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We introduce in this paper two time discretization schemes tailored for a range of Wasserstein gradient flows. These schemes are designed to preserve mass, positivity and to be uniquely solvable. In addition, they also ensure energy dissipation in many typical scenarios. Through extensive numerical experiments, we demonstrate the schemes' robustness, accuracy and efficiency.

From the perspective of control theory, convolutional layers (of neural networks) are 2-D (or N-D) linear time-invariant dynamical systems. The usual representation of convolutional layers by the convolution kernel corresponds to the representation of a dynamical system by its impulse response. However, many analysis tools from control theory, e.g., involving linear matrix inequalities, require a state space representation. For this reason, we explicitly provide a state space representation of the Roesser type for 2-D convolutional layers with $c_\mathrm{in}r_1 + c_\mathrm{out}r_2$ states, where $c_\mathrm{in}$/$c_\mathrm{out}$ is the number of input/output channels of the layer and $r_1$/$r_2$ characterizes the width/length of the convolution kernel. This representation is shown to be minimal for $c_\mathrm{in} = c_\mathrm{out}$. We further construct state space representations for dilated, strided, and N-D convolutions.

This work aims to improve texture inpainting after clutter removal in scanned indoor meshes. This is achieved with a new UV mapping pre-processing step which leverages semantic information of indoor scenes to more accurately match the UV islands with the 3D representation of distinct structural elements like walls and floors. Semantic UV Mapping enriches classic UV unwrapping algorithms by not only relying on geometric features but also visual features originating from the present texture. The segmentation improves the UV mapping and simultaneously simplifies the 3D geometric reconstruction of the scene after the removal of loose objects. Each segmented element can be reconstructed separately using the boundary conditions of the adjacent elements. Because this is performed as a pre-processing step, other specialized methods for geometric and texture reconstruction can be used in the future to improve the results even further.

Trigger-Action Programming (TAP) is a popular end-user programming framework in the home automation (HA) system, which eases users to customize home automation and control devices as expected. However, its simplified syntax also introduces new safety threats to HA systems through vulnerable rule interactions. Accurately fixing these vulnerabilities by logically and physically eliminating their root causes is essential before rules are deployed. However, it has not been well studied. In this paper, we present TAPFixer, a novel framework to automatically detect and repair rule interaction vulnerabilities in HA systems. It extracts TAP rules from HA profiles, translates them into an automaton model with physical and latency features, and performs model checking with various correctness properties. It then uses a novel negated-property reasoning algorithm to automatically infer a patch via model abstraction and refinement and model checking based on negated-properties. We evaluate TAPFixer on market HA apps (1177 TAP rules and 53 properties) and find that it can achieve an 86.65% success rate in repairing rule interaction vulnerabilities. We additionally recruit 23 HA users to conduct a user study that demonstrates the usefulness of TAPFixer for vulnerability repair in practical HA scenarios.

In this work we study the numerical approximation of a class of ergodic Backward Stochastic Differential Equations. These equations are formulated in an infinite horizon framework and provide a probabilistic representation for elliptic Partial Differential Equations of ergodic type. In order to build our numerical scheme, we put forward a new representation of the PDE solution by using a classical probabilistic representation of the gradient. Then, based on this representation, we propose a fully implementable numerical scheme using a Picard iteration procedure, a grid space discretization and a Monte-Carlo approximation. Up to a limiting technical condition that guarantee the contraction of the Picard procedure, we obtain an upper bound for the numerical error. We also provide some numerical experiments that show the efficiency of this approach for small dimensions.

In this paper, we compute stiffness matrix of the nonlocal Laplacian discretized by the piecewise linear finite element on nonuniform meshes, and implement the FEM in the Fourier transformed domain. We derive useful integral expressions of the entries that allow us to explicitly or semi-analytically evaluate the entries for various interaction kernels. Moreover, the limiting cases of the nonlocal stiffness matrix when the interactional radius $\delta\rightarrow0$ or $\delta\rightarrow\infty$ automatically lead to integer and fractional FEM stiffness matrices, respectively, and the FEM discretisation is intrinsically compatible. We conduct ample numerical experiments to study and predict some of its properties and test on different types of nonlocal problems. To the best of our knowledge, such a semi-analytic approach has not been explored in literature even in the one-dimensional case.

A common method for estimating the Hessian operator from random samples on a low-dimensional manifold involves locally fitting a quadratic polynomial. Although widely used, it is unclear if this estimator introduces bias, especially in complex manifolds with boundaries and nonuniform sampling. Rigorous theoretical guarantees of its asymptotic behavior have been lacking. We show that, under mild conditions, this estimator asymptotically converges to the Hessian operator, with nonuniform sampling and curvature effects proving negligible, even near boundaries. Our analysis framework simplifies the intensive computations required for direct analysis.

In this paper, we provide explicit formulas for the exact inverses of the symmetric tridiagonal near-Toeplitz matrices characterized by weak diagonal dominance in the Toeplitz part. Furthermore, these findings extend to scenarios where the corners of the near Toeplitz matrices lack diagonal dominance ($|\widetilde{b}| < 1$). Additionally, we compute the row sums and traces of the inverse matrices, thereby deriving upper bounds for their infinite norms. To demonstrate the practical applicability of our theoretical results, we present numerical examples addressing numerical solution of the Fisher problem using the fixed point method. Our findings reveal that the convergence rates of fixed-point iterations closely align with the expected rates, and there is minimal disparity between the upper bounds and the infinite norm of the inverse matrix. Specifically, this observation holds true for $|b| = 2$ with $|\widetilde{b}| \geq 1$. In other cases, there exists potential to enhance the obtained upper bounds.

We discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and differentiation matrices with optimal complexity and compute analysis and synthesis operations in quasi-optimal complexity. We investigate a particular application of these results to constructing orthogonal polynomials in annuli, called the generalised Zernike annular polynomials, which lead to sparse discretisations of partial differential equations. We compare against a scaled-and-shifted Chebyshev--Fourier series showing that in general the annular polynomials converge faster when approximating smooth functions and have better conditioning. We also construct a sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data.

We consider the motion of incompressible viscous fluid in a rectangle, imposing the periodicity condition in one direction and the no-slip boundary condition in the other. Assuming that the flow is subject to an external random force, white in time and regular in space, we construct an estimator for the viscosity using only observations of the enstrophy. The goal of the paper is to prove that the estimator is strongly consistent and asymptotically normal. The proof of consistency is based on the explicit formula for the estimator and some bounds for trajectories, while that of asymptotic normality uses in addition mixing properties of the Navier-Stokes flow.