A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and No
bile in 2018 (SIAM J. Numer. Anal., 56, 3121--3143) to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available online.
Classic mechanism design often assumes that a bidders action is restricted to report a type or a signal, possibly untruthfully. In todays digital economy, bidders are holding increasing amount of private information about the auctioned items. And due
to legal or ethical concerns, they would demand to reveal partial but truthful information, as opposed to report untrue signal or misinformation. To accommodate such bidder behaviors in auction design, we propose and study a novel mechanism design setup where each bidder holds two kinds of information: (1) private emph{value type}, which can be misreported; (2) private emph{information variable}, which the bidder may want to conceal or partially reveal, but importantly, emph{not} to misreport. We show that in this new setup, it is still possible to design mechanisms that are both emph{Incentive and Information Compatible} (IIC). We develop two different black-box transformations, which convert any mechanism $mathcal{M}$ for classic bidders to a mechanism $mathcal{M}$ for strategically reticent bidders, based on either outcome of expectation or expectation of outcome, respectively. We identify properties of the original mechanism $mathcal{M}$ under which the transformation leads to IIC mechanisms $mathcal{M}$. Interestingly, as corollaries of these results, we show that running VCG with expected bidder values maximizes welfare whereas the mechanism using expected outcome of Myersons auction maximizes revenue. Finally, we study how regulation on the auctioneers usage of information may lead to more robust mechanisms.
In this paper, we consider a generalized forward-backward splitting (G-FBS) operator for solving the monotone inclusions, and analyze its nonexpansive properties in a context of arbitrary variable metric. Then, for the associated fixed-point iteratio
ns (i.e. the G-FBS algorithms), the global ergodic and pointwise convergence rates of metric distance are obtained from the nonexpansiveness. The convergence in terms of objective function value is also investigated, when the G-FBS operator is applied to a minimization problem. A main contribution of this paper is to show that the G-FBS operator provides a simplifying and unifying framework to model and analyze a great variety of operator splitting algorithms, where the convergence behaviours can be easily described by the fixed-point construction of this simple operator.
Modeling the microstructure evolution of a material embedded in a device often involves integral boundary conditions. Here we propose a modified Nitsches method to solve the Poisson equation with an integral boundary condition, which is coupled to ph
ase-field equations of the microstructure evolution of a strongly correlated material undergoing metal-insulator transitions. Our numerical experiments demonstrate that the proposed method achieves optimal convergence rate while the rate of convergence of the conventional Lagrange multiplier method is not optimal. Furthermore, the linear system derived from the modified Nitsches method can be solved by an iterative solver with algebraic multigrid preconditioning. The modified Nitsches method can be applied to other physical boundary conditions mathematically similar to this electric integral boundary condition.
In this paper, we study the nonexpansive properties of metric resolvent, and present a convergence rate analysis for the associated fixed-point iterations (Banach-Picard and Krasnoselskii-Mann types). Equipped with a variable metric, we develop the g
lobal ergodic and non-ergodic iteration-complexity bounds in terms of both solution distance and objective value. A byproduct of our expositions also extends the proximity operator and Moreaus decomposition identity to arbitrary variable metric. It is further shown that many classes of the first-order operator splitting algorithms, including alternating direction methods of multipliers, primal-dual hybrid gradient and Bregman iterations, can be expressed by the fixed-point iterations of a simple metric resolvent, and thus, the convergence can be analyzed within this unified framework.
In this note, we study the nonexpansive properties based on arbitrary variable metric and explore the connections between firm nonexpansiveness, cocoerciveness and averagedness. A convergence rate analysis for the associated fixed-point iterations is
presented by developing the global ergodic and non-ergodic iteration-complexity bounds in terms of metric distances. The obtained results are finally exemplified with the metric resolvent, which provides a unified framework for many existing first-order operator splitting algorithms.
This paper is a further investigation of the generalized $N$-urn Ehrenfest model introduced in cite{Xue2020}. A moderate deviation principle from the hydrodynamic limit of the model is derived. The proof of this main result follows a routine procedur
e introduced in cite{Kipnis1989}, where a replacement lemma plays the key role. To prove the replacement lemma, the large deviation principle of the model given in cite{Xue2020} is utilized.
In this Letter, we present for the first time a calculation of the complete next-to-leading order corrections to the $gg to ZH$ process. We use the method of small mass expansion to tackle the most challenging two-loop virtual amplitude, in which the
top quark mass dependence is retained throughout the calculations. We show that our method provides reliable numeric results in all kinematic regions, and present phenomenological predictions for the total and differential cross sections at the Large Hadron Collider and its future upgrades. Our results are necessary ingredients towards reducing the theoretical uncertainties of the $pp to ZH$ cross sections down to the percent-level, and provide important theoretical inputs for future precision experimental collider programs.
Recently, a so-called E-MS algorithm was developed for model selection in the presence of missing data. Specifically, it performs the Expectation step (E step) and Model Selection step (MS step) alternately to find the minimum point of the observed g
eneralized information criteria (GIC). In practice, it could be numerically infeasible to perform the MS-step for high dimensional settings. In this paper, we propose a more simple and feasible generalized EMS (GEMS) algorithm which simply requires a decrease in the observed GIC in the MS-step and includes the original EMS algorithm as a special case. We obtain several numerical convergence results of the GEMS algorithm under mild conditions. We apply the proposed GEMS algorithm to Gaussian graphical model selection and variable selection in generalized linear models and compare it with existing competitors via numerical experiments. We illustrate its application with three real data sets.
Topological band theory has achieved great success in the high-throughput search for topological band structures both in paramagnetic and magnetic crystal materials. However, a significant proportion of materials are topologically trivial insulators
at the Fermi level. In this paper, we show that, remarkably, for a subset of the topologically trivial insulators, knowing only their electron number and the Wyckoff positions of the atoms we can separate them into two groups: the obstructed atomic insulator (OAI) and the atomic insulator (AI). The interesting group, the OAI, have a center of charge not localized on the atoms. Using the theory of topological quantum chemistry, in this work we first derive the necessary and sufficient conditions for a topologically trivial insulator to be a filling enforced obstructed atomic insulator (feOAI) in the 1651 Shubnikov space groups. Remarkably, the filling enforced criteria enable the identification of obstructed atomic bands without knowing the representations of the band structures. Hence, no ab-initio calculations are needed for the filling enforced criteria, although they are needed to obtain the band gaps. With the help of the Topological Quantum Chemistry website, we have performed a high-throughput search for feOAIs and have found that 957 ICSD entries (638 unique materials) are paramagnetic feOAIs, among which 738 (475) materials have an indirect gap. The metallic obstructed surface states of feOAIs are also showcased by several material examples.
