In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift-Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta-Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.
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Integration：Integration, the VLSI Journal。
Explanation：集成，VLSI雜志。
Publisher：Elsevier。
SIT：
In this article, we study the Fekete problem in segmental and combined nodal-segmental univariate polynomial interpolation by investigating sets of segments, or segments combined with nodes, such that the Vandermonde determinant for the respective polynomial interpolation problem is maximized. For particular families of segments, we will be able to find explicit solutions of the corresponding maximization problem. The quality of the Fekete segments depends hereby strongly on the utilized normalization of the segmental information in the Vandermonde matrix. To measure the quality of the Fekete segments in interpolation, we analyse the asymptotic behaviour of the generalized Lebesgue constant linked to the interpolation problem. For particular sets of Fekete segments we will get, similar to the nodal case, a favourable logarithmic growth of this constant.
We consider the general class of time-homogeneous stochastic dynamical systems, both discrete and continuous, and study the problem of learning a representation of the state that faithfully captures its dynamics. This is instrumental to learning the transfer operator or the generator of the system, which in turn can be used for numerous tasks, such as forecasting and interpreting the system dynamics. We show that the search for a good representation can be cast as an optimization problem over neural networks. Our approach is supported by recent results in statistical learning theory, highlighting the role of approximation error and metric distortion in the learning problem. The objective function we propose is associated with projection operators from the representation space to the data space, overcomes metric distortion, and can be empirically estimated from data. In the discrete-time setting, we further derive a relaxed objective function that is differentiable and numerically well-conditioned. We compare our method against state-of-the-art approaches on different datasets, showing better performance across the board.
The recent paper (IEEE Trans. IT 69, 1680) introduced an analytical method for calculating the channel capacity without the need for iteration. This method has certain limitations that restrict its applicability. Furthermore, the paper does not provide an explanation as to why the channel capacity can be solved analytically in this particular case. In order to broaden the scope of this method and address its limitations, we turn our attention to the reverse em-problem, proposed by Toyota (Information Geometry, 3, 1355 (2020)). This reverse em-problem involves iteratively applying the inverse map of the em iteration to calculate the channel capacity, which represents the maximum mutual information. However, several open problems remained unresolved in Toyota's work. To overcome these challenges, we formulate the reverse em-problem based on Bregman divergence and provide solutions to these open problems. Building upon these results, we transform the reverse em-problem into em-problems and derive a non-iterative formula for the reverse em-problem. This formula can be viewed as a generalization of the aforementioned analytical calculation method. Importantly, this derivation sheds light on the information geometrical structure underlying this special case. By effectively addressing the limitations of the previous analytical method and providing a deeper understanding of the underlying information geometrical structure, our work significantly expands the applicability of the proposed method for calculating the channel capacity without iteration.
Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has also been proved mathematically, i.e., it has been shown that the number of parameters of the approximating DNN increases at most polynomially in both the PDE dimension $d\in \mathbb{N}$ and the reciprocal of the prescribed accuracy $\epsilon\in (0,1)$. The main contribution of this paper is to rigorously prove for the first time that deep neural networks can also overcome the curse dimensionality in the approximation of a certain class of nonlinear PDEs with gradient-dependent nonlinearities.
In this paper, we present a~generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued \L{}ukasiewicz logics. To this end, we provide proof systems that augment Avron's Frege system for \L{}ukasiewicz three-valued logic with nested and general versions of the disjunction elimination rule, respectively. For these systems we provide upper bounds on speed-ups w.r.t.\ both the number of steps in proofs and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus for 3-valued \L{}ukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued \L{}ukasiewicz logics.
In this paper, we present and analyze fully discrete finite difference schemes designed for solving the initial value problem associated with the fractional Korteweg-de Vries (KdV) equation involving the fractional Laplacian. We design the scheme by introducing the discrete fractional Laplacian operator which is consistent with the continuous operator, and posses certain properties which are instrumental for the convergence analysis. Assuming the initial data (u_0 \in H^{1+\alpha}(\mathbb{R})), where (\alpha \in [1,2)), our study establishes the convergence of the approximate solutions obtained by the fully discrete finite difference schemes to a classical solution of the fractional KdV equation. Theoretical results are validated through several numerical illustrations for various values of fractional exponent $\alpha$. Furthermore, we demonstrate that the Crank-Nicolson finite difference scheme preserves the inherent conserved quantities along with the improved convergence rates.
In this paper we analyze the weighted essentially non-oscillatory (WENO) schemes in the finite volume framework by examining the first step of the explicit third-order total variation diminishing Runge-Kutta method. The rationale for the improved performance of the finite volume WENO-M, WENO-Z and WENO-ZR schemes over WENO-JS in the first time step is that the nonlinear weights corresponding to large errors are adjusted to increase the accuracy of numerical solutions. Based on this analysis, we propose novel Z-type nonlinear weights of the finite volume WENO scheme for hyperbolic conservation laws. Instead of taking the difference of the smoothness indicators for the global smoothness indicator, we employ the logarithmic function with tuners to ensure that the numerical dissipation is reduced around discontinuities while the essentially non-oscillatory property is preserved. The proposed scheme does not necessitate substantial extra computational expenses. Numerical examples are presented to demonstrate the capability of the proposed WENO scheme in shock capturing.
In this paper, we consider a new nonlocal approximation to the linear Stokes system with periodic boundary conditions in two and three dimensional spaces . A relaxation term is added to the equation of nonlocal divergence free equation, which is reminiscent to the relaxation of local Stokes equation with small artificial compressibility. Our analysis shows that the well-posedness of the nonlocal system can be established under some mild assumptions on the kernel of nonlocal interactions. Furthermore, the new nonlocal system converges to the conventional, local Stokes system in second order as the horizon parameter of the nonlocal interaction goes to zero. The study provides more theoretical understanding to some numerical methods, such as smoothed particle hydrodynamics, for simulating incompressible viscous flows.
In this paper, we investigate hypothesis testing for the linear combination of mean vectors across multiple populations through the method of random integration. We have established the asymptotic distributions of the test statistics under both null and alternative hypotheses. Additionally, we provide a theoretical explanation for the special use of our test statistics in situations when the nonzero signal in the linear combination of the true mean vectors is weakly dense. Moreover, Monte-Carlo simulations are presented to evaluate the suggested test against existing high-dimensional tests. The findings from these simulations reveal that our test not only aligns with the performance of other tests in terms of size but also exhibits superior power.
In this paper, we harness a result in point process theory, specifically the expectation of the weighted $K$-function, where the weighting is done by the true first-order intensity function. This theoretical result can be employed as an estimation method to derive parameter estimates for a particular model assumed for the data. The underlying motivation is to avoid the difficulties associated with dealing with complex likelihoods in point process models and their maximization. The exploited result makes our method theoretically applicable to any model specification. In this paper, we restrict our study to Poisson models, whose likelihood represents the base for many more complex point process models. In this context, our proposed method can estimate the vector of local parameters that correspond to the points within the analyzed point pattern without introducing any additional complexity compared to the global estimation. We illustrate the method through simulation studies for both purely spatial and spatio-temporal point processes and show complex scenarios based on the Poisson model through the analysis of two real datasets concerning environmental problems.
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