Traditional methods for solving linear systems have quickly become impractical due to an increase in the size of available data. Utilizing massive amounts of data is further complicated when the data is incomplete or has missing entries. In this work, we address the obstacles presented when working with large data and incomplete data simultaneously. In particular, we propose to adapt the Stochastic Gradient Descent method to address missing data in linear systems. Our proposed algorithm, the Stochastic Gradient Descent for Missing Data method (mSGD), is introduced and theoretical convergence guarantees are provided. In addition, we include numerical experiments on simulated and real world data that demonstrate the usefulness of our method.

A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and analyzed in this paper. Using an auxiliary function, the truncated Wigner equation and its adjoint form are cast into integral formulations, which can be then reformulated into renewal-type equations with probabilistic interpretations. We prove that the first moment of a branching random walk is the solution for the adjoint equation. With the help of the additional degree of freedom offered by the auxiliary function, we are able to produce a weighted-particle implementation of the branching random walk. In contrast to existing signed-particle implementations, this weighted-particle one shows a key capacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors. Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings, and demonstrate the accuracy, the efficiency, and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.

Mixed triangular spectral element method using nodal basis on unstructured meshes is investigated in this paper. The method is based on equivalent first order system of the elliptic problem and  rectangle-triangle transforms. It fully enjoys the tensorial structure and flexibility in handling complex domains by using nodal basis and unstructured triangular mesh. Different from the usual Galerkin formulation, the mixed form is particularly advantageous in this context, since it can avoid the singularity induced by the rectangle-triangle transform in the calculation of the matrices, and does not require the evaluation of the stiffness matrix. An $hp$ a priori error estimate is presented for the proposed method. The implementation details and some numerical examples are provided to validate the accuracy and flexibility of the method.

We study the problem of a weighted integral of infinitely differentiable multivariate functions defined on the unit cube with the $L$_∞-norm of partial derivative of all orders bounded by 1. We consider the algorithms that use finitely many function values as information (called standard information). On the one hand, we obtained that the interpolatory quadratures based on the extended Chebyshev nodes of the second kind have almost the same quadrature weights. On the other hand, by using the  Smolyak algorithm with the above interpolatory quadratures, we proved that the weighted integral problem is of exponential convergence in the worst case setting.

A numerical method is proposed to approach the Approximate Inertial Manifolds (AIMs) in unsteady incompressible Navier-Stokes equations, using multilevel finite element method with hierarchical basis functions. Following AIMS, the unknown variables, velocity and pressure in the governing equations, are divided into two components, namely low modes and high modes. Then, the couplings between low modes and high modes, which are not accounted by standard Galerkin method, are considered by AIMs, to improve the accuracy of the numerical results. Further, the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way. As an example, the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented, and the comparisons show that there is a good agreement between the present method and experimental results. In particular, the proposed method takes less computing time than the traditional method. As a conclusion, the present method is efficient in numerical analysis of fluid dynamics, especially in computing time.

In this paper, we investigate a priori and a posteriori error estimates of fully discrete $H$¹-Galerkin mixed finite element methods for parabolic optimal control problems. The state variables and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. First, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. Second, by use of energy approach, we derive a posteriori error estimates for optimal control problems, assuming that only the underlying mesh is static. A numerical example is presented to verify the theoretical results on a priori error estimates.

As applying the Levenberg-Marquardt method to the  reformulation of linear complementarity problem, a modulus-based Levenberg-Marquardt method with non-monotone  line search is established and the global convergence result is presented. Numerical experiments show that the proposed method is efficient and outperforms the modulus-based matrix splitting iteration method.

This paper is concerned with numerical computations of a class of biological models on unbounded spatial domains. To overcome the unboundedness of spatial domain, we first construct efficient local absorbing boundary conditions (LABCs) to reformulate the Cauchy problem into an initial-boundary value (IBV) problem. After that, we construct a linearized finite difference scheme for the reduced IVB problem, and provide the corresponding error estimates and stability analysis. The delay-dependent dynamical properties on the Nicholson's blowflies equation and the Mackey-Glass equation are numerically investigated. Finally, numerical examples are given to demonstrate the efficiency of our LABCs and theoretical results of the numerical scheme.

This paper studies on the dimensions of spline spaces over some given $T$-meshes. Using the smoothing cofactor-conformality method, we study the instability in the dimensions of the spline spaces over $T$-meshes with 2-nested and 3-nested $T$-cycles. We define a singularity factor of each simple $T$-cycle, the instability and the structure's degeneration are associated with the singularity factors. In order to get a stable dimension formula over $T$-mesh with a $N$-nested $T$-cycle, a constraint on the $T$-mesh is introduced. Finally, a possible degeneration for a case of parallel $T$-cycles is illustrated.

In this paper we propose a mixed regularization method for ill-posed problems. This method combines iterative regularization methods and continuous regularization methods effectively. First it applies iterative regularization methods in which there is no continuous regularization parameter to solve the normal equation of the ill-posed problem. Then  continuous regularization methods are applied to solve its residual problem. The presented mixed regularization algorithm is a general framework. Any iterative regularization method and continuous regularization method can be combined together to construct a mixed regularization method. Our theoretical analysis shows that the new mixed regularization method is with optimal order of error estimation and can reach the optimal order under a much wider range of the regularization parameter than the continuous regularization method such as Tikhobov regularization. Moreover, the new mixed regularization method can reduce the sensitivity of the regularization parameter and improve the solution of continuous regularization methods or iterative regularization methods. This advantage is helpful when the optimal regularization parameter is hard to choose. The numerical computations illustrate the effectiveness of our new mixed regularization method.

We consider the problem of a posteriori error estimates and adaptivity for three typical force-based atomistic-to-continuum coupling methods. Combining the residual and the stability estimates, we derive computable a posteriori error estimators for the three methods in the energy norm and formulate adaptive algorithms using these estimators. Our numerical experiments show optimal convergence rates of these algorithms. The efficiency of the estimators is also demonstrated numerically.

Diagonalized Chebyshev rational spectral methods for solving second-order elliptic problems on the half/whole line are proposed. Some Sobolev bi-orthogonal rational basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like Chebyshev rational series. Numerical results demonstrate the effectiveness of the suggested approaches.

In this paper, we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the $L$¹- and $L$²-Euler's elastica energy respectively as the regularization for image segmentation. To capture contour curvature more reliably, we develop novel augmented Lagrangian functionals that ensure the segmentation level set function to be signed distance functions, which avoids the reinitialization of segmentation function during the iterative process. With the proposed algorithm and with the same initial contours, we compare the performance of these two high-order segmentation models and numerically verify the different properties of the two models.

The inconsistent accuracy and truncation error in the treatment of boundary usually leads to performance defects, such as decreased accuracy and even numerical instability, of the entire computational method, especially for higher order methods. In this work, we construct a consistent fourth-order compact finite difference scheme for solving two-dimensional incompressible Navier-Stokes (N-S) equations. In the proposed method, the main truncation error term of the boundary scheme is kept the same as that of the interior compact finite difference scheme. With such a feature, the numerical stability and accuracy of the entire computation can be maintained the same as the interior compact finite difference scheme. Numerical examples show the effectiveness and accuracy of the present consistent compact high order scheme in $L$^∞. Its application to two dimensional lid-driven cavity flow problem further exhibits that under the same condition, the computed solution with the present scheme is much close to the benchmark in comparison to those from the 4^$th$ order explicit scheme. The compact finite difference method equipped with the present consistent boundary technique improves much the stability of the whole computation and shows its potential application to incompressible flow of high Reynolds number.