In this paper, with the help of finite element space on the coarse mesh, we design a type of efficient augmented subspace method to solve the large-scale algebraic eigenvalue problem resulting from finite element discretization of symmetric elliptic eigenvalue problems. The augmented subspace method may efficiently compute interior multiple and clustered eigenvalues of symmetric elliptic operators. It can transform solution of eigenvalue problems defined on fine mesh into several linear source problems on the same mesh, solved by some steps of multigrid or domain decomposition iterations, and a small-dimensional eigenvalue problem in the augmented subspace on every outer iteration. Using some new estimates we give a rigorous theoretical analysis for the proposed method, which shows that the convergence factor of the augmented subspace method is independent of fine mesh size and the gaps among the target eigenvalues (including multiple and clustered eigenvalues), depending on the distance from both ends of the target eigenvalues to others. Numerical examples are finally provided to validate the proposed method.
Abstract This paper addresses large-scale optimization problems defined over graph structures, specifically considering the estimation of parameters and states in hydraulic networks modeling river basins. The objective function is minimized over the Cartesian product of convex sets associated with different groups of variables. The proposed method, Accelerated Block Coordinate Descent (ABCD), employs Spectral Projected Gradient for the subproblems. We prove that, assuming only the continuity of the gradient, every accumulation point of the ABCD method is stationary. The ABCD approach is especially useful when variables in different blocks affect the objective function in dissimilar ways or when the computational cost of evaluating the function varies significantly between blocks. In hydraulic modeling, this allows for the separate treatment of flows, elevations and parameters. Since continuity equations are linear, we exploit potential accelerations based solely on the minimization of such equations. Numerical experiments indicate that the method can efficiently solve very large problems, exceeding the scale of river basins for which actual data are currently available.
Abstract We propose and analyse a novel semi-implicit low-regularity integrator for the dissipative surface quasi-geostrophic equations. The semi-implicit treatment yields an exact discrete transport cancellation, and hence unconditional energy stability, while retaining the efficiency of an exponential treatment of the fractional dissipation. To overcome the suboptimal regularity requirements typical of standard Sobolev-based error analysis for active scalars we introduce a general numerical Besov space framework. Within this framework we establish a sharp $B^{0}_{\infty ,1}$-error estimate and prove first-order temporal convergence under the endpoint assumption $\theta \in L^\infty (0,T; B^{2}_{\infty ,1})$. The gain is the removal of the additional fractional derivative loss in the Sobolev analysis; it is not an inclusion-wise weakening of the $W^{2,2+\epsilon }$ assumption. We present rigorous proofs for the stability and error bounds, complemented by extensive numerical experiments including convergence tests, vanishing viscosity limit validations and complex vortex dynamics, to demonstrate the robustness and accuracy of the method under low-regularity conditions.
Abstract This paper is devoted to investigating numerical integral $$\begin{align*} &\operatorname{Int}_{\mathbb S^d,w}(f):=\int_{\mathbb S^d}f(\textbf{x})w(\textbf{x}){\text{ d}}\sigma(\textbf{x}) \end{align*}$$ with respect to a weighted measure $w({\textbf{ x}})\,\text{d}\sigma (\textbf{x})$ for weighted function classes on the unit sphere $\mathbb S^{d}\subset \mathbb R^{d+1}$ in the deterministic and randomized settings. Specifically, we study weighted Sobolev classes $BW_{p,w}^{r}(\mathbb S^{d}),1\le p\le \infty$ with a Dunkl weight $w$, and weighted Besov classes $BB_\gamma ^\varTheta (L_{p,w}(\mathbb S^{d})),1\le p\le \infty$ with a doubling weight $w$ and a generalized smoothness index $\varTheta$ satisfying certain conditions. We derive asymptotic orders of optimal quadrature errors for $BW_{p,w}^{r}(\mathbb S^{d})$ in both settings. Moreover, we obtain upper bounds of optimal quadrature errors for $BB_\gamma ^\varTheta (L_{p,w}(\mathbb S^{d}))$ that are shown to be optimal when $w$ is an $A_\infty$ weight in the deterministic setting or a product weight in the randomized setting. Furthermore, asymptotic orders of sampling numbers for these classes are also proved. For upper bounds, we employ positive quadrature rules in the deterministic setting, and develop the weighted least $\ell _{p}$ approximation algorithms (operators) in the randomized setting, which are applicable to general doubling weights. Lower bounds are established by constructing proper ‘fooling’ functions. Similar results are also obtained on the unit ball and the standard simplex in $\mathbb R^{d}$.
Abstract Iterative methods based on tensors have emerged as powerful tools for solving tensor equations and have significantly advanced across multiple disciplines. In this study, we propose two-step tensor-based iterative methods to solve the tensor equations $\mathscr{A}*_{M}\mathscr{X}*_{M}\mathscr{B}=\mathscr{C}$ by incorporating preconditioning techniques and parametric optimization to enhance convergence properties. The theoretical results are complemented by comprehensive numerical experiments that demonstrated the computational efficiency of the proposed two-step parametrized iterative methods. The convergence criterion for parameter selection has been studied and a few numerical experiments have been conducted for optimal parameter selection. Effective algorithms were proposed to compute iterative methods based on two-step parametrized tensors, and the results are promising. In addition, we discuss the solution of the Sylvester equations and a regularized least-squares solution for image deblurring problems.
