We consider the equilibrium equations for a linearized Cosserat material. We identify their structure in terms of a differential complex, which is isomorphic to six copies of the de Rham complex through an algebraic isomorphism. Moreover, we show how the Cosserat materials can be analyzed by inheriting results from linearized elasticity. Both perspectives give rise to mixed finite element methods, which we refer to as strongly and weaky coupled, respectively. We prove convergence of both classes of methods, with particular attention to improved convergence rate estimates, and stability in the limit of vanishing Cosserat material parameters. The theoretical results are fully reflected in the actual performance of the methods, as shown by the numerical verifications.
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We obtain several inequalities on the generalized means of dependent p-values. In particular, the weighted harmonic mean of p-values is strictly sub-uniform under several dependence assumptions of p-values, including independence, weak negative association, the class of extremal mixture copulas, and some Clayton copulas. Sub-uniformity of the harmonic mean of p-values has an important implication in multiple hypothesis testing: It is statistically invalid to merge p-values using the harmonic mean unless a proper threshold or multiplier adjustment is used, and this invalidity applies across all significance levels. The required multiplier adjustment on the harmonic mean explodes as the number of p-values increases, and hence there does not exist a constant multiplier that works for any number of p-values, even under independence.
We study the weak convergence behaviour of the Leimkuhler--Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean--Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption. Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate $3/2$) than the standard Euler method (of weak order $1$). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests.
Implicit solvers for atmospheric models are often accelerated via the solution of a preconditioned system. For block preconditioners this typically involves the factorisation of the (approximate) Jacobian for the coupled system into a Helmholtz equation for some function of the pressure. Here we present a preconditioner for the compressible Euler equations with a flux form representation of the potential temperature on the Lorenz grid using mixed finite elements. This formulation allows for spatial discretisations that conserve both energy and potential temperature variance. By introducing the dry thermodynamic entropy as an auxiliary variable for the solution of the algebraic system, the resulting preconditioner is shown to have a similar block structure to an existing preconditioner for the material form transport of potential temperature on the Charney-Phillips grid, and to be more efficient and stable than either this or a previous Helmholtz preconditioner for the flux form transport of density weighted potential temperature on the Lorenz grid for a one dimensional thermal bubble configuration. The new preconditioner is further verified against standard two dimensional test cases in a vertical slice geometry.
In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the $L^2$-norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.
We solve the PBW-like problem of normal ordering for enveloping algebras of direct sums.
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations. This family of schemes has a simple form and can systemically achieve arbitrary high-order accuracy in theory. Moreover, these schemes can also overcome the non-physical issues that arise from many traditional numerical schemes. Due to their preservation of physical nature, these schemes can be straightforwardly used as backbones for further developing randomized and quantum algorithms in simulating Lindblad equations. In this work, we systematically study these methods and perform a detailed error analysis, which is validated through numerical examples.
Efficiently enumerating all the extreme points of a polytope identified by a system of linear inequalities is a well-known challenge issue. We consider a special case and present an algorithm that enumerates all the extreme points of a bisubmodular polyhedron in $\mathcal{O}(n^4|V|)$ time and $\mathcal{O}(n^2)$ space complexity, where $n$ is the dimension of underlying space and $V$ is the set of outputs. We use the reverse search and signed poset linked to extreme points to avoid the redundant search. Our algorithm is a generalization of enumerating all the extreme points of a base polyhedron which comprises some combinatorial enumeration problems.
We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of giving a description of the correctable erasure patterns in a maximally recoverable tensor code with the problem of describing bipartite rigid graphs or low-rank completable matrix patterns. Additionally, we relate dependencies among symmetric products of generic vectors to graph rigidity and symmetric matrix completion. With an eye toward applications to computer science, we study the dependency of these matroids on the characteristic by giving new combinatorial descriptions in several cases, including the first description of the correctable patterns in an (m, n, a=2, b=2) maximally recoverable tensor code.
We propose a high precision algorithm for solving the Gelfand-Levitan-Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson's type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations.
When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time $t^{n+\beta}$ with $\beta > 1$ being a tunable parameter. These new schemes, with a suitable $\beta$, allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes, and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.
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