In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence analysis of collective smoothers and collective two-level methods. The multigrid algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size is proportional to the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. The multigrid method is tested both as a stationary method and as a preconditioner for GMRES on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used as an inner solver within a semismooth Newton iteration; a risk-averse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.
相關內容
One of the hallmark achievements of the theory of graphical models and Bayesian model selection is the celebrated greedy equivalence search (GES) algorithm due to Chickering and Meek. GES is known to consistently estimate the structure of directed acyclic graph (DAG) models in various special cases including Gaussian and discrete models, which are in particular curved exponential families. A general theory that covers general nonparametric DAG models, however, is missing. Here, we establish the consistency of greedy equivalence search for general families of DAG models that satisfy smoothness conditions on the Markov factorization, and hence may not be curved exponential families, or even parametric. The proof leverages recent advances in nonparametric Bayes to construct a test for comparing misspecified DAG models that avoids arguments based on the Laplace approximation. Nonetheless, when the Laplace approximation is valid and a consistent scoring function exists, we recover the classical result. As a result, we obtain a general consistency theorem for GES applied to general DAG models.
In this paper, we consider an inverse space-dependent source problem for a time-fractional diffusion equation. To deal with the ill-posedness of the problem, we transform the problem into an optimal control problem with total variational (TV) regularization. In contrast to the classical Tikhonov model incorporating $L^2$ penalty terms, the inclusion of a TV term proves advantageous in reconstructing solutions that exhibit discontinuities or piecewise constancy. The control problem is approximated by a fully discrete scheme, and convergence results are provided within this framework. Furthermore, a lineraed primal-dual iterative algorithm is proposed to solve the discrete control model based on an equivalent saddle-point reformulation, and several numerical experiments are presented to demonstrate the efficiency of the algorithm.
The present work concerns the derivation of a numerical scheme to approximate weak solutions of the Euler equations with a gravitational source term. The designed scheme is proved to be fully well-balanced since it is able to exactly preserve all moving equilibrium solutions, as well as the corresponding steady solutions at rest obtained when the velocity vanishes. Moreover, the proposed scheme is entropy-preserving since it satisfies all fully discrete entropy inequalities. In addition, in order to satisfy the required admissibility of the approximate solutions, the positivity of both approximate density and pressure is established. Several numerical experiments attest the relevance of the developed numerical method.
The state vector-based simulation offers a convenient approach to developing and validating quantum algorithms with noise-free results. However, limited by the absence of cache-aware implementations and unpolished circuit optimizations, the past simulators were severely constrained in performance, leading to stagnation in quantum computing. In this paper, we present an innovative quantum circuit simulation toolkit comprising gate optimization and simulation modules to address these performance challenges. For the performance, scalability, and comprehensive evaluation, we conduct a series of particular circuit benchmarks and strong scaling tests on a DGX-A100 workstation and achieve averaging 9 times speedup compared to state-of-the-art simulators, including QuEST, IBM-Aer, and NVIDIA-cuQuantum. Moreover, the critical performance metric FLOPS increases by up to a factor of 8-fold, and arithmetic intensity experiences a remarkable 96x enhancement. We believe the proposed toolkit paves the way for faster quantum circuit simulations, thereby facilitating the development of novel quantum algorithms.
We consider weak convergence of one-step schemes for solving stochastic differential equations (SDEs) with one-sided Lipschitz conditions. It is known that the super-linear coefficients may lead to a blowup of moments of solutions and their numerical solutions. When solutions to SDEs have all finite moments, weak convergence of numerical schemes has been investigated in [Wang et al (2023), Weak error analysis for strong approximation schemes of SDEs with super-linear coefficients, IMA Journal numerical analysis]. Some modified Euler schemes have been analyzed for weak convergence. In this work, we present a family of explicit schemes of first and second-order weak convergence based on classical schemes for SDEs. We explore the effects of limited moments on these schemes. We provide a systematic but simple way to establish weak convergence orders for schemes based on approximations/modifications of drift and diffusion coefficients. We present several numerical examples of these schemes and show their weak convergence orders.
Complex conjugate matrix equations (CCME) have aroused the interest of many researchers because of computations and antilinear systems. Existing research is dominated by its time-invariant solving methods, but lacks proposed theories for solving its time-variant version. Moreover, artificial neural networks are rarely studied for solving CCME. In this paper, starting with the earliest CCME, zeroing neural dynamics (ZND) is applied to solve its time-variant version. Firstly, the vectorization and Kronecker product in the complex field are defined uniformly. Secondly, Con-CZND1 model and Con-CZND2 model are proposed and theoretically prove convergence and effectiveness. Thirdly, three numerical experiments are designed to illustrate the effectiveness of the two models, compare their differences, highlight the significance of neural dynamics in the complex field, and refine the theory related to ZND.
Making inference with spatial extremal dependence models can be computationally burdensome since they involve intractable and/or censored likelihoods. Building on recent advances in likelihood-free inference with neural Bayes estimators, that is, neural networks that approximate Bayes estimators, we develop highly efficient estimators for censored peaks-over-threshold models that {use data augmentation techniques} to encode censoring information in the neural network {input}. Our new method provides a paradigm shift that challenges traditional censored likelihood-based inference methods for spatial extremal dependence models. Our simulation studies highlight significant gains in both computational and statistical efficiency, relative to competing likelihood-based approaches, when applying our novel estimators to make inference with popular extremal dependence models, such as max-stable, $r$-Pareto, and random scale mixture process models. We also illustrate that it is possible to train a single neural Bayes estimator for a general censoring level, precluding the need to retrain the network when the censoring level is changed. We illustrate the efficacy of our estimators by making fast inference on hundreds-of-thousands of high-dimensional spatial extremal dependence models to assess extreme particulate matter 2.5 microns or less in diameter (${\rm PM}_{2.5}$) concentration over the whole of Saudi Arabia.
PEPit is a Python package aiming at simplifying the access to worst-case analyses of a large family of first-order optimization methods possibly involving gradient, projection, proximal, or linear optimization oracles, along with their approximate, or Bregman variants. In short, PEPit is a package enabling computer-assisted worst-case analyses of first-order optimization methods. The key underlying idea is to cast the problem of performing a worst-case analysis, often referred to as a performance estimation problem (PEP), as a semidefinite program (SDP) which can be solved numerically. To do that, the package users are only required to write first-order methods nearly as they would have implemented them. The package then takes care of the SDP modeling parts, and the worst-case analysis is performed numerically via a standard solver.
We propose a new parallel-in-time algorithm for solving optimal control problems constrained bypartial differential equations. Our approach, which is based on a deeper understanding of ParaExp,considers an overlapping time-domain decomposition in which we combine the solution of homogeneous problems using exponential propagation with the local solutions of inhomogeneous problems.The algorithm yields a linear system whose matrix-vector product can be fully performed in parallel.We then propose a preconditioner to speed up the convergence of GMRES in the special cases ofthe heat and wave equations. Numerical experiments are provided to illustrate the efficiency of ourpreconditioners.
The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191