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In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering a time discretization with a positive step size $h$ an error bound of size $h$ can be proved for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size $k$ an error bound of size $O(k/h)$ can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under similar assumptions to those of the time discrete case, that the error of the fully discrete case is in fact $O(h+k)$ which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behaviour $1/h$ from the bound $O(k/h)$ have not been observed.

We introduce a filtering technique for Discontinuous Galerkin approximations of hyperbolic problems. Following an approach already proposed for the Hamilton-Jacobi equations by other authors, we aim at reducing the spurious oscillations that arise in presence of discontinuities when high order spatial discretizations are employed. This goal is achieved using a filter function that keeps the high order scheme when the solution is regular and switches to a monotone low order approximation if it is not. The method has been implemented in the framework of the $deal.II$ numerical library, whose mesh adaptation capabilities are also used to reduce the region in which the low order approximation is used. A number of numerical experiments demonstrate the potential of the proposed filtering technique.

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, roughly studies the convergence order thus find that the low-order error of GFDM makes the convergence order of GFDM lower than that of FDM when node spacing is small, and points out the significant effect of the symmetry or uniformity of the node collocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied.

Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.

We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modelling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in space and time with inf-sup stable pairs of finite elements for the spatial approximation of the unknowns are investigated. Optimal order error estimates of energy-type are proven. Superconvergence at the time nodes is addressed briefly. The error analysis can be extended to discontinuous and enriched Galerkin space discretizations. The error estimates are confirmed by numerical experiments.

Multigrid is a powerful solver for large-scale linear systems arising from discretized partial differential equations. The convergence theory of multigrid methods for symmetric positive definite problems has been well developed over the past decades, while, for nonsymmetric problems, such theory is still not mature. As a foundation for multigrid analysis, two-grid convergence theory plays an important role in motivating multigrid algorithms. Regarding two-grid methods for nonsymmetric problems, most previous works focus on the spectral radius of iteration matrix or rely on convergence measures that are typically difficult to compute in practice. Moreover, the existing results are confined to two-grid methods with exact solution of the coarse-grid system. In this paper, we analyze the convergence of a two-grid method for nonsymmetric positive definite problems (e.g., linear systems arising from the discretizations of convection-diffusion equations). In the case of exact coarse solver, we establish an elegant identity for characterizing two-grid convergence factor, which is measured by a smoother-induced norm. The identity can be conveniently used to derive a class of optimal restriction operators and analyze how the convergence factor is influenced by restriction. More generally, we present some convergence estimates for an inexact variant of the two-grid method, in which both linear and nonlinear coarse solvers are considered.

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

We present a new sublinear time algorithm for approximating the spectral density (eigenvalue distribution) of an $n\times n$ normalized graph adjacency or Laplacian matrix. The algorithm recovers the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 distance in $O(n\cdot \text{poly}(1/\epsilon))$ time given sample access to the graph. This result compliments recent work by David Cohen-Steiner, Weihao Kong, Christian Sohler, and Gregory Valiant (2018), which obtains a solution with runtime independent of $n$, but exponential in $1/\epsilon$. We conjecture that the trade-off between dimension dependence and accuracy is inherent. Our method is simple and works well experimentally. It is based on a Chebyshev polynomial moment matching method that employees randomized estimators for the matrix trace. We prove that, for any Hermitian $A$, this moment matching method returns an $\epsilon$ approximation to the spectral density using just $O({1}/{\epsilon})$ matrix-vector products with $A$. By leveraging stability properties of the Chebyshev polynomial three-term recurrence, we then prove that the method is amenable to the use of coarse approximate matrix-vector products. Our sublinear time algorithm follows from combining this result with a novel sampling algorithm for approximating matrix-vector products with a normalized graph adjacency matrix. Of independent interest, we show a similar result for the widely used \emph{kernel polynomial method} (KPM), proving that this practical algorithm nearly matches the theoretical guarantees of our moment matching method. Our analysis uses tools from Jackson's seminal work on approximation with positive polynomial kernels.