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We introduce the multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficient $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\boldsymbol{y}) = \sum_{j\ge1} y_j\,\phi_j$ with $y_j\sim\mathcal{N}(0,1)$ and a given sequence of functions $\{\phi_j\}_{j\ge1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo (QMC) methods, and for which we use the finite element method (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces, taking into account the truncation error. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-d'/\lambda}) = O(\epsilon^{-(p^*+d'/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-d'/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $d' = d \, (1+\delta')$ for some $\delta' \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2+d'/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j\ge1}$.

We revisit the $k$-Hessian eigenvalue problem on a smooth, bounded, $(k-1)$-convex domain in $\mathbb R^n$. First, we obtain a spectral characterization of the $k$-Hessian eigenvalue as the infimum of the first eigenvalues of linear second-order elliptic operators whose coefficients belong to the dual of the corresponding G\r{a}rding cone. Second, we introduce a non-degenerate inverse iterative scheme to solve the eigenvalue problem for the $k$-Hessian operator. We show that the scheme converges, with a rate, to the $k$-Hessian eigenvalue for all $k$. When $2\leq k\leq n$, we also prove a local $L^1$ convergence of the Hessian of solutions of the scheme. Hyperbolic polynomials play an important role in our analysis.

In this paper, we propose a multirate iterative scheme with multiphysics finite element method for a fluid-saturated poroelasticity model. Firstly, we reformulate the original model into a fluid coupled problem to apply the multiphysics finite element method for the discretization of the space variables, and we design a multirate iterative scheme on the time scale which solve a generalized Stokes problem in the coarse time size and solve the diffusion problem in the finer time size according to the characteristics of the poroelasticity problem. Secondly, we prove that the multirate iterative scheme is stable and the numerical solution satisfies some energy conservation laws, which are important to ensure the uniqueness of solution to the decoupled computing problem. Also, we analyze the error estimates to prove that the proposed numerical method doesn't reduce the precision of numerical solution and greatly reduces the computational cost. Finally, we give the numerical tests to verify the theoretical results and draw a conclusion to summary the main results in this paper.

We propose a novel algorithmic method for constructing invariant variational schemes of systems of ordinary differential equations that are the Euler-Lagrange equations of a variational principle. The method is based on the invariantization of standard, non-invariant discrete Lagrangian functionals using equivariant moving frames. The invariant variational schemes are given by the Euler-Lagrange equations of the corresponding invariantized discrete Lagrangian functionals. We showcase this general method by constructing invariant variational schemes of ordinary differential equations that preserve variational and divergence symmetries of the associated continuous Lagrangians. Noether's theorem automatically implies that the resulting schemes are exactly conservative. Numerical simulations are carried out and show that these invariant variational schemes outperform standard numerical discretizations.

Mixed-precision algorithms combine low- and high-precision computations in order to benefit from the performance gains of reduced-precision without sacrificing accuracy. In this work, we design mixed-precision Runge-Kutta-Chebyshev (RKC) methods, where high precision is used for accuracy, and low precision for stability. Generally speaking, RKC methods are low-order explicit schemes with a stability domain growing quadratically with the number of function evaluations. For this reason, most of the computational effort is spent on stability rather than accuracy purposes. In this paper, we show that a na\"ive mixed-precision implementation of any Runge-Kutta scheme can harm the convergence order of the method and limit its accuracy, and we introduce a new class of mixed-precision RKC schemes that are instead unaffected by this limiting behaviour. We present three mixed-precision schemes: a first- and a second-order RKC method, and a first-order multirate RKC scheme for multiscale problems. These schemes perform only the few function evaluations needed for accuracy (1 or 2 for first- and second-order methods respectively) in high precision, while the rest are performed in low precision. We prove that while these methods are essentially as cheap as their fully low-precision equivalent, they retain the convergence order of their high-precision counterpart. Indeed, numerical experiments confirm that these schemes are as accurate as the corresponding high-precision method.

We consider asymptotically exact inference on the leading canonical correlation directions and strengths between two high dimensional vectors under sparsity restrictions. In this regard, our main contribution is the development of a loss function, based on which, one can operationalize a one-step bias-correction on reasonable initial estimators. Our analytic results in this regard are adaptive over suitable structural restrictions of the high dimensional nuisance parameters, which, in this set-up, correspond to the covariance matrices of the variables of interest. We further supplement the theoretical guarantees behind our procedures with extensive numerical studies.

Neural networks are susceptible to artificially designed adversarial perturbations. Recent efforts have shown that imposing certain modifications on classification layer can improve the robustness of the neural networks. In this paper, we explicitly construct a dense orthogonal weight matrix whose entries have the same magnitude, thereby leading to a novel robust classifier. The proposed classifier avoids the undesired structural redundancy issue in previous work. Applying this classifier in standard training on clean data is sufficient to ensure the high accuracy and good robustness of the model. Moreover, when extra adversarial samples are used, better robustness can be further obtained with the help of a special worst-case loss. Experimental results show that our method is efficient and competitive to many state-of-the-art defensive approaches. Our code is available at \url{//github.com/MTandHJ/roboc}.

Many systems, e.g. biological dynamics, are governed by diffusion-reaction equations on interface-separated domains. Interface conditions are typically described by systems of ordinary differential equations that provide fluxes across the interfaces. Using cellular calcium dynamics as an example of this class of ODE-flux boundary interface problems we prove the existence, uniqueness and boundedness of the solutions by applying comparison theorem, fundamental solution of the parabolic operator and a strategy used in Picard's existence theorem. Then we propose and analyze an efficient implicit-explicit finite element scheme which is implicit for the parabolic operator and explicit for the nonlinear terms. We show that the stability does not depend on the spatial mesh size. Also the optimal convergence rate in $H^1$ norm is obtained. Numerical experiments illustrate the theoretical results.

Provably stable flux reconstruction (FR) schemes are derived for partial differential equations cast in curvilinear coordinates. Specifically, energy stable flux reconstruction (ESFR) schemes are considered as they allow for design flexibility as well as stability proofs for the linear advection problem on affine elements. Additionally, split forms are examined as they enable the development of energy stability proofs. The first critical step proves, that in curvilinear coordinates, the discontinuous Galerkin (DG) conservative and non-conservative forms are inherently different--even under exact integration and analytically exact metric terms. This analysis demonstrates that the split form is essential to developing provably stable DG schemes on curvilinear coordinates and motivates the construction of metric dependent ESFR correction functions in each element. Furthermore, the provably stable FR schemes differ from schemes in the literature that only apply the ESFR correction functions to surface terms or on the conservative form, and instead incorporate the ESFR correction functions on the full split form of the equations. It is demonstrated that the scheme is divergent when the correction functions are only used for surface reconstruction in curvilinear coordinates. We numerically verify the stability claims for our proposed FR split forms and compare them to ESFR schemes in the literature. Lastly, the newly proposed provably stable FR schemes are shown to obtain optimal orders of convergence. The scheme loses the orders of accuracy at the equivalent correction parameter value c as that of the one-dimensional ESFR scheme.

Temporal modeling still remains challenging for action recognition in videos. To mitigate this issue, this paper presents a new video architecture, termed as Temporal Difference Network (TDN), with a focus on capturing multi-scale temporal information for efficient action recognition. The core of our TDN is to devise an efficient temporal module (TDM) by explicitly leveraging a temporal difference operator, and systematically assess its effect on short-term and long-term motion modeling. To fully capture temporal information over the entire video, our TDN is established with a two-level difference modeling paradigm. Specifically, for local motion modeling, temporal difference over consecutive frames is used to supply 2D CNNs with finer motion pattern, while for global motion modeling, temporal difference across segments is incorporated to capture long-range structure for motion feature excitation. TDN provides a simple and principled temporal modeling framework and could be instantiated with the existing CNNs at a small extra computational cost. Our TDN presents a new state of the art on the Something-Something V1 and V2 datasets and is on par with the best performance on the Kinetics-400 dataset. In addition, we conduct in-depth ablation studies and plot the visualization results of our TDN, hopefully providing insightful analysis on temporal difference operation. We release the code at //github.com/MCG-NJU/TDN.