We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie--Poisson system on the dual of the magnetic extension Lie algebra $\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie--Poisson systems on the dual of semidirect product Lie algebras of the form $\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$, where $\mathfrak{g}$ is a $J$-quadratic Lie algebra. Critically, the time integration method is free of computationally costly matrix exponentials. The full method preserves the underlying geometry, namely the Lie--Poisson structure and all the Casimirs, and nearly preserves the Hamiltonian function in the sense of backward error analysis. To showcase the method, we apply it to two models for magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model. For the latter, our simulations reveal the formation of large scale vortex condensates, indicating a backward energy cascade analogous to two-dimensional turbulence.
相關內容
Multivariate spatio-temporal models are widely applicable, but specifying their structure is complicated and may inhibit wider use. We introduce the R package tinyVAST from two viewpoints: the software user and the statistician. From the user viewpoint, tinyVAST adapts a widely used formula interface to specify generalized additive models, and combines this with arguments to specify spatial and spatio-temporal interactions among variables. These interactions are specified using arrow notation (from structural equation models), or an extended arrow-and-lag notation that allows simultaneous, lagged, and recursive dependencies among variables over time. The user also specifies a spatial domain for areal (gridded), continuous (point-count), or stream-network data. From the statistician viewpoint, tinyVAST constructs sparse precision matrices representing multivariate spatio-temporal variation, and parameters are estimated by specifying a generalized linear mixed model (GLMM). This expressive interface encompasses vector autoregressive, empirical orthogonal functions, spatial factor analysis, and ARIMA models. To demonstrate, we fit to data from two survey platforms sampling corals, sponges, rockfishes, and flatfishes in the Gulf of Alaska and Aleutian Islands. We then compare eight alternative model structures using different assumptions about habitat drivers and survey detectability. Model selection suggests that towed-camera and bottom trawl gears have spatial variation in detectability but sample the same underlying density of flatfishes and rockfishes, and that rockfishes are positively associated with sponges while flatfishes are negatively associated with corals. We conclude that tinyVAST can be used to test complicated dependencies representing alternative structural assumptions for research and real-world policy evaluation.
Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations.
We consider the problem of efficiently simulating stochastic models of chemical kinetics. The Gillespie Stochastic Simulation algorithm (SSA) is often used to simulate these models, however, in many scenarios of interest, the computational cost quickly becomes prohibitive. This is further exasperated in the Bayesian inference context when estimating parameters of chemical models, as the intractability of the likelihood requires multiple simulations of the underlying system. To deal with issues of computational complexity in this paper, we propose a novel hybrid $\tau$-leap algorithm for simulating well-mixed chemical systems. In particular, the algorithm uses $\tau$-leap when appropriate (high population densities), and SSA when necessary (low population densities, when discrete effects become non-negligible). In the intermediate regime, a combination of the two methods, which leverages the properties of the underlying Poisson formulation, is employed. As illustrated through a number of numerical experiments the hybrid $\tau$ offers significant computational savings when compared to SSA without however sacrificing the overall accuracy. This feature is particularly welcomed in the Bayesian inference context, as it allows for parameter estimation of stochastic chemical kinetics at reduced computational cost.
We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $\sigma\in(0,2)$ since they involve fractional Laplace operators $(-\Delta)^{\sigma/2}$. They arise e.g.~in control and game theory as dynamic programming equations, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $\sigma$. The accuracy of previous approximations depend on $\sigma$ and are worse when $\sigma$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We present several numerical examples.
This paper studies the convergence of a spatial semidiscretization of a three-dimensional stochastic Allen-Cahn equation with multiplicative noise. For non-smooth initial values, the regularity of the mild solution is investigated, and an error estimate is derived with the spatial $ L^2 $-norm. For smooth initial values, two error estimates with the general spatial $ L^q $-norms are established.
We present a novel discontinuous Galerkin finite element method for numerical simulations of the rotating thermal shallow water equations in complex geometries using curvilinear meshes, with arbitrary accuracy. We derive an entropy functional which is convex, and which must be preserved in order to preserve model stability at the discrete level. The numerical method is provably entropy stable and conserves mass, buoyancy, vorticity, and energy. This is achieved by using novel entropy stable numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the use of chain rule at the discrete level. Numerical simulations on a cubed sphere mesh are presented to verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilisation.
This work focuses on the numerical approximations of random periodic solutions of stochastic differential equations (SDEs). Under non-globally Lipschitz conditions, we prove the existence and uniqueness of random periodic solutions for the considered equations and its numerical approximations generated by the stochastic theta (ST) methods with theta within (1/2,1]. It is shown that the random periodic solution of each ST method converges strongly in the mean square sense to that of SDEs for all step size. More precisely, the mean square convergence order is 1/2 for SDEs with multiplicative noise and 1 for SDEs with additive noise. Numerical results are finally reported to confirm these theoretical findings.
We develop a novel and efficient discontinuous Galerkin spectral element method (DG-SEM) for the spherical rotating shallow water equations in vector invariant form. We prove that the DG-SEM is energy stable, and discretely conserves mass, vorticity, and linear geostrophic balance on general curvlinear meshes. These theoretical results are possible due to our novel entropy stable numerical DG fluxes for the shallow water equations in vector invariant form. We experimentally verify these results on a cubed sphere mesh. Additionally, we show that our method is robust, that is can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence without the need for artificial stabilisation.
Quantitative analysis of pseudo-diffusion in diffusion-weighted magnetic resonance imaging (DWI) data shows potential for assessing fetal lung maturation and generating valuable imaging biomarkers. Yet, the clinical utility of DWI data is hindered by unavoidable fetal motion during acquisition. We present IVIM-morph, a self-supervised deep neural network model for motion-corrected quantitative analysis of DWI data using the Intra-voxel Incoherent Motion (IVIM) model. IVIM-morph combines two sub-networks, a registration sub-network, and an IVIM model fitting sub-network, enabling simultaneous estimation of IVIM model parameters and motion. To promote physically plausible image registration, we introduce a biophysically informed loss function that effectively balances registration and model-fitting quality. We validated the efficacy of IVIM-morph by establishing a correlation between the predicted IVIM model parameters of the lung and gestational age (GA) using fetal DWI data of 39 subjects. IVIM-morph exhibited a notably improved correlation with gestational age (GA) when performing in-vivo quantitative analysis of fetal lung DWI data during the canalicular phase. IVIM-morph shows potential in developing valuable biomarkers for non-invasive assessment of fetal lung maturity with DWI data. Moreover, its adaptability opens the door to potential applications in other clinical contexts where motion compensation is essential for quantitative DWI analysis. The IVIM-morph code is readily available at: //github.com/TechnionComputationalMRILab/qDWI-Morph.
Clinical neuroimaging data is naturally hierarchical. Different magnetic resonance imaging (MRI) sequences within a series, different slices covering the head, and different regions within each slice all confer different information. In this work we present a hierarchical attention network for abnormality detection using MRI scans obtained in a clinical hospital setting. The proposed network is suitable for non-volumetric data (i.e. stacks of high-resolution MRI slices), and can be trained from binary examination-level labels. We show that this hierarchical approach leads to improved classification, while providing interpretability through either coarse inter- and intra-slice abnormality localisation, or giving importance scores for different slices and sequences, making our model suitable for use as an automated triaging system in radiology departments.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191