This paper is devoted to the study of an extended global algorithm on computing the top eigenpairs of a large class of matrices. Three versions of the algorithm are presented that includes a preliminary version for real matrices, one for complex matrices, and one for large scale sparse real matrix. Some examples are illustrated as powerful applications of the algorithms. The main contributions of the paper are two localized estimation techniques, plus the use of a machine learning inspired approach in terms of a modified power iteration. Based on these new tools, the proposed algorithm successfully employs the inverse iteration with varying shifts (a very fast “cubic algorithm”) to achieve a superior estimation accuracy and computation efficiency to existing approaches under the general setup considered in this work.

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We prove mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.

We develop in this paper efficient numerical schemes for solving the micropolar Navier-Stokes equations by combining the SAV approach and pressure-correction method. Our first- and second-order semi-discrete schemes enjoy remarkable properties such as (i) unconditional energy stability with a modified energy, and (ii) only a sequence of decoupled linear equations with constant coefficients need to be solved at each time step. We also construct fully discrete versions of these schemes with a special spectral discretization which preserve the essential properties of the semi-discrete schemes. Numerical experiments are presented to validate the proposed schemes.

In this paper, we propose a residual-based a posteriori error estimator of embedded–hybridized discontinuous Galerkin finite element methods for the Stokes problems in two and three dimensions. The piecewise polynomials of degree $k (k≥1)$ and $k−1$ are used to approximate the velocity and pressure in the interior of elements, and the piecewise polynomials of degree $k$ are utilized to approximate the velocity and pressure on the inter-element boundaries. The attractive properties, named divergence-free and $H$(div)-conforming, are satisfied by the approximate velocity field. We prove that the a posteriori error estimator is robust in the sense that the ratio of the upper and lower bounds is independent of the mesh size and the viscosity. Finally, we provide several numerical examples to verify the theoretical results.

Hyperspectral unmixing (HU) plays an important role in terrain classification, agricultural monitoring, mineral recognition and quantification, and military surveillance. The existing model of the linear HU requires the observed vector to be a linear combination of the vertices. Due to the presence of noise, or any other perturbation source, we relax this linear constraint and penalize it to the objective function. The obtained model is solved by a sequence of gradient type steps which contain a projection onto the simplex constraint. We propose two gradient type algorithms for the linear HU, which can find vertices of the minimum volume simplex containing the observed hyper-spectral vectors. When the number of given pixels is huge, the computational time and complexity are so large that solving HU efficiently is usually challenging. A key observation is that our objective function is a summation of many similar simple functions. Then the computational time and complexity can be reduced by selecting a small portion of data points randomly. Furthermore, a stochastic variance reduction strategy is used. Preliminary numerical results showed that our new algorithms outperformed state-of-the-art algorithms on both synthetic and real data.

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated by a finite-dimensional constrained non-convex minimization problem after an appropriate spatial discretization. To efficiently solve the above optimization problem, we propose a so-called adaptive block Bregman proximal gradient (AB-BPG) algorithm that fully exploits the problem’s block structure. The proposed method updates each order parameter alternatively, and the update order of blocks can be chosen in a deterministic or random manner. Besides, we choose the step size by developing a practical linear search approach such that the generated sequence either keeps energy dissipation or has a controllable subsequence with energy dissipation. The convergence property of the proposed method is established without the requirement of global Lipschitz continuity of the derivative of the bulk energy part by using the Bregman divergence. The numerical results on computing stationary ordered structures in binary, ternary, and quinary component coupled-mode Swift-Hohenberg models have shown a significant acceleration over many existing methods.

We analyze a stochastic modified method of characteristics (MMOC) modeling advective-diffusive transport in randomly heterogeneous porous media. Under the log-normal assumption of the porous media and the finite-dimensional noise assumption that leads to unbounded diffusivity, we prove an optimal-order error estimate for the stochastic MMOC scheme. Numerical experiments are presented to substantiate the numerical analysis.