In this paper, we aim at developing scalable neural network-type learning systems. Motivated by the idea of constructive neural networks in approximation theory, we focus on constructing rather than training feed-forward neural networks (FNNs) for learning, and propose a novel FNNs learning system called the constructive FNN (CFN). Theoretically, we prove that the proposed method not only overcomes the classical saturation problem for constructive FNN approximation, but also reaches the optimal learning rate when the regression function is smooth, while the state-of-the-art learning rates established for traditional FNNs are only near optimal (up to a logarithmic factor). A series of numerical simulations are provided to show the efficiency and feasibility of CFN.

We construct a new family of high genus examples of free boundary minimal surfaces in the Euclidean unit 3-ball by desingularizing the intersection of a coaxial pair of a critical catenoid and an equatorial disk. The surfaces are constructed by singular perturbation methods and have three boundary components. They are the free boundary analogue of the Costa-Hoffman-Meeks surfaces and the surfaces constructed by Kapouleas by desingularizing coaxial catenoids and planes. It is plausible that the minimal surfaces we constructed here are the same as the ones obtained recently by Ketover using the min-max method.

The computation of a rigid body transformation which optimally aligns a set of measurement points with a surface and related registration problems are studied from the viewpoint of geometry and optimization.We provide a convergence analysis for widely used registration algorithms such as ICP, using either closest points (Besl and McKay [2]) or tangent planes at closest points (Chen and Medioni [4]), and for a recently developed approach based on quadratic approximants of the squared distance function [24]. ICP based on closest points exhibits local linear convergence only. Its counterpart which minimizes squared distances to the tangent planes at closest points is a Gauss-Newton iteration; it achieves local quadratic convergence for a zero residual problem and - if enhanced by regularization and step size control - comes close to quadratic convergence in many realistic scenarios. Quadratically convergent algorithms are based on the approach in [24]. The theoretical results are supported by a number of experiments; there, we also compare the algorithms with respect to global convergence behavior, stability and running time.

We propose second order accurate discontinuous Galerkin (DG) schemes which satisfy a strict maximum principle for general nonlinear convection-diffusion equations on unstructured triangular meshes. Motivated by genuinely high order maximum-principle-satisfying DG schemes for hyperbolic conservation laws, we prove that under suitable time step restriction for forward Euler time stepping, for general nonlinear convection-diffusion equations, the same scaling limiter coupled with second order DG methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy. Similar to the purely convection cases, the limiters are mass conservative and easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. Following the idea in earlier work, we extend the schemes to two-dimensional convection-diffusion equations on triangular meshes. There are no geometric constraints on the mesh such as angle acuteness. Numerical results including incompressible Navier-Stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods.

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