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Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.

This paper establishes a structure-preserving numerical scheme for the Cahn--Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn--Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under $H^{-1}$-gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly reduces the computational complexity while preserves original structural properties. Numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Finally, by applying the proposed structure-preserving scheme, we numerically demonstrate that surface diffusion can be approximated by the Cahn--Hilliard equation with degenerate mobility and Flory--Huggins potential when the absolute temperature is sufficiently low, which agrees well with the theoretical result by using formal asymptotic analysis.wn theoretically by formal matched asymptotics.

Motivated by the discrete dipole approximation (DDA) for the scattering of electromagnetic waves by a dielectric obstacle that can be considered as a simple discretization of a Lippmann-Schwinger style volume integral equation for time-harmonic Maxwell equations, we analyze an analogous discretization of convolution operators with strongly singular kernels. For a class of kernel functions that includes the finite Hilbert transformation in 1D and the principal part of the Maxwell volume integral operator used for DDA in dimensions 2 and 3, we show that the method, which does not fit into known frameworks of projection methods, can nevertheless be considered as a finite section method for an infinite block Toeplitz matrix. The symbol of this matrix is given by a Fourier series that does not converge absolutely. We use Ewald's method to obtain an exponentially fast convergent series representation of this symbol and show that it is a bounded function, thereby allowing to describe the spectrum and the numerical range of the matrix. It turns out that this numerical range includes the numerical range of the integral operator, but that it is in some cases strictly larger. In these cases the discretization method does not provide a spectrally correct approximation, and while it is stable for a large range of the spectral parameter $\lambda$, there are values of $\lambda$ for which the singular integral equation is well posed, but the discretization method is unstable.

The Set Packing problem is, given a collection of sets $\mathcal{S}$ over a ground set $\mathcal{U}$, to find a maximum collection of sets that are pairwise disjoint. The problem is among the most fundamental NP-hard optimization problems that have been studied extensively in various computational regimes. The focus of this work is on parameterized complexity, Parameterized Set Packing (PSP): Given $r \in {\mathbb N}$, is there a collection $ \mathcal{S}' \subseteq \mathcal{S}: |\mathcal{S}'| = r$ such that the sets in $\mathcal{S}'$ are pairwise disjoint? Unfortunately, the problem is not fixed parameter tractable unless $\mathsf{W[1] = FPT}$, and, in fact, an "enumeration" running time of $|\mathcal{S}|^{\Omega(r)}$ is required unless the exponential time hypothesis (ETH) fails. This paper is a quest for tractable instances of Set Packing from parameterized complexity perspectives. We say that the input $(\mathcal{U},\mathcal{S})$ is "compact" if $|\mathcal{U}| = f(r)\cdot\Theta(\textsf{poly}( \log |\mathcal{S}|))$, for some $f(r) \ge r$. In the Compact Set Packing problem, we are given a compact instance of PSP. In this direction, we present a "dichotomy" result of PSP: When $|\mathcal{U}| = f(r)\cdot o(\log |\mathcal{S}|)$, PSP is in $\textsf{FPT}$, while for $|\mathcal{U}| = r\cdot\Theta(\log (|\mathcal{S}|))$, the problem is $W[1]$-hard; moreover, assuming ETH, Compact PSP does not even admit $|\mathcal{S}|^{o(r/\log r)}$ time algorithm. Although certain results in the literature imply hardness of compact versions of related problems such as Set $r$-Covering and Exact $r$-Covering, these constructions fail to extend to Compact PSP. A novel contribution of our work is the identification and construction of a gadget, which we call Compatible Intersecting Set System pair, that is crucial in obtaining the hardness result for Compact PSP.

Conditional local independence is an asymmetric independence relation among continuous time stochastic processes. It describes whether the evolution of one process is directly influenced by another process given the histories of additional processes, and it is important for the description and learning of causal relations among processes. We develop a model-free framework for testing the hypothesis that a counting process is conditionally locally independent of another process. To this end, we introduce a new functional parameter called the Local Covariance Measure (LCM), which quantifies deviations from the hypothesis. Following the principles of double machine learning, we propose an estimator of the LCM and a test of the hypothesis using nonparametric estimators and sample splitting or cross-fitting. We call this test the (cross-fitted) Local Covariance Test ((X)-LCT), and we show that its level and power can be controlled uniformly, provided that the nonparametric estimators are consistent with modest rates. We illustrate the theory by an example based on a marginalized Cox model with time-dependent covariates, and we show in simulations that when double machine learning is used in combination with cross-fitting, then the test works well without restrictive parametric assumptions.

Stochastic gradient descent (SGD) is a scalable and memory-efficient optimization algorithm for large datasets and stream data, which has drawn a great deal of attention and popularity. The applications of SGD-based estimators to statistical inference such as interval estimation have also achieved great success. However, most of the related works are based on i.i.d. observations or Markov chains. When the observations come from a mixing time series, how to conduct valid statistical inference remains unexplored. As a matter of fact, the general correlation among observations imposes a challenge on interval estimation. Most existing methods may ignore this correlation and lead to invalid confidence intervals. In this paper, we propose a mini-batch SGD estimator for statistical inference when the data is $\phi$-mixing. The confidence intervals are constructed using an associated mini-batch bootstrap SGD procedure. Using ``independent block'' trick from \cite{yu1994rates}, we show that the proposed estimator is asymptotically normal, and its limiting distribution can be effectively approximated by the bootstrap procedure. The proposed method is memory-efficient and easy to implement in practice. Simulation studies on synthetic data and an application to a real-world dataset confirm our theory.

Uncertain fractional differential equation (UFDE) is a kind of differential equation about uncertain process. As an significant mathematical tool to describe the evolution process of dynamic system, UFDE is better than the ordinary differential equation with integer derivatives because of its hereditability and memorability characteristics. However, in most instances, the precise analytical solutions of UFDE is difficult to obtain due to the complex form of the UFDE itself. Up to now, there is not plenty of researches about the numerical method of UFDE, as for the existing numerical algorithms, their accuracy is also not high. In this research, derive from the interval weighting method, a class of fractional adams method is innovatively proposed to solve UFDE. Meanwhile, such fractional adams method extends the traditional predictor-corrector method to higher order cases. The stability and truncation error limit of the improved algorithm are analyzed and deduced. As the application, several numerical simulations (including $\alpha$-path, extreme value and the first hitting time of the UFDE) are provided to manifest the higher accuracy and efficiency of the proposed numerical method.

We prove a convergence theorem for U-statistics of degree two, where the data dimension $d$ is allowed to scale with sample size $n$. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite $n$ and $d$, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with $d$ and the bandwidth.

Exponential decay estimates of a general linear weakly damped wave equation are studied with decay rate lying in a range. Based on the $C^0$-conforming finite element method to discretize spatial variables keeping temporal variable continuous, a semidiscrete system is analysed, and uniform decay estimates are derived with precisely the same decay rate as in the continuous case. Optimal error estimates with minimal smoothness assumptions on the initial data are established, which preserve exponential decay rate, and for a 2D problem, the maximum error bound is also proved. The present analysis is then generalized to include the problems with non-homogeneous forcing function, space-dependent damping, and problems with compensator. It is observed that decay rates are improved with large viscous damping and compensator. Finally, some numerical experiments are performed to validate the theoretical results established in this paper.

We present a structure-preserving scheme based on a recently-proposed mixed formulation for incompressible hyperelasticity formulated in principal stretches. Although there exist Hamiltonians introduced for quasi-incompressible elastodynamics based on different variational formulations, the one in the fully incompressible regime has yet been identified in the literature. The adopted mixed formulation naturally provides a new Hamiltonian for fully incompressible elastodynamics. Invoking the discrete gradient formula, we are able to design fully-discrete schemes that preserve the Hamiltonian and momenta. The scaled mid-point formula, another popular option for constructing algorithmic stresses, is analyzed and demonstrated to be non-robust numerically. The generalized Taylor-Hood element based on the spline technology conveniently provides a higher-order, robust, and inf-sup stable spatial discretization option for finite strain analysis. To enhance the element performance in volume conservation, the grad-div stabilization, a technique initially developed in computational fluid dynamics, is introduced here for elastodynamics. It is shown that the stabilization term does not impose additional restrictions for the algorithmic stress to respect the invariants, leading to an energy-decaying and momentum-conserving fully discrete scheme. A set of numerical examples is provided to justify the claimed properties. The grad-div stabilization is found to enhance the discrete mass conservation effectively. Furthermore, in contrast to conventional algorithms based on Cardano's formula and perturbation techniques, the spectral decomposition algorithm developed by Scherzinger and Dohrmann is robust and accurate to ensure the discrete conservation laws and is thus recommended for stretch-based material modeling.