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A numerical algorithm for regularization of the solution of the source problem for the diffusion-logistic model based on information about the process at fixed moments of time of integral type has been developed. The peculiarity of the problem under study is the discrete formulation in space and impossibility to apply classical algorithms for its numerical solution. The regularization of the problem is based on the application of A.N. Tikhonov's approach and a priori information about the source of the process. The problem was formulated in a variational formulation and solved by the global tensor optimization method. It is shown that in the case of noisy data regularization improves the accuracy of the reconstructed source.

High order schemes are known to be unstable in the presence of shock discontinuities or under-resolved solution features for nonlinear conservation laws. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi-discrete entropy inequality independently of discretization parameters. This work extends high order entropy stable schemes to the quasi-1D shallow water equations and the quasi-1D compressible Euler equations, which model one-dimensional flows through channels or nozzles with varying width. We introduce new non-symmetric entropy conservative finite volume fluxes for both sets of quasi-1D equations, as well as a generalization of the entropy conservation condition to non-symmetric fluxes. When combined with an entropy stable interface flux, the resulting schemes are high order accurate, conservative, and semi-discretely entropy stable. For the quasi-1D shallow water equations, the resulting schemes are also well-balanced.

We introduce a new type of influence function, the asymptotic expected sensitivity function, which is often equivalent to but mathematically more tractable than the traditional one based on the Gateaux derivative. To illustrate, we study the robustness of some important rank correlations, including Spearman's and Kendall's correlations, and the recently developed Chatterjee's correlation.

The Zps-additive codes of length n are subgroups of Zps^n , and can be seen as a generalization of linear codes over Z2, Z4, or more general over Z2s . In this paper, we show two methods for computing a parity-check matrix of a Zps-additive code from a generator matrix of the code in standard form. We also compare the performance of our results implemented in Magma with the current available function in Magma for codes over finite rings in general. A time complexity analysis is also shown.

We investigate a second-order accurate time-stepping scheme for solving a time-fractional diffusion equation with a Caputo derivative of order~$\alpha \in (0,1)$. The basic idea of our scheme is based on local integration followed by linear interpolation. It reduces to the standard Crank--Nicolson scheme in the classical diffusion case, that is, as $\alpha\to 1$. Using a novel approach, we show that the proposed scheme is $\alpha$-robust and second-order accurate in the $L^2(L^2)$-norm, assuming a suitable time-graded mesh. For completeness, we use the Galerkin finite element method for the spatial discretization and discuss the error analysis under reasonable regularity assumptions on the given data. Some numerical results are presented at the end.

By using the stochastic particle method, the truncated Euler-Maruyama (TEM) method is proposed for numerically solving McKean-Vlasov stochastic differential equations (MV-SDEs), possibly with both drift and diffusion coefficients having super-linear growth in the state variable. Firstly, the result of the propagation of chaos in the L^q (q\geq 2) sense is obtained under general assumptions. Then, the standard 1/2-order strong convergence rate in the L^q sense of the proposed method corresponding to the particle system is derived by utilizing the stopping time analysis technique. Furthermore, long-time dynamical properties of MV-SDEs, including the moment boundedness, stability, and the existence and uniqueness of the invariant probability measure, can be numerically realized by the TEM method. Additionally, it is proven that the numerical invariant measure converges to the underlying one of MV-SDEs in the L^2-Wasserstein metric. Finally, the conclusions obtained in this paper are verified through examples and numerical simulations.

We consider a logic with truth values in the unit interval and which uses aggregation functions instead of quantifiers, and we describe a general approach to asymptotic elimination of aggregation functions and, indirectly, of asymptotic elimination of Mostowski style generalized quantifiers, since such can be expressed by using aggregation functions. The notion of ``local continuity'' of an aggregation function, which we make precise in two (related) ways, plays a central role in this approach.

We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph $G=(V,E)$ with edge weights $w:E \rightarrow \mathbb{R}$, two terminals $s$ and $t$ in $G$, find two internally vertex-disjoint paths between $s$ and $t$ with minimum total weight. As shown recently by Schlotter and Seb\H{o} (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, there are no cycles in $G$ with negative total weight. We propose a polynomial-time algorithm that solves the Shortest Two Disjoint Paths problem for conservative weights in the case when the negative-weight edges form a constant number of trees in $G$.

We study the approximation capacity of some variation spaces corresponding to shallow ReLU$^k$ neural networks. It is shown that sufficiently smooth functions are contained in these spaces with finite variation norms. For functions with less smoothness, the approximation rates in terms of the variation norm are established. Using these results, we are able to prove the optimal approximation rates in terms of the number of neurons for shallow ReLU$^k$ neural networks. It is also shown how these results can be used to derive approximation bounds for deep neural networks and convolutional neural networks (CNNs). As applications, we study convergence rates for nonparametric regression using three ReLU neural network models: shallow neural network, over-parameterized neural network, and CNN. In particular, we show that shallow neural networks can achieve the minimax optimal rates for learning H\"older functions, which complements recent results for deep neural networks. It is also proven that over-parameterized (deep or shallow) neural networks can achieve nearly optimal rates for nonparametric regression.

The present work provides a comprehensive study of symmetric-conjugate operator splitting methods in the context of linear parabolic problems and demonstrates their additional benefits compared to symmetric splitting methods. Relevant applications include nonreversible systems and ground state computations for linear Schr\"odinger equations based on the imaginary time propagation. Numerical examples confirm the favourable error behaviour of higher-order symmetric-conjugate splitting methods and illustrate the usefulness of a time stepsize control, where the local error estimation relies on the computation of the imaginary parts and thus requires negligible costs.