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Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. In this paper, we introduce a new class of structured tensors called W-tensors, which not only extends thewell-studied nonnegative tensors by allowing negative entries but also covers several important tensors arising naturally from spectral hypergraph theory.We then show that finding the maximum H-eigenvalue of an even-order symmetric W-tensor is equivalent to solving a structured semidefinite program and hence can be validated in polynomial time. This yields a highly efficient semidefinite program algorithm for computing themaximumH-eigenvalue of W-tensors and is based on a new structured sums-of-squares decomposition result for a nonnegative polynomial induced by W-tensors. Numerical experiments illustrate that the proposed algorithm can successfully find the maximum H-eigenvalue of W-tensors with dimension up to 10,000, subject to machine precision. As applications, we provide a polynomial time algorithm for computing the maximum H-eigenvalues of large-size Laplacian tensors of hyperstars and hypertrees, where the algorithm can be up to 13 times faster than the state-of-the-art numerical method introduced by Ng, Qi, and Zhou in 2009. Finally, we also show that the proposed algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended Z-tensors,whose order may be even or odd.

Discontinuous Galerkin (DG) methods combine features in finite element methods (weak formulation, finite dimensional solution and test function spaces) and in finite volume methods (numerical fluxes, nonlinear limiters) and are particularly suitable for simulating convection dominated problems, such as linear and nonlinear waves including shock waves. In this article we will give a brief survey of DG methods, emphasizing their applications in computational fluid dynamics (CFD). We will discuss essential ingredients and properties of DG methods, and will also give a few examples of recent developments of DG methods which have facilitated their applications in CFD.

Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over $(3 × 10^{12})^2$ near the point of the singularity, we are able to advance the solution up to $\tau_2 = 0.003505$ and predict a singularity time of $t_s ≈ 0.0035056$, while achieving a pointwise relative error of $O(10^{−4})$ in the vorticity vector ω and observing a $(3 × 10^8)$-fold increase in the maximum vorticity $\|ω\|_{\infty}$. The numerical data are checked against all major blowup/non-blowup criteria, including Beale–Kato–Majda, Constantin–Fefferman–Majda, and Deng–Hou–Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

Motivated by big data applications, first-order methods have been extremely popular in recent years. However, naive gradient methods generally converge slowly. Hence, much efforts have been made to accelerate various first-order methods. This paper proposes two accelerated methods towards solving structured linearly constrained convex programming, for which we assume composite convex objective that is the sum of a differentiable function and a possibly nondifferentiable one. The first method is the accelerated linearized augmented Lagrangian method (LALM). At each update to the primal variable, it allows linearization to the differentiable function and also the augmented term, and thus it enables easy subproblems. Assuming merely convexity, we show that LALM owns O(1/t) convergence if parameters are kept fixed during all the iterations and can be accelerated to O(1/t^2) if the parameters are adapted, where t is the number of total iterations. The second method is the accelerated linearized alternating direction method of multipliers (LADMM). In addition to the composite convexity, it further assumes two-block structure on the objective. Different from classic ADMM, our method allows linearization to the objective and also augmented term to make the update simple. Assuming strong convexity on one block variable, we show that LADMM also enjoys O(1=t2) convergence with adaptive parameters. This result is a significant improvement over that in [Goldstein et. al, SIIMS'14], which requires strong convexity on both block variables and no linearization to the objective or augmented term. Numerical experiments are performed on quadratic programming, image denoising, and support vector machine. The proposed accelerated methods are compared to nonaccelerated ones and also existing accelerated methods. The results demonstrate the validness of acceleration and superior performance of the proposed methods over existing ones.