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We consider the estimation of the cumulative hazard function, and equivalently the distribution function, with censored data under a setup that preserves the privacy of the survival database. This is done through a $\alpha$-locally differentially private mechanism for the failure indicators and by proposing a non-parametric kernel estimator for the cumulative hazard function that remains consistent under the privatization. Under mild conditions, we also prove lowers bounds for the minimax rates of convergence and show that estimator is minimax optimal under a well-chosen bandwidth.

In this paper, we consider the numerical approximation of a time-fractional stochastic Cahn--Hilliard equation driven by an additive fractionally integrated Gaussian noise. The model involves a Caputo fractional derivative in time of order $\alpha\in(0,1)$ and a fractional time-integral noise of order $\gamma\in[0,1]$. The numerical scheme approximates the model by a piecewise linear finite element method in space and a convolution quadrature in time (for both time-fractional operators), along with the $L^2$-projection for the noise. We carefully investigate the spatially semidiscrete and fully discrete schemes, and obtain strong convergence rates by using clever energy arguments. The temporal H\"older continuity property of the solution played a key role in the error analysis. Unlike the stochastic Allen--Cahn equation, the presence of the unbounded elliptic operator in front of the cubic nonlinearity in the underlying model adds complexity and challenges to the error analysis. To overcome these difficulties, several new techniques and error estimates are developed. The study concludes with numerical examples that validate the theoretical findings.

We propose a new class of finite element approximations to ideal compressible magnetohydrody- namic equations in smooth regime. Following variational approximations developed for fluid models in the last decade, our discretizations are built via a discrete variational principle mimicking the continuous Euler-Poincare principle, and to further exploit the geometrical structure of the prob- lem, vector fields are represented by their action as Lie derivatives on differential forms of any degree. The resulting semi-discrete approximations are shown to conserve the total mass, entropy and energy of the solutions for a wide class of finite element approximations. In addition, the divergence-free nature of the magnetic field is preserved in a pointwise sense and a time discretiza- tion is proposed, preserving those invariants and giving a reversible scheme at the fully discrete level. Numerical simulations are conducted to verify the accuracy of our approach and its ability to preserve the invariants for several test problems.

In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, our methods can be efficiently implemented via the (inverse) fast Fourier transforms, and the computational cost at each time step is ${\mathcal O}(N\log N)$ with $N$ the total number of spatial points. However, this fast algorithm fails in the variable-order cases due to the spatial dependence of the Fourier multiplier. On the other hand, the direct matrix-vector multiplication approach becomes impractical due to excessive memory requirements. To address this challenge, we proposed an accelerated matrix-free approach for the efficient computation of variable-order cases. The computational cost is ${\mathcal O}(MN\log N)$ and storage cost ${\mathcal O}(MN)$, where $M \ll N$. Moreover, our method can be easily parallelized to further enhance its efficiency. Numerical studies show that our methods are effective in solving the variable-order space fractional wave equations, especially in high-dimensional cases. Wave propagation in heterogeneous media is studied in comparison to homogeneous counterparts. We find that wave dynamics in fractional cases become more intricate due to nonlocal interactions. Specifically, dynamics in heterogeneous media are more complex than those in homogeneous media.

In this paper, we consider the hull of an algebraic geometry code, meaning the intersection of the code and its dual. We demonstrate how codes whose hulls are algebraic geometry codes may be defined using only rational places of Kummer extensions (and Hermitian function fields in particular). Our primary tool is explicitly constructing non-special divisors of degrees $g$ and $g-1$ on certain families of function fields with many rational places, accomplished by appealing to Weierstrass semigroups. We provide explicit algebraic geometry codes with hulls of specified dimensions, producing along the way linearly complementary dual algebraic geometric codes from the Hermitian function field (among others) using only rational places and an answer to an open question posed by Ballet and Le Brigand for particular function fields. These results complement earlier work by Mesnager, Tang, and Qi that use lower-genus function fields as well as instances using places of a higher degree from Hermitian function fields to construct linearly complementary dual (LCD) codes and that of Carlet, Mesnager, Tang, Qi, and Pellikaan to provide explicit algebraic geometry codes with the LCD property rather than obtaining codes via monomial equivalences.

The joint bidiagonalization (JBD) process iteratively reduces a matrix pair $\{A,L\}$ to two bidiagonal forms simultaneously, which can be used for computing a partial generalized singular value decomposition (GSVD) of $\{A,L\}$. The process has a nested inner-outer iteration structure, where the inner iteration usually can not be computed exactly. In this paper, we study the inaccurately computed inner iterations of JBD by first investigating influence of computational error of the inner iteration on the outer iteration, and then proposing a reorthogonalized JBD (rJBD) process to keep orthogonality of a part of Lanczos vectors. An error analysis of the rJBD is carried out to build up connections with Lanczos bidiagonalizations. The results are then used to investigate convergence and accuracy of the rJBD based GSVD computation. It is shown that the accuracy of computed GSVD components depend on the computing accuracy of inner iterations and condition number of $(A^T,L^T)^T$ while the convergence rate is not affected very much. For practical JBD based GSVD computations, our results can provide a guideline for choosing a proper computing accuracy of inner iterations in order to obtain approximate GSVD components with a desired accuracy. Numerical experiments are made to confirm our theoretical results.

In this paper we propose and analyze a novel multilevel version of Stein variational gradient descent (SVGD). SVGD is a recent particle based variational inference method. For Bayesian inverse problems with computationally expensive likelihood evaluations, the method can become prohibitive as it requires to evolve a discrete dynamical system over many time steps, each of which requires likelihood evaluations at all particle locations. To address this, we introduce a multilevel variant that involves running several interacting particle dynamics in parallel corresponding to different approximation levels of the likelihood. By carefully tuning the number of particles at each level, we prove that a significant reduction in computational complexity can be achieved. As an application we provide a numerical experiment for a PDE driven inverse problem, which confirms the speed up suggested by our theoretical results.

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation of the model problem. The two-way coupling mechanisms between the Herrmann formulation for linear elasticity and the reaction-diffusion equation (written in mixed form) consist of diffusion-induced active stress and stress-dependent diffusion. The two sub-problems are analysed using the extended Babu\v{s}ka--Brezzi--Braess theory for perturbed saddle-point problems. The well-posedness of the nonlinearly coupled system is established using a Banach fixed-point strategy under the smallness assumption on data. The virtual element formulations for the uncoupled sub-problems are proven uniquely solvable by a fixed-point argument in conjunction with appropriate projection operators. We derive the a priori error estimates, and test the accuracy and performance of the proposed method through computational simulations.

The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin (WG) mixed-FEM based on Banach spaces for the stationary Navier--Stokes equation in pseudostress-velocity formulation. More precisely, a modified pseudostress tensor, called $ \boldsymbol{\sigma} $, depending on the pressure, and the diffusive and convective terms has been introduced in the proposed technique, and a dual-mixed variational formulation has been derived where the aforementioned pseudostress tensor and the velocity, are the main unknowns of the system, whereas the pressure is computed via a post-processing formula. Thus, it is sufficient to provide a WG space for the tensor variable and a space of piecewise polynomial vectors of total degree at most 'k' for the velocity. Moreover, in order to define the weak discrete bilinear form, whose continuous version involves the classical divergence operator, the weak divergence operator as a well-known alternative for the classical divergence operator in a suitable discrete subspace is proposed. The well-posedness of the numerical solution is proven using a fixed-point approach and the discrete versions of the Babu\v{s}ka-Brezzi theory and the Banach-Ne\v{c}as-Babu\v{s}ka theorem. Additionally, an a priori error estimate is derived for the proposed method. Finally, several numerical results illustrating the method's good performance and confirming the theoretical rates of convergence are presented.