixels in its homogeneous neighborhood. This scheme prevents the diffusion between pixels belonging to different homogeneous regions. Thus, it preserves image edges quite well. The outlier removal is adopted to detect and suppress outliers appearing in the output of selective averaging. The experiments on both gray and color image denoising show that our method is feasible and effective for piecewise constant image restoration, and achieves superior performance among all the compared methods.

• Abstract

The second and all higher order moments of the $\beta$-stable Lévy process diverge, the feature of which is sometimes referred to as shortcoming of the model when applied to physical processes. So a parameter $\lambda$ is introduced to exponentially temper the  Lévy process. The generator of the new process is tempered fractional Laplacian $(\Delta+\lambda)^{\beta/2}$ [W. H. Deng, B. Y. Li, W. Y. Tian and P. W. Zhang, Multiscale Model. Simul., 16(1), 125-149, 2018]. In this paper, we first design the finite difference schemes for the tempered fractional Laplacian equation with the generalized Dirichlet type boundary condition, their accuracy depending on the regularity of the exact solution on $\bar{\Omega}$. Then the techniques of effectively solving the resulting algebraic equation are presented, and the performances of the schemes are demonstrated by several numerical examples.

• Abstract

In this paper we are concerned with plane wave discretizations of nonhomogeneous and anisotropic time-harmonic Maxwell's equations. Combined with local spectral element method, we design a plane wave discontinuous Galerkin method for the discretization of such three dimensional nonhomogeneous and anisotropic Maxwell's equations. The error estimates of the approximation solutions generated by the proposed discretization method are derived in one special case of the TE mode scattering. In the error estimates, some dependence of the error bounds on the condition number of the anisotropic matrix is explicitly given. Numerical results indicate that the resulting approximate solutions generated by the new method possess high accuracy and verify the validity of the theoretical results.

• Abstract

This paper is mainly concerned with the strong convergence analysis of the semi-implicit Euler method for a kind of stochastic Volterra integro-differential equations (SVIDEs). The solvability and the mean-square boundedness of numerical solutions are presented. In view of the properties of the Itô integral, different from the known stochastic problems, it is proved that the strong convergence order of the semi-implicit Euler method is 1, although the approximation order of the Itô integral is 0.5. The theoretical results are illustrated by extensive numerical examples.

• Abstract

The aim of this paper is to study the convergence rates of the trinomial tree methods (TTMs) and perturbed finite difference methods (PFDMs) for system parabolic variational inequalities which govern the value function of regime-switching American option. This paper has threefold contributions: (i) It establishes the higher-order equivalence between the TTMs and the PFDMs for the regime-switching American options; (ii) It proves the regularities of the solutions to the system of parabolic variational inequalities governing the price of the American options, and studies the comparison principles and the penalty methods. These results are used to prove the convergence rates of the PFDMs; (iii) It proves the convergence rates of the PFDMs for the system of parabolic variational inequalities governing the price of the American options. The convergence rates of the TTMs are obtained by the higher-order equivalence between the TTMs and the PFDMs and the convergence theory for the PFDMs.

• Abstract

When solving hyperbolic Boltzmann model equation with discrete velocity models (DVM), the strong discontinuity of the velocity distribution function can be captured well by utilizing the non-oscillatory and non-free parameter dissipation (NND) finite difference scheme. However, most NND scheme solvers march in time explicitly, which compromise the computation efficiency due to the limitation of stability condition, especially when solving unsteady problems. In order to improve the efficiency, an implicit scheme based on NND is presented in this paper. Linearization factors are introduced to construct the implicit scheme and to reduce the stencil size. With the help of dual time-stepping method, the convergence rate of unsteady rarefied flow simulation can be massively improved. Numerical tests of steady and unsteady supersonic flow around cylinders are computed in different flow regimes. Results are shown to prove the validity and efficiency of the  implicit scheme.

• Abstract

In this work, we propose and analyze a second-order accurate numerical scheme, both in time and space, for the multi-dimensional Poisson-Nernst-Planck system. Linearized stability analysis is developed, so that the second order accuracy is theoretically justified for the numerical scheme, in both temporal and spatial discretization. In particularly, the discrete $W^{1,4}$ estimate for the electric potential field, which plays a crucial role in the proof, are rigorously established. In addition, various numerical tests have confirmed the anticipated numerical accuracy, and further demonstrated the effectiveness and robustness of the numerical scheme in solving problems of practical interest.

• Abstract

In this note we prove a new theoretical estimate for the convergence rate of the maximal weighted residual Kaczmarz algorithm for solving  a consistent linear system. The estimate depends only on quantities that are easy to compute and not on the number of equations in the system. We compare the maximal weighted residual Kaczmarz algorithm and the greedy randomized Kaczmarz algorithm by two sets of examples. Numerical results show that the maximal weighted residual Kaczmarz algorithm requires almost the same number of iterations as that of the greedy randomized Kaczmarz algorithm for underdetermined linear systems and less iterations for overdetermined linear systems. Due to less computational cost in the row index selection strategy, the maximal weighted residual Kaczmarz algorithm is more efficient than the greedy randomized Kaczmarz algorithm.

• Abstract

An optimized two-step hybrid block method is presented for integrating general second order initial value problems numerically. The method considers two intra-step points which are selected adequately in order to optimize the local truncation errors of the main formulas for the solution and the first derivative at the final point of the block. The new proposed method is consistent, zero-stable and has seventh algebraic order of convergence. To illustrate the performance of the method, some numerical experiments are presented for solving this kind of problems, in comparison with methods of similar characteristics in the literature.