This paper proves error estimates for $H^2$ conforming finite elements for equations which model the flow of surfaces by different powers of the mean curvature (this includes mean curvature flow). for an adapted scheme originally proposed in [17] for the inverse mean curvature flow. The scheme is based on a known regularization procedure and produces different kinds of errors, a regularization error, a finite element discretization error for the regularized problems and a full error. While in the literature and own previous work different aspects of the aforementioned error types are treated, here, we solely and for the first time focus on the finite element discretization error in the $W^{2,μ}$ norm for the regularized equation analyzing also the dependencies from the regularization parameter.
Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work, we show a priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $H^1$ and $L^2$ error estimates for the corresponding regularized problem. As an …
In this paper, we study two kinds of structure-preserving splitting methods, including the Lie--Trotter type splitting method and the finite difference type method, for the stochasticlogarithmic Schrödinger equation (SlogS equation) via a regularized energy approximation. We first introduce a regularized SlogS equation with a small parameter $0<ε\ll1$ which approximates the SlogS equation and avoids the singularity near zero density. Then we present a priori estimates, the regularized entropy and energy, and the stochastic symplectic structure of the proposed numerical methods. Furthermore, we…
In this paper we consider the iteratively regularized Gauss-Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton / discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero, and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., $L^\infty$ as a preimage space. T…
Princeton University（普林斯顿大学）；Program for Applied and Computational Mathematics（应用与计算数学项目）；Department of Mathematics（数学系）
We consider binary and multi-class classification problems using hypothesis classes of neural networks. For a given hypothesis class, we use Rademacher complexity estimates and direct approximation theorems to obtain a priori error estimates for regularized loss functionals.
In this paper, we theoretically investigate the low-rank matrix recovery problem in the context of the unconstrained regularized nuclear norm minimization (RNNM) framework. Our theoretical findings show that, the RNNM method is able to provide a robust recovery of any matrix $X$ (not necessary to be exactly low-rank) from its few noisy measurements $\textbf{b}=\mathcal{A}(X)+\textbf{n}$ with a bounded constraint $\|\textbf{n}\|_{2}\leqε$, provided that the $tk$-order restricted isometry constant (RIC) of $\mathcal{A}$ satisfies a certain constraint related to $t>0$. Specifically, the obtained …
Fully-Discrete Finite Element Approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise: II. 2D and 3D Case
Department of Mathematics, University of Crete（克里特大学数学系）
This paper develops a robust three-level time split high-order Leapfrog/Crank-Nicolson technique for solving the two-dimensional unsteady sobolev and regularized long wave equations arising in fluid mechanics. A deep analysis of the stability and error estimates of the proposed approach is considered using the $L^{\infty}(0,T;H^{2})$-norm. Under a suitable time step requirement, the theoretical studies indicate that the constructed numerical scheme is strongly stable (in the sense of $L^{\infty}(0,T;H^{2})$-norm), temporal second-order accurate and convergence of order $O(h^{\frac{8}{3}})$ in …
This article develops a general theory for minimum norm interpolating estimators and regularized empirical risk minimizers (RERM) in linear models in the presence of additive, potentially adversarial, errors. In particular, no conditions on the errors are imposed. A quantitative bound for the prediction error is given, relating it to the Rademacher complexity of the covariates, the norm of the minimum norm interpolator of the errors and the size of the subdifferential around the true parameter. The general theory is illustrated for Gaussian features and several norms: The $\ell_1$, $\ell_2$, g…
Dynamic Proximal Gradient Algorithms for Schatten-$p$ Quasi-Norm Regularized Problems
Zhejiang Normal University（浙江师范大学）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院 H. 米尔顿·斯图尔特工业与系统工程学院）；School of Mathematical Sciences, Shenzhen University（深圳大学数学科学学院）；School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
We consider a regularization problem whose objective function consists of a convex fidelity term and a regularization term determined by the $\ell_1$ norm composed with a linear transform. Empirical results show that the regularization with the $\ell_1$ norm can promote sparsity of a regularized solution. It is the goal of this paper to understand theoretically the effect of the regularization parameter on the sparsity of the regularized solutions. We establish a characterization of the sparsity under the transform matrix of the solution. When the fidelity term has a special structure and the …
Moving object detection and its associated background-foreground separation have been widely used in a lot of applications, including computer vision, transportation and surveillance. Due to the presence of the static background, a video can be naturally decomposed into a low-rank background and a sparse foreground. Many regularization techniques, such as matrix nuclear norm, have been imposed on the background. In the meanwhile, sparsity or smoothness based regularizations, such as total variation and $\ell_1$, can be imposed on the foreground. Moreover, graph Laplacians are further imposed t…
McGill University（麦吉尔大学）；University of Minnesota（明尼苏达大学）
In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the curse of dimensionality, but it is independent of the number of layers in the networks. The adversarial robustness regularization combines adversarial training, which we show to be equivalent to Total Variation regularization, with Lipschitz regularization. We demonstrate empirically that the regularized models are more robust, and that gradient norms of images …
Xiamen University（厦门大学）；University of Science and Technology of China（中国科学技术大学）；Hong Kong University of Science and Technology（香港科技大学）；Xiamen University of Technology（厦门理工学院）
