In this article, we study the inconsistency of systems of $\min-\rightarrow$ fuzzy relational equations. We give analytical formulas for computing the Chebyshev distances $\nabla = \inf_{d \in \mathcal{D}} \Vert \beta - d \Vert$ associated to systems of $\min-\rightarrow$ fuzzy relational equations of the form $\Gamma \Box_{\rightarrow}^{\min} x = \beta$, where $\rightarrow$ is a residual implicator among the G\"odel implication $\rightarrow_G$, the Goguen implication $\rightarrow_{GG}$ or Lukasiewicz's implication $\rightarrow_L$ and $\mathcal{D}$ is the set of second members of consistent systems defined with the same matrix $\Gamma$. The main preliminary result that allows us to obtain these formulas is that the Chebyshev distance $\nabla$ is the lower bound of the solutions of a vector inequality, whatever the residual implicator used. Finally, we show that, in the case of the $\min-\rightarrow_{G}$ system, the Chebyshev distance $\nabla$ may be an infimum, while it is always a minimum for $\min-\rightarrow_{GG}$ and $\min-\rightarrow_{L}$ systems.
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$f$-divergences, which quantify discrepancy between probability distributions, are ubiquitous in information theory, machine learning, and statistics. While there are numerous methods for estimating $f$-divergences from data, a limit distribution theory, which quantifies fluctuations of the estimation error, is largely obscure. As limit theorems are pivotal for valid statistical inference, to close this gap, we develop a general methodology for deriving distributional limits for $f$-divergences based on the functional delta method and Hadamard directional differentiability. Focusing on four prominent $f$-divergences -- Kullback-Leibler divergence, $\chi^2$ divergence, squared Hellinger distance, and total variation distance -- we identify sufficient conditions on the population distributions for the existence of distributional limits and characterize the limiting variables. These results are used to derive one- and two-sample limit theorems for Gaussian-smoothed $f$-divergences, both under the null and the alternative. Finally, an application of the limit distribution theory to auditing differential privacy is proposed and analyzed for significance level and power against local alternatives.
Directional interpolation is a fast and efficient compression technique for high-frequency Helmholtz boundary integral equations, but it requires a very large amount of storage in its original form. Algebraic recompression can significantly reduce the storage requirements and speed up the solution process accordingly. During the recompression process, weight matrices are required to correctly measure the influence of different basis vectors on the final result, and for highly accurate approximations, these weight matrices require more storage than the final compressed matrix. We present a compression method for the weight matrices and demonstrate that it introduces only a controllable error to the overall approximation. Numerical experiments show that the new method leads to a significant reduction in storage requirements.
Neural ordinary differential equations (neural ODEs) are a popular family of continuous-depth deep learning models. In this work, we consider a large family of parameterized ODEs with continuous-in-time parameters, which include time-dependent neural ODEs. We derive a generalization bound for this class by a Lipschitz-based argument. By leveraging the analogy between neural ODEs and deep residual networks, our approach yields in particular a generalization bound for a class of deep residual networks. The bound involves the magnitude of the difference between successive weight matrices. We illustrate numerically how this quantity affects the generalization capability of neural networks.
In this paper we give the detailed error analysis of two algorithms (denoted as $W_1$ and $W_2$) for computing the symplectic factorization of a symmetric positive definite and symplectic matrix $A \in \mathbb R^{2n \times 2n}$ in the form $A=LL^T$, where $L \in \mathbb R^{2n \times 2n}$ is a symplectic block lower triangular matrix. Algorithm $W_1$ is an implementation of the $HH^T$ factorization from [Dopico et al., 2009]. Algorithm $W_2$, proposed in [Bujok et al., 2023], uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrix blocks that appear during the factorization. We prove that Algorithm $W_2$ is numerically stable for a broader class of symmetric positive definite matrices $A \in \mathbb R^{2n \times 2n}$, producing the computed factors $\tilde L$ in floating-point arithmetic with machine precision $u$, such that $||A-\tilde L {\tilde L}^T||_2= {\cal O}(u ||A||_2)$. However, Algorithm $W_1$ is unstable in general for symmetric positive definite and symplectic matrix $A$. This was confirmed by numerical experiments in [Bujok et al., 2023]. In this paper we give corresponding bounds also for Algorithm $W_1$ that are weaker, since we show that the factorization error depends on the size of the inverse of the principal submatrix $A_{11}$. The tests performed in MATLAB illustrate that our error bounds for considered algorithms are reasonably sharp.
This manuscript studies the numerical solution of the time-fractional Burgers-Huxley equation in a reproducing kernel Hilbert space. The analytical solution of the equation is obtained in terms of a convergent series with easily computable components. It is observed that the approximate solution uniformly converges to the exact solution for the aforementioned equation. Also, the convergence of the proposed method is investigated. Numerical examples are given to demonstrate the validity and applicability of the presented method. The numerical results indicate that the proposed method is powerful and effective with a small computational overhead.
This paper introduces a novel class of fair and interpolatory curves called $p\kappa$-curves. These curves are comprised of smoothly stitched B\'ezier curve segments, where the curvature distribution of each segment is made to closely resemble a parabola, resulting in an aesthetically pleasing shape. Moreover, each segment passes through an interpolated point at a parameter where the parabola has an extremum, encouraging the alignment of interpolated points with curvature extrema. To achieve these properties, we tailor an energy function that guides the optimization process to obtain the desired curve characteristics. Additionally, we develop an efficient algorithm and an initialization method, enabling interactive modeling of the $p\kappa$-curves without the need for global optimization. We provide various examples and comparisons with existing state-of-the-art methods to demonstrate the curve modeling capabilities and visually pleasing appearance of $p\kappa$-curves.
We proposed a divergence-free and $H(div)$-conforming embedded-hybridized discontinuous Galerkin (E-HDG) method for solving stationary incompressible viso-resistive magnetohydrodynamic (MHD) equations. In particular, the E-HDG method is computationally far more advantageous over the hybridized discontinuous Galerkin (HDG) counterpart in general. The benefit is even significant in the three-dimensional/high-order/fine mesh scenario. On a simplicial mesh, our method with a specific choice of the approximation spaces is proved to be well-posed for the linear case. Additionally, the velocity and magnetic fields are divergence-free and $H(div)$-conforming for both linear and nonlinear cases. Moreover, the results of well-posedness analysis, divergence-free property, and $H(div)$-conformity can be directly applied to the HDG version of the proposed approach. The HDG or E-HDG method for the linearized MHD equations can be incorporated into the fixed point Picard iteration to solve the nonlinear MHD equations in an iterative manner. We examine the accuracy and convergence of our E-HDG method for both linear and nonlinear cases through various numerical experiments including two- and three-dimensional problems with smooth and singular solutions. For smooth problems, the results indicate that convergence rates in the $L^2$ norm for the velocity and magnetic fields are optimal in the regime of low Reynolds number and magnetic Reynolds number. Furthermore, the divergence error is machine zero for both smooth and singular problems. Finally, we numerically demonstrated that our proposed method is pressure-robust.
We aim to establish Bowen's equations for upper capacity invariance pressure and Pesin-Pitskel invariance pressure of discrete-time control systems. We first introduce a new invariance pressure called induced invariance pressure on partitions that specializes the upper capacity invariance pressure on partitions, and then show that the two types of invariance pressures are related by a Bowen's equation. Besides, to establish Bowen's equation for Pesin-Pitskel invariance pressure on partitions we also introduce a new notion called BS invariance dimension on subsets. Moreover, a variational principle for BS invariance dimension on subsets is established.
In this paper, the strong formulation of the generalised Navier-Stokes momentum equation is investigated. Specifically, the formulation of shear-stress divergence is investigated, due to its effect on the performance and accuracy of computational methods. It is found that the term may be expressed in two different ways. While the first formulation is commonly used, the alternative derivation is found to be potentially more convenient for direct numerical manipulation. The alternative formulation relocates a part of strain information under the variable-coefficient Laplacian operator, thus making future computational schemes potentially simpler with larger time-step sizes.
The Reynolds equation, combined with the Elrod algorithm for including the effect of cavitation, resembles a nonlinear convection-diffusion-reaction (CDR) equation. Its solution by finite elements is prone to oscillations in convection-dominated regions, which are present whenever cavitation occurs. We propose a stabilized finite-element method that is based on the variational multiscale method and exploits the concept of orthogonal subgrid scales. We demonstrate that this approach only requires one additional term in the weak form to obtain a stable method that converges optimally when performing mesh refinement.
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