In this work, we propose a high-order multiscale method for an elliptic model problem with rough and possibly highly oscillatory coefficients. Convergence rates of higher order are obtained using the regularity of the right-hand side only. Hence, no restrictive assumptions on the coefficient, the domain, or the exact solution are required. In the spirit of the Localized Orthogonal Decomposition, the method constructs coarse problem-adapted ansatz spaces by solving auxiliary problems on local subdomains. More precisely, our approach is based on the strategy presented by Maier [SIAM J. Numer. Anal. 59(2), 2021]. The unique selling point of the proposed method is an improved localization strategy curing the effect of deteriorating errors with respect to the mesh size when the local subdomains are not large enough. We present a rigorous a priori error analysis and demonstrate the performance of the method in a series of numerical experiments.
相關內容
Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Gr\"{o}nwall inequality, optimal error estimates in $L^2$- and $H^1$-norms are derived for the problem with initial data $u_0 \in H_0^1(\Omega)\cap H^2(\Omega)$. Under higher regularity condition $u_0 \in \dot{H}^3(\Omega)$, a super convergence result is established and as a consequence, $L^\infty$ error estimate is obtained for 2D problems. Numerical experiments are presented to validate our theoretical findings.
Recently, the use of deep equilibrium methods has emerged as a new approach for solving imaging and other ill-posed inverse problems. While learned components may be a key factor in the good performance of these methods in practice, a theoretical justification from a regularization point of view is still lacking. In this paper, we address this issue by providing stability and convergence results for the class of equilibrium methods. In addition, we derive convergence rates and stability estimates in the symmetric Bregman distance. We strengthen our results for regularization operators with contractive residuals. Furthermore, we use the presented analysis to gain insight into the practical behavior of these methods, including a lower bound on the performance of the regularized solutions. In addition, we show that the convergence analysis leads to the design of a new type of loss function which has several advantages over previous ones. Numerical simulations are used to support our findings.
Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and accuracy of the multifidelity-based closure, we employ the recently developed "in-the-loop" training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging using the two-dimensional Navier-Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.
This work is concerned with the analysis of a space-time finite element discontinuous Galerkin method on polytopal meshes (XT-PolydG) for the numerical discretization of wave propagation in coupled poroelastic-elastic media. The mathematical model consists of the low-frequency Biot's equations in the poroelastic medium and the elastodynamics equation for the elastic one. To realize the coupling, suitable transmission conditions on the interface between the two domains are (weakly) embedded in the formulation. The proposed PolydG discretization in space is then coupled with a dG time integration scheme, resulting in a full space-time dG discretization. We present the stability analysis for both the continuous and the semidiscrete formulations, and we derive error estimates for the semidiscrete formulation in a suitable energy norm. The method is applied to a wide set of numerical test cases to verify the theoretical bounds. Examples of physical interest are also presented to investigate the capability of the proposed method in relevant geophysical scenarios.
The accurate prediction of aerodynamic drag on satellites orbiting in the upper atmosphere is critical to the operational success of modern space technologies, such as satellite-based communication or navigation systems, which have become increasingly popular in the last few years due to the deployment of constellations of satellites in low-Earth orbit. As a result, physics-based models of the ionosphere and thermosphere have emerged as a necessary tool for the prediction of atmospheric outputs under highly variable space weather conditions. This paper proposes a high-fidelity approach for physics-based space weather modeling based on the solution of the Navier-Stokes equations using a high-order discontinuous Galerkin method, combined with a matrix-free strategy suitable for high-performance computing on GPU architectures. The approach consists of a thermospheric model that describes a chemically frozen neutral atmosphere in non-hydrostatic equilibrium driven by the external excitation of the Sun. A novel set of variables is considered to treat the low densities present in the upper atmosphere and to accommodate the wide range of scales present in the problem. At the same time, and unlike most existing approaches, radial and angular directions are treated in a non-segregated approach. The study presents a set of numerical examples that demonstrate the accuracy of the approximation and validate the current approach against observational data along a satellite orbit, including estimates of established empirical and physics-based models of the ionosphere-thermosphere system. Finally, a 1D radial derivation of the physics-based model is presented and utilized for conducting a parametric study of the main thermal quantities under various solar conditions.
Energy and data-efficient online time series prediction for predicting evolving dynamical systems are critical in several fields, especially edge AI applications that need to update continuously based on streaming data. However, current DNN-based supervised online learning models require a large amount of training data and cannot quickly adapt when the underlying system changes. Moreover, these models require continuous retraining with incoming data making them highly inefficient. To solve these issues, we present a novel Continuous Learning-based Unsupervised Recurrent Spiking Neural Network Model (CLURSNN), trained with spike timing dependent plasticity (STDP). CLURSNN makes online predictions by reconstructing the underlying dynamical system using Random Delay Embedding by measuring the membrane potential of neurons in the recurrent layer of the RSNN with the highest betweenness centrality. We also use topological data analysis to propose a novel methodology using the Wasserstein Distance between the persistence homologies of the predicted and observed time series as a loss function. We show that the proposed online time series prediction methodology outperforms state-of-the-art DNN models when predicting an evolving Lorenz63 dynamical system.
We consider the linear contextual multi-class multi-period packing problem (LMMP) where the goal is to pack items such that the total vector of consumption is below a given budget vector and the total value is as large as possible. We consider the setting where the reward and the consumption vector associated with each action is a class-dependent linear function of the context, and the decision-maker receives bandit feedback. LMMP includes linear contextual bandits with knapsacks and online revenue management as special cases. We establish a new estimator which guarantees a faster convergence rate, and consequently, a lower regret in such problems. We propose a bandit policy that is a closed-form function of said estimated parameters. When the contexts are non-degenerate, the regret of the proposed policy is sublinear in the context dimension, the number of classes, and the time horizon $T$ when the budget grows at least as $\sqrt{T}$. We also resolve an open problem posed by Agrawal & Devanur (2016) and extend the result to a multi-class setting. Our numerical experiments clearly demonstrate that the performance of our policy is superior to other benchmarks in the literature.
We present a novel stabilized isogeometric formulation for the Stokes problem, where the geometry of interest is obtained via overlapping NURBS (non-uniform rational B-spline) patches, i.e., one patch on top of another in an arbitrary but predefined hierarchical order. All the visible regions constitute the computational domain, whereas independent patches are coupled through visible interfaces using Nitsche's formulation. Such a geometric representation inevitably involves trimming, which may yield trimmed elements of extremely small measures (referred to as bad elements) and thus lead to the instability issue. Motivated by the minimal stabilization method that rigorously guarantees stability for trimmed geometries [1], in this work we generalize it to the Stokes problem on overlapping patches. Central to our method is the distinct treatments for the pressure and velocity spaces: Stabilization for velocity is carried out for the flux terms on interfaces, whereas pressure is stabilized in all the bad elements. We provide a priori error estimates with a comprehensive theoretical study. Through a suite of numerical tests, we first show that optimal convergence rates are achieved, which consistently agrees with our theoretical findings. Second, we show that the accuracy of pressure is significantly improved by several orders using the proposed stabilization method, compared to the results without stabilization. Finally, we also demonstrate the flexibility and efficiency of the proposed method in capturing local features in the solution field.
In this work, we develop an efficient high order discontinuous Galerkin (DG) method for solving the Electrical Impedance Tomography (EIT). EIT is a highly nonlinear ill-posed inverse problem where the interior conductivity of an object is recovered from the surface measurements of voltage and current flux. We first propose a new optimization problem based on the recovery of the conductivity from the Dirichlet-to-Neumann map to minimize the mismatch between the predicted current and the measured current on the boundary. And we further prove the existence of the minimizer. Numerically the optimization problem is solved by a third order DG method with quadratic polynomials. Numerical results for several two-dimensional problems with both single and multiple inclusions are demonstrated to show the high {accuracy and efficiency} of the proposed high order DG method. Analysis and computation for discontinuous conductivities are also studied in this work.
Bayesian methods are commonly applied to solve image analysis problems such as noise-reduction, feature enhancement and object detection. A primary limitation of these approaches is the computational complexity due to the interdependence of neighboring pixels which limits the ability to perform full posterior sampling through Markov chain Monte Carlo (MCMC). To alleviate this problem, we develop a new posterior sampling method that is based on modeling the prior and likelihood in the space of the Fourier transform of the image. One advantage of Fourier-based methods is that many spatially correlated processes in image space can be represented via independent processes over Fourier space. A recent approach known as Bayesian Image Analysis in Fourier Space (or BIFS), has introduced parameter functions to describe prior expectations about image properties in Fourier space. To date BIFS has relied on Maximum a Posteriori (MAP) estimation for generating posterior estimates; providing just a single point estimate. The work presented here develops a posterior sampling approach for BIFS that can explore the full posterior distribution while continuing to take advantage of the independence modeling over Fourier space. As a result computational efficiency is improved over that for conventional Bayesian image analysis and mixing concerns that commonly have to be dealt with in high dimensional Markov chain Monte Carlo sampling problems are avoided. Implementation results and details are provided using simulated data.
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