Kanghun Cho, Roktaek Lim, Dongwoo Sheen
In this paper we propose an idea of constructing a macro--scale matrix system given a micro--scale matrix linear system. Then the macro--scale system is solved at cheaper computing costs. The method uses the idea of the generalized multiscale finite element method based. Some numerical results are presented.
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Heterogeneous Multiscale Method for elliptic problem without scale separation
该研究验证了异质多尺度方法可用于无尺度分离的椭圆问题，并指出局部正交方法是其特例，拓展了该多尺度方法的适用范围。
We investigate multiscale finite element methods for an elliptic distributed optimal control problem with rough coefficients. They are based on the (local) orthogonal decomposition methodology of Målqvist and Peterseim.
We introduce an algebraic multiscale method for two--dimensional problems. The method uses the generalized multiscale finite element method based on the quadrilateral nonconforming finite element spaces. Differently from the one--dimensional algebraic multiscale method, we apply the dimension reduction techniques to construct multiscale basis functions. Also moment functions are considered to impose continuity between local basis functions. Some representative numerical results are presented.
A Multiscale Finite Element Method for an Elliptic Distributed Optimal Control Problem with Rough Coefficients and Control Constraints
针对含粗糙系数和逐点控制约束的椭圆型分布最优控制问题，提出并分析一种多尺度有限元方法，证明其性能可媲美标准有限元在光滑问题上的表现，并用数值实验验证。
Multiscale discontinuous Petrov--Galerkin method for the multiscale elliptic problems
本文提出一种新的多尺度断续Petrov-Galerkin方法（MsDPGM），用于解决多尺度椭圆问题，通过结合多尺度Petrov-Galerkin方法和间断Galerkin方法，有效消除共振误差并降低计算复杂度。
Localized Orthogonal Decomposition Method with $H^1$ Interpolation for Multiscale Elliptic Problem
针对多尺度椭圆问题，提出带$H^1$插值的局部正交分解方法，无需尺度分离假设，给出先验误差估计并通过数值实验验证理论结果。
We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The method allows for suitable localization and does not rely on additional regularity assumptions on the domain, the diffusion coefficient, or the exact (weak) solution as typically required for high-order approaches. Rigorous a priori error estimates are presented with respect to the involved discretization parameters, and the interplay between these parameters as…
Edge Multiscale Methods for elliptic problems with heterogeneous coefficients
本文提出两种新的边多尺度方法，用于求解具有高对比度异质系数的偏微分方程：边谱多尺度有限元方法（ESMsFEM）和基于小波的边多尺度有限元方法（WEMsFEM）。通过大量数值实验验证了这些方法在高对比度异质系数椭圆问题中的收敛率，该收敛率与粗网格尺寸H、谱基函数数量以及小波空间的层次ℓ有关。
A reduced order Schwarz method for nonlinear multiscale elliptic equations based on two-layer neural networks
本文提出一种基于区域分解、加速Schwarz框架和两层神经网络的求解器，通过离线训练神经网络作为子域边界到边界映射的替代，高效求解多尺度完全非线性椭圆方程，在准确性与泛化性上表现良好。
Multiscale elliptic equations with scale separation are often approximated by the corresponding homogenized equations with slowly varying homogenized coefficients (the G-limit). The traditional homogenization techniques typically rely on the periodicity of the multiscale coefficients, thus finding the G-limits often requires sophisticated techniques in more general settings even when multiscale coefficient is known, if possible. Alternatively, we propose a simple approach to estimate the G-limits from (noisy-free or noisy) multiscale solution data, either from the existing forward multiscale s…
A model reduction method for multiscale elliptic PDEs with random coefficients using an optimization approach
本文提出了一种基于优化方法的多尺度椭圆PDEs模型降阶方法，通过构造局部多尺度数据驱动随机基函数，实现高效求解具有随机系数的多尺度问题，具有较高的计算效率和收敛性分析。
A Dilation-based Seamless Multiscale Method For Elliptic Problems
本文将多尺度动力系统中的无缝方法引入数值均匀化问题，通过基于局部膨胀的新方法近似高维多尺度椭圆算子，在无需完全分辨率下平衡了不可处理性与准确性，并给出了误差估计与数值验证。
Cluster-based Generalized Multiscale Finite Element Method for elliptic PDEs with random coefficients
本文提出一种基于聚类算法的广义多尺度有限元方法，用于求解具有随机系数的椭圆PDE，在多查询设置中实现高效求解，通过离线和在线阶段构造减少基函数并提升计算效率。
Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems
本文提出一种基于张量神经网络的机器学习方法，通过均匀化技术将椭圆多尺度问题转化为高维极限问题，并利用张量结构实现无需Monte Carlo的高精度高维积分，为高精度求解一般多尺度问题提供了新途径。
Domain Decomposition Preconditioners for a Discontinuous Galerkin Formulation of a Multiscale Elliptic Problem
针对系数高度变化的多尺度椭圆问题，提出基于间断伽辽金格式的非重叠和重叠区域分解方法，证明预处理代数系统的条件数在特定假设下与系数无关。
A combined finite element and multiscale finite element method for the multiscale elliptic problems
针对多尺度椭圆问题，提出一种在计算域不同区域分别使用标准有限元和超采样多尺度有限元的混合方法，并通过罚函数处理界面条件，在周期系数假设下给出收敛性分析，数值实验验证了精度和效率。
A multiscale HDG method for second order elliptic equations. Part I. Polynomial and homogenization-based multiscale spaces
本文提出一种用于高异性系数二阶椭圆方程数值上标的多尺度有限元方法，结合混合格式、域分解和混合不连续Galerkin方法，构建了三种尺度的基底空间，提供稳定的全局耦合和灵活的框架。
Localization of Elliptic Multiscale Problems
针对具有异质和高变系数的椭圆问题，构造了局部广义有限元基函数，通过顶点补片上的局部问题求解，证明了误差随补片层数指数衰减，且无需预渐近或共振效应即可保持线性收敛速度。
Localized subspace iteration methods for elliptic multiscale problems
本文提出局部子空间迭代方法，用于构建椭圆问题的广义有限元基函数，通过局部化原微分算子和对应的局部谱问题子空间迭代，有效处理多尺度系数问题。
Adaptive generalized multiscale finite element methods for H(curl)-elliptic problems with heterogeneous coefficients
本文提出了一种自适应多尺度方法，用于求解高异质介质中的H(curl)-椭圆问题，通过构建合适的快照空间和降维过程识别重要解模式，并分析后验误差指标及自适应算法，展示方法在异质性和对比度下的鲁棒收敛性。
A post-processed higher-order multiscale method for nondivergence-form elliptic equations
本文提出一种后处理的高阶多尺度方法，用于求解非散度形式的线性二阶椭圆偏微分方程，通过稳定对称公式和粗基函数构造多尺度方法，实现更高阶收敛性。
Multiscale methods with compactly supported radial basis functions for Galerkin approximation of elliptic PDEs
研究使用紧支撑径向基函数的多尺度伽辽金方法求解有界域上的椭圆PDE，分析了收敛性和条件数，并处理了Dirichlet边界条件及边界重叠支持问题。
Computational multiscale methods for nondivergence-form elliptic partial differential equations
本文提出基于局部正交分解的非散度型椭圆方程计算多尺度方法，通过细尺度局部单元问题构造算子自适应粗空间，并证明LOD方法在非散度型问题中仍保持超越尺度分离与周期性的适用性和精度。
A Bayesian numerical homogenization method for elliptic multiscale inverse problems
本文提出一种结合数值均质化与贝叶斯技术的新型方法，用于求解多尺度反问题，通过粗模型和模型降阶恢复微观高振荡张量，理论分析与数值实验验证了方法的有效性。
Mixed Multiscale Finite Volume Methods for Elliptic Problems in Two-phase Flow Simulations
针对多孔介质流动中的多尺度椭圆方程，提出混合多尺度有限体积方法框架，通过新型多尺度速度基函数结合全局信息，在粗网格上高效捕捉小尺度效应，并应用于非均质多孔介质两相流模拟，数值实验验证了精度和效率。
In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the computational domain. For example, in the simulation of steady flow transporting through highly heterogeneous porous media driven by extraction wells, the singularities lie in the near-well regions. The basic idea of the combined method is to utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and using th…
A homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such process numerically. In this paper we introduce a FEHMM scheme for multiscale elliptic problems based on nonconforming spaces. In particular we use the noconforming element with the periodic boundary condition introduced in the companion paper. Theoretical analysis derives a priori error estimates in the standard Sobolev norms. Several numerical results which …
We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on detailed analytical understanding of specific multiscale PDEs, in particular, their asymptotic limits. We present the application of the framework on two examples --- a linear kinetic equation and an elliptic equation with rough media. On these two examples, this framework achieves the asymptotic preserving property for the kinetic equations and numerical homo…
Tangent-Space Multiscale Manifold Methods for Nonlinear Elliptic Problems
针对非线性非均匀椭圆问题，提出切空间多尺度流形方法，通过非线性重构粗状态表示细尺度解，建立粗方程，还有网络插值变体，记录相关特性，有条件控制局部化缺陷并证明其衰减。
In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove the proposed multiscale method achieves nearly exponential convergence in the approximation error with respect to the computational degrees of freedom. Our strategy is to perform an energy orthogonal decomposition of the solution space into a coarse scale component comprising $a$-harmonic functions in each element of the mesh, and a fine scale component named the bubble part that can be computed lo…
An improved high-order method for elliptic multiscale problems
本文提出一种改进的高阶多尺度方法，通过改进局部化策略克服局部子域不足时的误差恶化问题，仅依赖右端项正则性即可获得高阶收敛率。
An algebraic multiscale preconditioner for large sparse SPD matrices
针对高异质系数椭圆问题，提出一种基于图划分和局部广义特征值求解的两网格代数多尺度预处理器，无需几何信息，具有并行性，数值实验表明对系数对比度和问题规模具有鲁棒性，优于标准代数多重网格。
In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, ``[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless march to larger and larger [bandwidths] continues''. This paper attempts to further the virtue of the Fast Fourier Transform (FFT) as not only bandwidth is pushed to its limits, but also the dimension of the problem. Instead of using the traditional FFT however, we make a key substitution: a high-dimensional, sparse Fourier transform (SFT) paired with rando…
A resourceful splitting technique with applications to deterministic and stochastic multiscale finite element methods
本文利用分裂技术开发新的多尺度基函数，应用于确定性和随机椭圆方程，显著降低随机问题的随机参数空间维度，并通过稀疏网格配点法减少确定性求解次数。
Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter $H$. This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximat…
We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from the data to achieve significant dimension reduction in the solution space. At the online stage, the extracted basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of low dimension structure is established by showing the high separability of the under…
An algebraic multiscale method for spatial network models
本研究提出一种基于局部正交分解（LOD）的全代数多尺度方法，可对空间网络模型进行可靠粗尺度近似，适用于复杂几何对象与椭圆偏微分方程离散，且经严格误差分析和数值算例验证。
A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities
本文提出了一种多尺度数值方法，用于求解具有局部不确定性和非线性的半线性椭圆随机偏微分方程，通过域分解方法将不同不确定性和非线性源包含在感兴趣的子域（称为补丁）中，并通过迭代算法求解一系列线性全局问题和非线性局部问题，利用自适应采样基于最小二乘法构建稀疏多项式近似来解决非线性局部问题，证明了算法的一致性、收敛性和鲁棒性，并通过数值实验展示了方法的性能。
Multiscale model reduction and two-level Schwarz preconditioner for H(curl) elliptic problems
针对异质 H(curl) 椭圆问题的旋度协调有限元线性系统，提出基于 MS-GFEM 的指数收敛模型降阶及两层限制加性 Schwarz 预条件子，并通过二维、三维实验验证其降维优势。
Optimal control for multiscale equations with rough coefficients
本文研究具有任意粗糙系数的多尺度椭圆方程凸最优控制问题，采用RPS及其推广GRPS数值均匀化技术实现粗尺度高效求解，仅需一次性细尺度预计算，显著降低迭代成本，数值实验验证了理论分析。
In this article we develop the Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for elliptic partial differential equations with inhomogeneous Dirichlet, Neumann, and Robin boundary conditions, and the high contrast property emerges from the coefficients of elliptic operators and Robin boundary conditions. By careful construction of multiscale bases of the CEM-GMsFEM, we introduce two operators $\mathcal{D}^m$ and $\mathcal{N}^m$ which are used to handle inhomogeneous Dirichlet and Neumann boundary values and are also proved to converge independently of co…
Randomized Radial Basis Function Neural Network for Solving Multiscale Elliptic Equations
本文提出随机径向基函数神经网络（RRNN），通过子域分解和随机配点耦合求解多尺度椭圆方程，仅训练输出层权重，数值实验验证其精度和效率提升。
Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast
本文提出一种新的数值上scaling方法，用于高对比率粗糙扩散系数的椭圆问题，通过局部可定位正交分解实现最优收敛性，无需预渐进行为。
Linearized Localized Orthogonal Decomposition for Quasilinear Nonmonotone Elliptic PDE
本文提出一种受 LOD 启发的线性化多尺度方法，用于含空间多尺度系数的非单调拟线性椭圆 PDE，并通过两种线性化技术的适定性、收敛分析与数值实验验证其有效性。
A localized orthogonal decomposition method for semi-linear elliptic problems
针对系数高度振荡的半线性椭圆问题，提出一种基于局部正交分解的多尺度方法，通过构造低维多尺度空间并证明H1误差关于粗网格尺寸线性收敛。
An algebraic multiscale framework for two-level overlapping Schwarz preconditioners
该研究提出一种代数多尺度框架构建两层重叠Schwarz预条件子的粗空间，其在三维标量扩散和线性弹性问题中表现良好，粗问题规模更小且鲁棒性佳。
There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled information used for approximation (in the latter). As such, both problems can be thought of as recovering a target function based on some coarse data that are either artificially chosen by an upscaling algorithm, or determined by some physical measurement process. The purpose of this paper is then to study that, under such a setup and for a specific e…
In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The co…
AlgMortar: a fully algebraic multiscale mortar preconditioner
针对二阶椭圆方程离散的线性系统求解难题，提出AlgMortar这一完全代数多尺度Mortar预条件器，利用细网格矩阵信息，经图划分等构建局部系统并耦合，实验证明其用于共轭梯度法时可扩展性好且竞争力强。
Generalized Multiscale Finite Element Methods. Nonlinear Elliptic Equations
本文利用广义多尺度有限元方法框架解决具有高对比系数的非线性椭圆方程，通过线性化方程构建连续伽辽金和不连续伽辽金粗网格空间，展示CG和DG方法的有效性。
I-MCHM: Interface Multicontinuum Homogenization for Multiscale Elliptic Problems
针对高对比度多尺度椭圆问题，提出I-MCHM方法，通过构造双边约束局部基与界面修正的粗双线性形式，实现无逐点传输律的精准升尺度，经多几何数值实验验证有效性。
Optimal solutions employing an algebraic Variational Multiscale approach Part II: Application to Navier-Stokes
本文提出了一种基于变分多尺度方法的非线性扩展框架，用于求解二维不可压缩纳维-斯托克斯方程，通过最佳投射概念实现高阶精度和守恒性质。
Adaptive Iterative Numerical Homogenization for Quasilinear Nonmonotone Elliptic PDE
针对多尺度拟线性非单调椭圆问题，提出基于Kačanov迭代与局部正交分解的自适应迭代数值均匀化方法，通过局部误差指示器自适应更新多尺度基函数，证明了收敛性并给出了先验误差估计。
Multiscale Finite Element approach for "weakly" random problems and related issues
针对随机系数为多尺度确定性问题扰动的椭圆问题，提出通过修改经典有限元基为确定性多尺度有限元基的方法，并证明其近似性质与直接基于随机问题生成的多尺度有限元基相同，数值实验显示高效性和计算加速。
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analysed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is pr…
Localized Harmonic Characteristic Basis Functions for Multiscale Finite Element Methods
本文提出一种多尺度有限元方法，通过局部谐波特征基函数高效求解高各向异性材料中的椭圆方程，数值实验验证了方法的有效性。
Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains
本文提出一种基于量化张量列车（QTT）的求解器，用于求解具有异质系数场的线性椭圆方程，能够实现极细网格的全场仿真，其自由度数量可达经典求解器的20个数量级，且在$ \LL^2 $范数下准确恢复解及其梯度。该求解器利用了QTT的指数压缩特性，从而显著提升了现有文献的性能。
Hybrid Localized Spectral Decomposition for multiscale problems
本文提出一种基于原始混合公式的有限元方法，通过空间分解和局部特征值问题处理高对比度系数的椭圆方程，获得了准最优和最优的先验误差估计。
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
针对多尺度椭圆型 PDEs 均匀化系数计算中的共振误差问题，提出基于抛物线型和椭圆型单元问题的两种新方法，实现共振误差的指数级衰减。
Multiscale modeling for problems with high contrast heterogeneous coefficients by the CEM-GMsFEM
综述约束能量最小化广义多尺度有限元法(CEM-GMsFEM)，通过谱辅助空间和过采样构造多尺度基函数，实现局部计算和指数误差衰减，用于求解高对比异质系数椭圆PDE。
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