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In this paper, we develop sixth-order hybrid finite difference methods (FDMs) for the elliptic interface problem $-\nabla \cdot( a\nabla u)=f$ in $\Omega\backslash \Gamma$, where $\Gamma$ is a smooth interface inside $\Omega$. The variable scalar coefficient $a>0$ and source $f$ are possibly discontinuous across $\Gamma$. The hybrid FDMs utilize a 9-point compact stencil at any interior regular point of the grid and a 13-point stencil at irregular points near $\Gamma$. For interior regular points away from $\Gamma$, we obtain a sixth-order 9-point compact FDM satisfying the M-matrix property. Consequently, for the elliptic problem without interface (i.e., $\Gamma$ is empty), our compact FDM satisfies the discrete maximum principle, which guarantees the theoretical sixth-order convergence. We also derive sixth-order compact (4-point for corners and 6-point for edges) FDMs having the M-matrix property at any boundary point subject to (mixed) Dirichlet/Neumann/Robin boundary conditions. For irregular points near $\Gamma$, we propose fifth-order 13-point FDMs, whose stencil coefficients can be effectively calculated by recursively solving several small linear systems. Theoretically, the proposed high order FDMs use high order (partial) derivatives of the coefficient $a$, the source term $f$, the interface curve $\Gamma$, the two jump functions along $\Gamma$, and the functions on $\partial \Omega$. Numerically, we always use function values to approximate all required high order (partial) derivatives in our hybrid FDMs without losing accuracy. Our proposed FDMs are independent of the choice representing $\Gamma$ and are also applicable if the jump conditions on $\Gamma$ only depend on the geometry (e.g., curvature) of the curve $\Gamma$. Our numerical experiments confirm the sixth-order convergence in the $l_{\infty}$ norm of the proposed hybrid FDMs for the elliptic interface problem.

It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $\lambda \to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrt{\lambda}$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $\lambda$ could be linked to the discretization parameters, yielding appropriate error estimates.

Deep learning methods have gained considerable interest in the numerical solution of various partial differential equations (PDEs). One particular focus is on physics-informed neural networks (PINNs), which integrate physical principles into neural networks. This transforms the process of solving PDEs into optimization problems for neural networks. In order to address a collection of advection-diffusion equations (ADE) in a range of difficult circumstances, this paper proposes a novel network structure. This architecture integrates the solver, which is a multi-scale deep neural network (MscaleDNN) utilized in the PINN method, with a hard constraint technique known as HCPINN. This method introduces a revised formulation of the desired solution for advection-diffusion equations (ADE) by utilizing a loss function that incorporates the residuals of the governing equation and penalizes any deviations from the specified boundary and initial constraints. By surpassing the boundary constraints automatically, this method improves the accuracy and efficiency of the PINN technique. To address the ``spectral bias'' phenomenon in neural networks, a subnetwork structure of MscaleDNN and a Fourier-induced activation function are incorporated into the HCPINN, resulting in a hybrid approach called SFHCPINN. The effectiveness of SFHCPINN is demonstrated through various numerical experiments involving advection-diffusion equations (ADE) in different dimensions. The numerical results indicate that SFHCPINN outperforms both standard PINN and its subnetwork version with Fourier feature embedding. It achieves remarkable accuracy and efficiency while effectively handling complex boundary conditions and high-frequency scenarios in ADE.

We consider approximating the solution of the Helmholtz exterior Dirichlet problem for a nontrapping obstacle, with boundary data coming from plane-wave incidence, by the solution of the corresponding boundary value problem where the exterior domain is truncated and a local absorbing boundary condition coming from a Pad\'e approximation (of arbitrary order) of the Dirichlet-to-Neumann map is imposed on the artificial boundary (recall that the simplest such boundary condition is the impedance boundary condition). We prove upper- and lower-bounds on the relative error incurred by this approximation, both in the whole domain and in a fixed neighbourhood of the obstacle (i.e. away from the artificial boundary). Our bounds are valid for arbitrarily-high frequency, with the artificial boundary fixed, and show that the relative error is bounded away from zero, independent of the frequency, and regardless of the geometry of the artificial boundary.

Following the traditional paradigm of convolutional neural networks (CNNs), modern CNNs manage to keep pace with more recent, for example transformer-based, models by not only increasing model depth and width but also the kernel size. This results in large amounts of learnable model parameters that need to be handled during training. While following the convolutional paradigm with the according spatial inductive bias, we question the significance of \emph{learned} convolution filters. In fact, our findings demonstrate that many contemporary CNN architectures can achieve high test accuracies without ever updating randomly initialized (spatial) convolution filters. Instead, simple linear combinations (implemented through efficient $1\times 1$ convolutions) suffice to effectively recombine even random filters into expressive network operators. Furthermore, these combinations of random filters can implicitly regularize the resulting operations, mitigating overfitting and enhancing overall performance and robustness. Conversely, retaining the ability to learn filter updates can impair network performance. Lastly, although we only observe relatively small gains from learning $3\times 3$ convolutions, the learning gains increase proportionally with kernel size, owing to the non-idealities of the independent and identically distributed (\textit{i.i.d.}) nature of default initialization techniques.

In this paper, we propose an efficient quadratic interpolation formula utilizing solution gradients computed and stored at nodes and demonstrate its application to a third-order cell-centered finite-volume discretization on tetrahedral grids. The proposed quadratic formula is constructed based on an efficient formula of computing a projected derivative. It is efficient in that it completely eliminates the need to compute and store second derivatives of solution variables or any other quantities, which are typically required in upgrading a second-order cell-centered unstructured-grid finite-volume discretization to third-order accuracy. Moreover, a high-order flux quadrature formula, as required for third-order accuracy, can also be simplified by utilizing the efficient projected-derivative formula, resulting in a numerical flux at a face centroid plus a curvature correction not involving second derivatives of the flux. Similarly, a source term can be integrated over a cell to high-order in the form of a source term evaluated at the cell centroid plus a curvature correction, again, not requiring second derivatives of the source term. The discretization is defined as an approximation to an integral form of a conservation law but the numerical solution is defined as a point value at a cell center, leading to another feature that there is no need to compute and store geometric moments for a quadratic polynomial to preserve a cell average. Third-order accuracy and improved second-order accuracy are demonstrated and investigated for simple but illustrative test cases in three dimensions.

Understanding the effect of a feature vector $x \in \mathbb{R}^d$ on the response value (label) $y \in \mathbb{R}$ is the cornerstone of many statistical learning problems. Ideally, it is desired to understand how a set of collected features combine together and influence the response value, but this problem is notoriously difficult, due to the high-dimensionality of data and limited number of labeled data points, among many others. In this work, we take a new perspective on this problem, and we study the question of assessing the difference of influence that the two given features have on the response value. We first propose a notion of closeness for the influence of features, and show that our definition recovers the familiar notion of the magnitude of coefficients in the parametric model. We then propose a novel method to test for the closeness of influence in general model-free supervised learning problems. Our proposed test can be used with finite number of samples with control on type I error rate, no matter the ground truth conditional law $\mathcal{L}(Y |X)$. We analyze the power of our test for two general learning problems i) linear regression, and ii) binary classification under mixture of Gaussian models, and show that under the proper choice of score function, an internal component of our test, with sufficient number of samples will achieve full statistical power. We evaluate our findings through extensive numerical simulations, specifically we adopt the datamodel framework (Ilyas, et al., 2022) for CIFAR-10 dataset to identify pairs of training samples with different influence on the trained model via optional black box training mechanisms.

A long-standing goal in scene understanding is to obtain interpretable and editable representations that can be directly constructed from a raw monocular RGB-D video, without requiring specialized hardware setup or priors. The problem is significantly more challenging in the presence of multiple moving and/or deforming objects. Traditional methods have approached the setup with a mix of simplifications, scene priors, pretrained templates, or known deformation models. The advent of neural representations, especially neural implicit representations and radiance fields, opens the possibility of end-to-end optimization to collectively capture geometry, appearance, and object motion. However, current approaches produce global scene encoding, assume multiview capture with limited or no motion in the scenes, and do not facilitate easy manipulation beyond novel view synthesis. In this work, we introduce a factored neural scene representation that can directly be learned from a monocular RGB-D video to produce object-level neural presentations with an explicit encoding of object movement (e.g., rigid trajectory) and/or deformations (e.g., nonrigid movement). We evaluate ours against a set of neural approaches on both synthetic and real data to demonstrate that the representation is efficient, interpretable, and editable (e.g., change object trajectory). Code and data are available at: $\href{//geometry.cs.ucl.ac.uk/projects/2023/factorednerf/}{\text{//geometry.cs.ucl.ac.uk/projects/2023/factorednerf/}}$.

We present a new mimetic finite difference method for diffusion problems that converges on grids with \textit{curved} (i.e., non-planar) faces. Crucially, it gives a symmetric discrete problem that uses only one discrete unknown per curved face. The principle at the core of our construction is to abandon the standard definition of local consistency of mimetic finite difference methods. Instead, we exploit the novel and global concept of $P_{0}$-consistency. Numerical examples confirm the consistency and the optimal convergence rate of the proposed mimetic method for cubic grids with randomly perturbed nodes as well as grids with curved boundaries.

Monocular visual-inertial odometry (VIO) is a low-cost solution to provide high-accuracy, low-drifting pose estimation. However, it has been meeting challenges in vehicular scenarios due to limited dynamics and lack of stable features. In this paper, we propose Ground-VIO, which utilizes ground features and the specific camera-ground geometry to enhance monocular VIO performance in realistic road environments. In the method, the camera-ground geometry is modeled with vehicle-centered parameters and integrated into an optimization-based VIO framework. These parameters could be calibrated online and simultaneously improve the odometry accuracy by providing stable scale-awareness. Besides, a specially designed visual front-end is developed to stably extract and track ground features via the inverse perspective mapping (IPM) technique. Both simulation tests and real-world experiments are conducted to verify the effectiveness of the proposed method. The results show that our implementation could dramatically improve monocular VIO accuracy in vehicular scenarios, achieving comparable or even better performance than state-of-art stereo VIO solutions. The system could also be used for the auto-calibration of IPM which is widely used in vehicle perception. A toolkit for ground feature processing, together with the experimental datasets, would be made open-source (//github.com/GREAT-WHU/gv_tools).