查看更多>>摘要：Anderson acceleration(AA)is an extrapolation technique designed to speed up fixed-point iterations.For optimization problems,we propose a novel algorithm by combining the AA with the energy adaptive gradient method(AEGD)[arXiv:2010.05109].The feasibility of our algorithm is ensured in light of the convergence theory for AEGD,though it is not a fixed-point iteration.We provide rigorous convergence rates of AA for gradient descent(GD)by an acceleration factor of the gain at each implementation of AA-GD.Our experi-mental results show that the proposed AA-AEGD algorithm requires little tuning of hyper-parameters and exhibits superior fast convergence.
查看更多>>摘要：Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock cap-turing numerical schemes.Because they do not involve any special data structure,and do not induce savings in memory requirements,they are easily implemented on existing codes and are recommended for 1D and 2D simulations when intensive testing is required.The multilevel technique can also be applied to balance laws,but in this case,numerical errors may be induced by the technique.We present a series of numerical tests that point out that the use of monotonicity-preserving interpolatory techniques eliminates the numerical errors observed when using the usual 4-point centered Lagrange interpolation,and leads to a more robust multilevel code for balance laws,while maintaining the efficiency rates observed for hyperbolic conservation laws.
查看更多>>摘要：The reconstruction of images from their corresponding noisy Radon transform is a typical example of an ill-posed linear inverse problem as arising in the application of computer-ized tomography(CT).As the(naïve)solution does not depend on the measured data con-tinuously,regularization is needed to reestablish a continuous dependence.In this work,we investigate simple,but yet still provably convergent approaches to learning linear regu-larization methods from data.More specifically,we analyze two approaches:one generic linear regularization that learns how to manipulate the singular values of the linear opera-tor in an extension of our previous work,and one tailored approach in the Fourier domain that is specific to CT-reconstruction.We prove that such approaches become convergent regularization methods as well as the fact that the reconstructions they provide are typi-cally much smoother than the training data they were trained on.Finally,we compare the spectral as well as the Fourier-based approaches for CT-reconstruction numerically,discuss their advantages and disadvantages and investigate the effect of discretization errors at dif-ferent resolutions.
查看更多>>摘要：In this paper,we design an efficient,multi-stage image segmentation framework that incor-porates a weighted difference of anisotropic and isotropic total variation(AITV).The seg-mentation framework generally consists of two stages:smoothing and thresholding,thus referred to as smoothing-and-thresholding(SaT).In the first stage,a smoothed image is obtained by an AITV-regularized Mumford-Shah(MS)model,which can be solved effi-ciently by the alternating direction method of multipliers(ADMMs)with a closed-form solution of a proximal operator of the l1-al2 regularizer.The convergence of the ADMM algorithm is analyzed.In the second stage,we threshold the smoothed image by K-means clustering to obtain the final segmentation result.Numerical experiments demonstrate that the proposed segmentation framework is versatile for both grayscale and color images,effi-cient in producing high-quality segmentation results within a few seconds,and robust to input images that are corrupted with noise,blur,or both.We compare the AITV method with its original convex TV and nonconvex TVp(0＜p＜1)counterparts,showcasing the qualitative and quantitative advantages of our proposed method.
查看更多>>摘要：In this work,we propose a second-order model for image denoising by employing a novel potential function recently developed in Zhu(J Sci Comput 88:46,2021)for the design of a regularization term.Due to this new second-order derivative based regularizer,the model is able to alleviate the staircase effect and preserve image contrast.The augmented Lagran-gian method(ALM)is utilized to minimize the associated functional and convergence analysis is established for the proposed algorithm.Numerical experiments are presented to demonstrate the features of the proposed model.
查看更多>>摘要：Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimensions.In this paper,we provide analytical solutions to certain optimal con-trol problems whose running cost depends on the state variable and with constraints on the control.We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with state-dependent Hamiltonians.Addition-ally,we present an efficient,grid-free numerical solver based on our representation formulas,which is shown to scale linearly with the state dimension,and thus,to overcome the curse of dimensionality.Using existing optimization methods and the min-plus technique,we extend our numerical solvers to address more general classes of convex and nonconvex initial costs.We demonstrate the capabilities of our numerical solvers using implementations on a central processing unit(CPU)and a field-programmable gate array(FPGA).In several cases,our FPGA implementation obtains over a 10 times speedup compared to the CPU,which demon-strates the promising performance boosts FPGAs can achieve.Our numerical results show that our solvers have the potential to serve as a building block for solving broader classes of high-dimensional optimal control problems in real-time.
查看更多>>摘要：We consider the inverse problem of finding guiding pattern shapes that result in desired self-assembly morphologies of block copolymer melts.Specifically,we model polymer self-assembly using the self-consistent field theory and derive,in a non-parametric setting,the sensitivity of the dissimilarity between the desired and the actual morphologies to arbitrary perturbations in the guiding pattern shape.The sensitivity is then used for the optimization of the confining pattern shapes such that the dissimilarity between the desired and the actual morphologies is minimized.The efficiency and robustness of the proposed gradient-based algorithm are demonstrated in a number of examples related to templating vertical intercon-nect accesses(VIA).
查看更多>>摘要：Machine learning has been widely used for solving partial differential equations(PDEs)in recent years,among which the random feature method(RFM)exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency.Potentially,the optimization problem in the RFM is more difficult to solve than those that arise in traditional methods.Unlike the broader machine-learning research,which frequently targets tasks within the low-precision regime,our study focuses on the high-precision regime crucial for solving PDEs.In this work,we study this problem from the following aspects:(i)we analyze the coefficient matrix that arises in the RFM by studying the distribution of singular values;(ii)we investigate whether the continuous training causes the overfitting issue;(iii)we test direct and iterative methods as well as randomized methods for solving the optimization problem.Based on these results,we find that direct methods are superior to other methods if memory is not an issue,while iterative methods typically have low accuracy and can be improved by preconditioning to some extent.
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