Coupled systems of free flow and porous media arise in a variety of technical and environmental applications. For laminar flow regimes, such systems are described by the Stokes equations in the free-flow region and Darcy's law in the porous medium. An appropriate set of coupling conditions is needed on the fluid-porous interface. Discretisations of the Stokes-Darcy problems yield large, sparse, ill-conditioned, and, depending on the interface conditions, non-symmetric linear systems. Therefore, robust and efficient preconditioners are needed to accelerate convergence of the applied Krylov method. In this work, we consider the second order MAC scheme for the coupled Stokes-Darcy problems and develop and investigate block diagonal, block triangular and constraint preconditioners. We apply two classical sets of coupling conditions considering the Beavers-Joseph and the Beavers-Joseph-Saffman condition for the tangential velocity. For the Beavers-Joseph interface condition, the resulting system is non-symmetric, therefore GMRES method is used for both cases. Spectral analysis is conducted for the exact versions of the preconditioners identifying clusters and bounds. Furthermore, for practical use we develop efficient inexact versions of the preconditioners. We demonstrate effectiveness and robustness of the proposed preconditioners in numerical experiments.
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One of main obstacles in verifying the energy dissipation laws of implicit-explicit Runge-Kutta (IERK) methods for phase field equations is to establish the uniform boundedness of stage solutions without the global Lipschitz continuity assumption of nonlinear bulk. With the help of discrete orthogonal convolution kernels, an updated time-space splitting technique is developed to establish the uniform boundedness of stage solutions for a refined class of IERK methods in which the associated differentiation matrices and the average dissipation rates are always independent of the time-space discretization meshes. This makes the refined IERK methods highly advantageous in self-adaptive time-stepping procedures as some larger adaptive step-sizes in actual simulations become possible. From the perspective of optimizing the average dissipation rate, we construct some parameterized refined IERK methods up to third-order accuracy, in which the involved diagonally implicit Runge-Kutta methods for the implicit part have an explicit first stage and allow a stage-order of two such that they are not necessarily algebraically stable. Then we are able to establish, for the first time, the original energy dissipation law and the unconditional $L^2$ norm convergence. Extensive numerical tests are presented to support our theory.
Data scarcity and data imbalance have attracted a lot of attention in many fields. Data augmentation, explored as an effective approach to tackle them, can improve the robustness and efficiency of classification models by generating new samples. This paper presents REPRINT, a simple and effective hidden-space data augmentation method for imbalanced data classification. Given hidden-space representations of samples in each class, REPRINT extrapolates, in a randomized fashion, augmented examples for target class by using subspaces spanned by principal components to summarize distribution structure of both source and target class. Consequently, the examples generated would diversify the target while maintaining the original geometry of target distribution. Besides, this method involves a label refinement component which allows to synthesize new soft labels for augmented examples. Compared with different NLP data augmentation approaches under a range of data imbalanced scenarios on four text classification benchmark, REPRINT shows prominent improvements. Moreover, through comprehensive ablation studies, we show that label refinement is better than label-preserving for augmented examples, and that our method suggests stable and consistent improvements in terms of suitable choices of principal components. Moreover, REPRINT is appealing for its easy-to-use since it contains only one hyperparameter determining the dimension of subspace and requires low computational resource.
We study the identification of binary choice models with fixed effects. We provide a condition called sign saturation and show that this condition is sufficient for the identification of the model. In particular, we can guarantee identification even when all the regressors are bounded, including multiple discrete regressors. We also show that without this condition, the model is not identified unless the error distribution belongs to a special class. The same sign saturation condition is also essential for identifying the sign of treatment effects. A test is provided to check the sign saturation condition and can be implemented using existing algorithms for the maximum score estimator.
This paper leverages various philosophical and ontological frameworks to explore the concept of embodied artificial general intelligence (AGI), its relationship to human consciousness, and the key role of the metaverse in facilitating this relationship. Several theoretical frameworks underpin this exploration, such as embodied cognition, Michael Levin's computational boundary of a "Self," Donald D. Hoffman's Interface Theory of Perception, and Bernardo Kastrup's analytical idealism, which lead to considering our perceived outer reality as a symbolic representation of alternate inner states of being, and where AGI could embody a different form of consciousness with a larger computational boundary. The paper further discusses the developmental stages of AGI, the requirements for the emergence of an embodied AGI, the importance of a calibrated symbolic interface for AGI, and the key role played by the metaverse, decentralized systems, open-source blockchain technology, as well as open-source AI research. It also explores the idea of a feedback loop between AGI and human users in metaverse spaces as a tool for AGI calibration, as well as the role of local homeostasis and decentralized governance as preconditions for achieving a stable embodied AGI. The paper concludes by emphasizing the importance of achieving a certain degree of harmony in human relations and recognizing the interconnectedness of humanity at a global level, as key prerequisites for the emergence of a stable embodied AGI.
We introduce generator matching, a modality-agnostic framework for generative modeling using arbitrary Markov processes. Generators characterize the infinitesimal evolution of a Markov process, which we leverage for generative modeling in a similar vein to flow matching: we construct conditional generators which generate single data points, then learn to approximate the marginal generator which generates the full data distribution. We show that generator matching unifies various generative modeling methods, including diffusion models, flow matching and discrete diffusion models. Furthermore, it provides the foundation to expand the design space to new and unexplored Markov processes such as jump processes. Finally, generator matching enables the construction of superpositions of Markov generative processes and enables the construction of multimodal models in a rigorous manner. We empirically validate our method on protein and image structure generation, showing that superposition with a jump process improves image generation.
This paper considers estimating the parameters in a regime-switching stochastic differential equation(SDE) driven by Normal Inverse Gaussian(NIG) noise. The model under consideration incorporates a continuous-time finite state Markov chain to capture regime changes, enabling a more realistic representation of evolving market conditions or environmental factors. Although the continuous dynamics are typically observable, the hidden nature of the Markov chain introduces significant complexity, rendering standard likelihood-based methods less effective. To address these challenges, we propose an estimation algorithm designed for discrete, high-frequency observations, even when the Markov chain is not directly observed. Our approach integrates the Expectation-Maximization (EM) algorithm, which iteratively refines parameter estimates in the presence of latent variables, with a quasi-likelihood method adapted to NIG noise. Notably, this method can simultaneously estimate parameters within both the SDE coefficients and the driving noise. Simulation results are provided to evaluate the performance of the algorithm. These experiments demonstrate that the proposed method provides reasonable estimation under challenging conditions.
The randomized singular value decomposition (SVD) has become a popular approach to computing cheap, yet accurate, low-rank approximations to matrices due to its efficiency and strong theoretical guarantees. Recent work by Boull\'e and Townsend (FoCM, 2023) presents an infinite-dimensional analog of the randomized SVD to approximate Hilbert-Schmidt operators. However, many applications involve computing low-rank approximations to symmetric positive semi-definite matrices. In this setting, it is well-established that the randomized Nystr\"om approximation is usually preferred over the randomized SVD. This paper explores an infinite-dimensional analog of the Nystr\"om approximation to compute low-rank approximations to non-negative self-adjoint trace-class operators. We present an analysis of the method and, along the way, improve the existing infinite-dimensional bounds for the randomized SVD. Our analysis yields bounds on the expected value and tail bounds for the Nystr\"om approximation error in the operator, trace, and Hilbert-Schmidt norms. Numerical experiments on integral operators arising from Gaussian process sampling and Bayesian inverse problems are used to validate the proposed infinite-dimensional Nystr\"om algorithm.
This paper introduces fast R updating algorithms designed for statistical applications, including regression, filtering, and model selection, where data structures change frequently. Although traditional QR decomposition is essential for matrix operations, it becomes computationally intensive when dynamically updating the design matrix in statistical models. The proposed algorithms efficiently update the R matrix without recalculating Q, significantly reducing computational costs. These algorithms provide a scalable solution for high-dimensional regression models, enhancing the feasibility of large-scale statistical analyses and model selection in data-intensive fields. Comprehensive simulation studies and real-world data applications reveal that the methods significantly reduce computational time while preserving accuracy. An extensive discussion highlights the versatility of fast R updating algorithms, illustrating their benefits across a wide range of models and applications in statistics and machine learning.
We consider the problem of estimating the error when solving a system of differential algebraic equations. Richardson extrapolation is a classical technique that can be used to judge when computational errors are irrelevant and estimate the discretization error. We have simulated molecular dynamics with constraints using the GROMACS library and found that the output is not always amenable to Richardson extrapolation. We derive and illustrate Richardson extrapolation using a variety of numerical experiments. We identify two necessary conditions that are not always satisfied by the GROMACS library.
Mass lumping techniques are commonly employed in explicit time integration schemes for problems in structural dynamics and both avoid solving costly linear systems with the consistent mass matrix and increase the critical time step. In isogeometric analysis, the critical time step is constrained by so-called "outlier" frequencies, representing the inaccurate high frequency part of the spectrum. Removing or dampening these high frequencies is paramount for fast explicit solution techniques. In this work, we propose mass lumping and outlier removal techniques for nontrivial geometries, including multipatch and trimmed geometries. Our lumping strategies provably do not deteriorate (and often improve) the CFL condition of the original problem and are combined with deflation techniques to remove persistent outlier frequencies. Numerical experiments reveal the advantages of the method, especially for simulations covering large time spans where they may halve the number of iterations with little or no effect on the numerical solution.
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