This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection-diffusion-reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of non-symmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection-)diffusion-reaction problems.
相關內容
Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure's displacement have been extensively studied for incompressible fluid-structure interaction (FSI) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions to the FSI problems in the standard $L^2$ norm. In this article, we propose a new kinematically coupled scheme for incompressible FSI thin-structure model and establish a new framework for the numerical analysis of FSI problems in terms of a newly introduced coupled non-stationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the $L^2$ norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.
We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.
We introduce two hybridizable discontinuous Galerkin (HDG) methods for numerically solving the Monge-Ampere equation. The first HDG method is devised to solve the nonlinear elliptic Monge-Ampere equation by using Newton's method. The second HDG method is devised to solve a sequence of the Poisson equation until convergence to a fixed-point solution of the Monge-Ampere equation is reached. Numerical examples are presented to demonstrate the convergence and accuracy of the HDG methods. Furthermore, the HDG methods are applied to r-adaptive mesh generation by redistributing a given scalar density function via the optimal transport theory. This r-adaptivity methodology leads to the Monge-Ampere equation with a nonlinear Neumann boundary condition arising from the optimal transport of the density function to conform the resulting high-order mesh to the boundary. Hence, we extend the HDG methods to treat the nonlinear Neumann boundary condition. Numerical experiments are presented to illustrate the generation of r-adaptive high-order meshes on planar and curved domains.
We present and analyse a hybridized discontinuous Galerkin method for incompressible flow problems using non-affine cells, proving that it preserves a key invariance property that illudes most methods, namely that any irrotational component of the prescribed force is exactly balanced by the pressure gradient and does not influence the velocity field. This invariance property can be preserved in the discrete problem if the incompressibility constraint is satisfied in a sufficiently strong sense. We derive sufficient conditions to guarantee discretely divergence-free functions are exactly divergence-free, and give examples of divergence-free finite elements on meshes containing triangular, quadrilateral, tetrahedral, or hexahedral cells generated by a (possibly non-affine) map from their respective reference cells. In the case of quadrilateral cells, we prove an optimal error estimate for the velocity field that does not depend on the pressure approximation. Our theoretical analysis is supported by numerical results.
We show that the two-stage minimum description length (MDL) criterion widely used to estimate linear change-point (CP) models corresponds to the marginal likelihood of a Bayesian model with a specific class of prior distributions. This allows results from the frequentist and Bayesian paradigms to be bridged together. Thanks to this link, one can rely on the consistency of the number and locations of the estimated CPs and the computational efficiency of frequentist methods, and obtain a probability of observing a CP at a given time, compute model posterior probabilities, and select or combine CP methods via Bayesian posteriors. Furthermore, we adapt several CP methods to take advantage of the MDL probabilistic representation. Based on simulated data, we show that the adapted CP methods can improve structural break detection compared to state-of-the-art approaches. Finally, we empirically illustrate the usefulness of combining CP detection methods when dealing with long time series and forecasting.
Species-sampling problems (SSPs) refer to a vast class of statistical problems calling for the estimation of (discrete) functionals of the unknown species composition of an unobservable population. A common feature of SSPs is their invariance with respect to species labelling, which is at the core of the Bayesian nonparametric (BNP) approach to SSPs under the popular Pitman-Yor process (PYP) prior. In this paper, we develop a BNP approach to SSPs that are not "invariant" to species labelling, in the sense that an ordering or ranking is assigned to species' labels. Inspired by the population genetics literature on age-ordered alleles' compositions, we study the following SSP with ordering: given an observable sample from an unknown population of individuals belonging to species (alleles), with species' labels being ordered according to weights (ages), estimate the frequencies of the first $r$ order species' labels in an enlarged sample obtained by including additional unobservable samples. By relying on an ordered PYP prior, we obtain an explicit posterior distribution of the first $r$ order frequencies, with estimates being of easy implementation and computationally efficient. We apply our approach to the analysis of genetic variation, showing its effectiveness in estimating the frequency of the oldest allele, and then we discuss other potential applications.
The solution of probabilistic inverse problems for which the corresponding forward problem is constrained by physical principles is challenging. This is especially true if the dimension of the inferred vector is large and the prior information about it is in the form of a collection of samples. In this work, a novel deep learning based approach is developed and applied to solving these types of problems. The approach utilizes samples of the inferred vector drawn from the prior distribution and a physics-based forward model to generate training data for a conditional Wasserstein generative adversarial network (cWGAN). The cWGAN learns the probability distribution for the inferred vector conditioned on the measurement and produces samples from this distribution. The cWGAN developed in this work differs from earlier versions in that its critic is required to be 1-Lipschitz with respect to both the inferred and the measurement vectors and not just the former. This leads to a loss term with the full (and not partial) gradient penalty. It is shown that this rather simple change leads to a stronger notion of convergence for the conditional density learned by the cWGAN and a more robust and accurate sampling strategy. Through numerical examples it is shown that this change also translates to better accuracy when solving inverse problems. The numerical examples considered include illustrative problems where the true distribution and/or statistics are known, and a more complex inverse problem motivated by applications in biomechanics.
In this paper we present a new perspective on error analysis of Legendre approximations for differentiable functions. We start by introducing a sequence of Legendre-Gauss-Lobatto polynomials and prove their theoretical properties, such as an explicit and optimal upper bound. We then apply these properties to derive a new and explicit bound for the Legendre coefficients of differentiable functions and establish some explicit and optimal error bounds for Legendre projections in the $L^2$ and $L^{\infty}$ norms. Illustrative examples are provided to demonstrate the sharpness of our new results.
Robust Markov decision processes (RMDPs) provide a promising framework for computing reliable policies in the face of model errors. Many successful reinforcement learning algorithms build on variations of policy-gradient methods, but adapting these methods to RMDPs has been challenging. As a result, the applicability of RMDPs to large, practical domains remains limited. This paper proposes a new Double-Loop Robust Policy Gradient (DRPG), the first generic policy gradient method for RMDPs. In contrast with prior robust policy gradient algorithms, DRPG monotonically reduces approximation errors to guarantee convergence to a globally optimal policy in tabular RMDPs. We introduce a novel parametric transition kernel and solve the inner loop robust policy via a gradient-based method. Finally, our numerical results demonstrate the utility of our new algorithm and confirm its global convergence properties.
A parametric class of trust-region algorithms for unconstrained nonconvex optimization is considered where the value of the objective function is never computed. The class contains a deterministic version of the first-order Adagrad method typically used for minimization of noisy function, but also allows the use of (possibly approximate) second-order information when available. The rate of convergence of methods in the class is analyzed and is shown to be identical to that known for first-order optimization methods using both function and gradients values, recovering existing results for purely-first order variants and improving the explicit dependence on problem dimension. This rate is shown to be essentially sharp. A new class of methods is also presented, for which a slightly worse and essentially sharp complexity result holds. Limited numerical experiments show that the new methods' performance may be comparable to that of standard steepest descent, despite using significantly less information, and that this performance is relatively insensitive to noise.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191