查看更多>>摘要：We propose a novel framework for learning a low-dimensional representation of data based on nonlinear dynamical systems,which we call the dynamical dimension reduction(DDR).In the DDR model,each point is evolved via a nonlinear flow towards a lower-dimensional subspace;the projection onto the subspace gives the low-dimensional embedding.Training the model involves identifying the nonlinear flow and the subspace.Following the equation discovery method,we represent the vector field that defines the flow using a linear combi-nation of dictionary elements,where each element is a pre-specified linear/nonlinear candi-date function.A regularization term for the average total kinetic energy is also introduced and motivated by the optimal transport theory.We prove that the resulting optimization problem is well-posed and establish several properties of the DDR method.We also show how the DDR method can be trained using a gradient-based optimization method,where the gradients are computed using the adjoint method from the optimal control theory.The DDR method is implemented and compared on synthetic and example data sets to other dimension reduction methods,including the PCA,t-SNE,and Umap.
查看更多>>摘要：Computing tasks may often be posed as optimization problems.The objective functions for real-world scenarios are often nonconvex and/or nondifferentiable.State-of-the-art methods for solving these problems typically only guarantee convergence to local minima.This work presents Hamilton-Jacobi-based Moreau adaptive descent(HJ-MAD),a zero-order algorithm with guaranteed convergence to global minima,assuming continuity of the objective func-tion.The core idea is to compute gradients of the Moreau envelope of the objective(which is"piece-wise convex")with adaptive smoothing parameters.Gradients of the Moreau envelope(i.e.,proximal operators)are approximated via the Hopf-Lax formula for the viscous Hamil-ton-Jacobi equation.Our numerical examples illustrate global convergence.
查看更多>>摘要：We present a class of preconditioners for the linear systems resulting from a finite ele-ment or discontinuous Galerkin discretizations of advection-dominated problems.These preconditioners are designed to treat the case of geometrically localized stiffness,where the convergence rates of iterative methods are degraded in a localized subregion of the mesh.Slower convergence may be caused by a number of factors,including the mesh size,ani-sotropy,highly variable coefficients,and more challenging physics.The approach taken in this work is to correct well-known preconditioners such as the block Jacobi and the block incomplete LU(ILU)with an adaptive inner subregion iteration.The goal of these precon-ditioners is to reduce the number of costly global iterations by accelerating the convergence in the stiff region by iterating on the less expensive reduced problem.The tolerance for the inner iteration is adaptively chosen to minimize subregion-local work while guarantee-ing global convergence rates.We present analysis showing that the convergence of these preconditioners,even when combined with an adaptively selected tolerance,is independ-ent of discretization parameters(e.g.,the mesh size and diffusion coefficient)in the sub-region.We demonstrate significant performance improvements over black-box precondi-tioners when applied to several model convection-diffusion problems.Finally,we present performance results of several variations of iterative subregion correction preconditioners applied to the Reynolds number 2.25 x 106 fluid flow over the NACA 0012 airfoil,as well as massively separated flow at 30° angle of attack.
查看更多>>摘要：While much progress has been made in capturing high-quality facial performances using motion capture markers and shape-from-shading,high-end systems typically also rely on rotoscope curves hand-drawn on the image.These curves are subjective and difficult to draw consistently;moreover,ad-hoc procedural methods are required for generating match-ing rotoscope curves on synthetic renders embedded in the optimization used to determine three-dimensional(3D)facial pose and expression.We propose an alternative approach whereby these curves and other keypoints are detected automatically on both the image and the synthetic renders using trained neural networks,eliminating artist subjectivity,and the ad-hoc procedures meant to mimic it.More generally,we propose using machine learning networks to implicitly define deep energies which when minimized using classical optimi-zation techniques lead to 3D facial pose and expression estimation.
查看更多>>摘要：We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media with-out scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiff-ness matrix can be used repeatedly in multi-query scenarios.
查看更多>>摘要：Signal decomposition and multiscale signal analysis provide many useful tools for time-frequency analysis.We proposed a random feature method for analyzing time-series data by constructing a sparse approximation to the spectrogram.The randomization is both in the time window locations and the frequency sampling,which lowers the overall sampling and computational cost.The sparsification of the spectrogram leads to a sharp separation between time-frequency clusters which makes it easier to identify intrinsic modes,and thus leads to a new data-driven mode decomposition.The applications include signal represen-tation,outlier removal,and mode decomposition.On benchmark tests,we show that our approach outperforms other state-of-the-art decomposition methods.
查看更多>>摘要：Higher order finite difference weighted essentially non-oscillatory(WENO)schemes have been constructed for conservation laws.For multidimensional problems,they offer a high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of the comparable accuracy.This makes them quite attractive for several science and engineering applications.But,to the best of our knowledge,such schemes have not been extended to non-linear hyperbolic systems with non-conservative products.In this paper,we perform such an extension which improves the domain of the applicability of such schemes.The extension is carried out by writing the scheme in fluctuation form.We use the HLLI Rie-mann solver of Dumbser and Balsara(J.Comput.Phys.304:275-319,2016)as a building block for carrying out this extension.Because of the use of an HLL building block,the resulting scheme has a proper supersonic limit.The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme,thus expanding its domain of the applicability.Our new finite difference WENO formulation uses the same WENO recon-struction that was used in classical versions,making it very easy for users to transition over to the present formulation.For conservation laws,the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO,with two major advantages:(ⅰ)It can capture jumps in stationary linearly degenerate wave families exactly.(ⅱ)It only requires the reconstruction to be applied once.Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of the accuracy for smooth multidimensional flows.Stringent Riemann problems and several novel multidimensional problems that are drawn from compressible Baer-Nunziato multiphase flow,multiphase debris flow and two-layer shallow water equations are also shown to document the robustness of the method.For some test problems that require well-balancing we have even been able to apply the scheme without any modification and obtain good results.Many useful PDEs may have stiff relaxation source terms for which the finite difference formulation of WENO is shown to provide some genuine advantages.
查看更多>>摘要：We present a class of arbitrarily high order fully explicit kinetic numerical methods in compressible fluid dynamics,both in time and space,which include the relaxation schemes by Jin and Xin.These methods can use the CFL number larger or equal to unity on regular Cartesian meshes for the multi-dimensional case.These kinetic models depend on a small parameter that can be seen as a"Knudsen"number.The method is asymptotic preserving in this Knudsen number.Also,the computational costs of the method are of the same order of a fully explicit scheme.This work is the extension of Abgrall et al.(2022)[3]to multi-dimensional systems.We have assessed our method on several problems for two-dimen-sional scalar problems and Euler equations and the scheme has proven to be robust and to achieve the theoretically predicted high order of accuracy on smooth solutions.
查看更多>>摘要：In this paper,a new finite element and finite difference(FE-FD)method has been devel-oped for anisotropic parabolic interface problems with a known moving interface using Cartesian meshes.In the spatial discretization,the standard P1 FE discretization is applied so that the part of the coefficient matrix is symmetric positive definite,while near the inter-face,the maximum principle preserving immersed interface discretization is applied.In the time discretization,a modified Crank-Nicolson discretization is employed so that the hybrid FE-FD is stable and second order accurate.Correction terms are needed when the interface crosses grid lines.The moving interface is represented by the zero level set of a Lipschitz continuous function.Numerical experiments presented in this paper confirm sec-ond order convergence.
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