Communications on Applied Mathematics and Computation

Preface

Andrea Bertozzi, Ron Fedkiw, Frederic Gibou, Chiu-Yen Kao, Chi-Wang Shu, Richard Tsai, Wotao Yin, Hong-Kai Zhao

2024, 6(2): 755-756. doi:10.1007/s42967-024-00387-w

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A Dynamical System-Based Framework for Dimension Reduction

Ryeongkyung Yoon, Braxton Osting

2024, 6(2): 757-789. doi:10.1007/s42967-022-00234-w

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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 combination of dictionary elements, where each element is a pre-specified linear/nonlinear candidate 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.

Global Solutions to Nonconvex Problems by Evolution of Hamilton-Jacobi PDEs

Howard Heaton, Samy Wu Fung, Stanley Osher

2024, 6(2): 790-810. doi:10.1007/s42967-022-00239-5

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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 function. 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 Hamilton-Jacobi equation. Our numerical examples illustrate global convergence.

Iterative Subregion Correction Preconditioners with Adaptive Tolerance for Problems with Geometrically Localized Stiffness

Michael Franco, Per-Olof Persson, Will Pazner

2024, 6(2): 811-836. doi:10.1007/s42967-023-00254-0

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We present a class of preconditioners for the linear systems resulting from a finite element 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, anisotropy, 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 preconditioners 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 guaranteeing global convergence rates. We present analysis showing that the convergence of these preconditioners, even when combined with an adaptively selected tolerance, is independent of discretization parameters (e.g., the mesh size and diffusion coefficient) in the subregion. We demonstrate significant performance improvements over black-box preconditioners 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 2.25×10⁶ fluid flow over the NACA 0012 airfoil, as well as massively separated flow at 30° angle of attack.

Deep Energies for Estimating Three-Dimensional Facial Pose and Expression

Jane Wu, Michael Bao, Xinwei Yao, Ronald Fedkiw

2024, 6(2): 837-861. doi:10.1007/s42967-023-00256-y

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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 matching 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 optimization techniques lead to 3D facial pose and expression estimation.

Exponentially Convergent Multiscale Finite Element Method

Yifan Chen, Thomas Y. Hou, Yixuan Wang

2024, 6(2): 862-878. doi:10.1007/s42967-023-00260-2

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We provide a concise review of the exponentially convergent multiscale finite element method (ExpMsFEM) for efficient model reduction of PDEs in heterogeneous media without 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 stiffness matrix can be used repeatedly in multi-query scenarios.

SRMD: Sparse Random Mode Decomposition

Nicholas Richardson, Hayden Schaeffer, Giang Tran

2024, 6(2): 879-906. doi:10.1007/s42967-023-00273-x

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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 representation, outlier removal, and mode decomposition. On benchmark tests, we show that our approach outperforms other state-of-the-art decomposition methods.

Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-conservative Products

Dinshaw S. Balsara, Deepak Bhoriya, Chi-Wang Shu, Harish Kumar

2024, 6(2): 907-962. doi:10.1007/s42967-023-00275-9

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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 Riemann 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 reconstruction 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.

An Arbitrarily High Order and Asymptotic Preserving Kinetic Scheme in Compressible Fluid Dynamic

Rémi Abgrall, Fatemeh Nassajian Mojarrad

2024, 6(2): 963-991. doi:10.1007/s42967-023-00274-w

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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-dimensional 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.

A Stable FE-FD Method for Anisotropic Parabolic PDEs with Moving Interfaces

Baiying Dong, Zhilin Li, Juan Ruiz-álvarez

2024, 6(2): 992-1012. doi:10.1007/s42967-023-00281-x

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In this paper, a new finite element and finite difference (FE-FD) method has been developed 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 interface, 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 second order convergence.

Batch Active Learning for Multispectral and Hyperspectral Image Segmentation Using Similarity Graphs

Bohan Chen, Kevin Miller, Andrea L. Bertozzi, Jon Schwenk

2024, 6(2): 1013-1033. doi:10.1007/s42967-023-00284-8

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Graph learning, when used as a semi-supervised learning (SSL) method, performs well for classification tasks with a low label rate. We provide a graph-based batch active learning pipeline for pixel/patch neighborhood multi- or hyperspectral image segmentation. Our batch active learning approach selects a collection of unlabeled pixels that satisfy a graph local maximum constraint for the active learning acquisition function that determines the relative importance of each pixel to the classification. This work builds on recent advances in the design of novel active learning acquisition functions (e.g., the Model Change approach in arXiv:2110.07739) while adding important further developments including patch-neighborhood image analysis and batch active learning methods to further increase the accuracy and greatly increase the computational efficiency of these methods. In addition to improvements in the accuracy, our approach can greatly reduce the number of labeled pixels needed to achieve the same level of the accuracy based on randomly selected labeled pixels.

An Improved Coupled Level Set and Continuous Moment-of-Fluid Method for Simulating Multiphase Flows with Phase Change

Zhouteng Ye, Cody Estebe, Yang Liu, Mehdi Vahab, Zeyu Huang, Mark Sussman, Alireza Moradikazerouni, Kourosh Shoele, Yongsheng Lian, Mitsuhiro Ohta, M. Yousuff Hussaini

2024, 6(2): 1034-1069. doi:10.1007/s42967-023-00286-6

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An improved algorithm for computing multiphase flows is presented in which the multimaterial Moment-of-Fluid (MOF) algorithm for multiphase flows, initially described by Li et al. (2015), is enhanced addressing existing MOF difficulties in computing solutions to problems in which surface tension forces are crucial for understanding salient flow mechanisms. The Continuous MOF (CMOF) method is motivated in this article. The CMOF reconstruction method inherently removes the “checkerboard instability” that persists when using the MOF method on surface tension driven multiphase (multimaterial) flows. The CMOF reconstruction algorithm is accelerated by coupling the CMOF method to the level set method and coupling the CMOF method to a decision tree machine learning (ML) algorithm. Multiphase flow examples are shown in the two-dimensional (2D), three-dimensional (3D) axisymmetric “RZ”, and 3D coordinate systems. Examples include two material and three material multiphase flows: bubble formation, the impingement of a liquid jet on a gas bubble in a cryogenic fuel tank, freezing, and liquid lens dynamics.

Piecewise Acoustic Source Imaging with Unknown Speed of Sound Using a Level-Set Method

Guanghui Huang, Jianliang Qian, Yang Yang

2024, 6(2): 1070-1095. doi:10.1007/s42967-023-00291-9

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We investigate the following inverse problem: starting from the acoustic wave equation, reconstruct a piecewise constant passive acoustic source from a single boundary temporal measurement without knowing the speed of sound. When the amplitudes of the source are known a priori, we prove a unique determination result of the shape and propose a level set algorithm to reconstruct the singularities. When the singularities of the source are known a priori, we show unique determination of the source amplitudes and propose a least-squares fitting algorithm to recover the source amplitudes. The analysis bridges the low-frequency source inversion problem and the inverse problem of gravimetry. The proposed algorithms are validated and quantitatively evaluated with numerical experiments in 2D and 3D.

Meta-Auto-Decoder: a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations

Zhanhong Ye, Xiang Huang, Hongsheng Liu, Bin Dong

2024, 6(2): 1096-1130. doi:10.1007/s42967-023-00293-7

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Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc. Typical reduced order modeling techniques accelerate the solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the offline stage. These methods often need a predefined mesh as well as a series of precomputed solution snapshots, and may struggle to balance between the efficiency and accuracy due to the limitation of the linear ansatz. Utilizing the nonlinear representation of neural networks (NNs), we propose the Meta-Auto-Decoder (MAD) to construct a nonlinear trial manifold, whose best possible performance is measured theoretically by the decoder width. Based on the meta-learning concept, the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage. Fast adaptation to new (possibly heterogeneous) PDE parameters is enabled by searching on this trial manifold, and optionally fine-tuning the trial manifold at the same time. Extensive numerical experiments show that the MAD method exhibits a faster convergence speed without losing the accuracy than other deep learning-based methods.

Nearest Neighbor Sampling of Point Sets Using Rays

Liangchen Liu, Louis Ly, Colin B. Macdonald, Richard Tsai

2024, 6(2): 1131-1174. doi:10.1007/s42967-023-00318-1

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We propose a new framework for the sampling, compression, and analysis of distributions of point sets and other geometric objects embedded in Euclidean spaces. Our approach involves constructing a tensor called the RaySense sketch, which captures nearest neighbors from the underlying geometry of points along a set of rays. We explore various operations that can be performed on the RaySense sketch, leading to different properties and potential applications. Statistical information about the data set can be extracted from the sketch, independent of the ray set. Line integrals on point sets can be efficiently computed using the sketch. We also present several examples illustrating applications of the proposed strategy in practical scenarios.

Convergence of Hyperbolic Neural Networks Under Riemannian Stochastic Gradient Descent

Wes Whiting, Bao Wang, Jack Xin

2024, 6(2): 1175-1188. doi:10.1007/s42967-023-00302-9

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We prove, under mild conditions, the convergence of a Riemannian gradient descent method for a hyperbolic neural network regression model, both in batch gradient descent and stochastic gradient descent. We also discuss a Riemannian version of the Adam algorithm. We show numerical simulations of these algorithms on various benchmarks.

A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem on Implicit Surfaces

Young Kyu Lee, Shingyu Leung

2024, 6(2): 1189-1216. doi:10.1007/s42967-023-00303-8

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We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces. The approach follows an embedding approach for solving the surface eikonal equation. We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood. Our proposed algorithm is easy to implement and efficient. We will give some two- and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.

Optimization in Machine Learning: a Distribution-Space Approach

Yongqiang Cai, Qianxiao Li, Zuowei Shen

2024, 6(2): 1217-1240. doi:10.1007/s42967-023-00322-5

摘要 ( 288 )

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We present the viewpoint that optimization problems encountered in machine learning can often be interpreted as minimizing a convex functional over a function space, but with a non-convex constraint set introduced by model parameterization. This observation allows us to repose such problems via a suitable relaxation as convex optimization problems in the space of distributions over the training parameters. We derive some simple relationships between the distribution-space problem and the original problem, e.g., a distribution-space solution is at least as good as a solution in the original space. Moreover, we develop a numerical algorithm based on mixture distributions to perform approximate optimization directly in the distribution space. Consistency of this approximation is established and the numerical efficacy of the proposed algorithm is illustrated in simple examples. In both theory and practice, this formulation provides an alternative approach to large-scale optimization in machine learning.

Adaptive State-Dependent Diffusion for Derivative-Free Optimization

Bj?rn Engquist, Kui Ren, Yunan Yang

2024, 6(2): 1241-1269. doi:10.1007/s42967-023-00324-3

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This paper develops and analyzes a stochastic derivative-free optimization strategy. A key feature is the state-dependent adaptive variance. We prove global convergence in probability with algebraic rate and give the quantitative results in numerical examples. A striking fact is that convergence is achieved without explicit information of the gradient and even without comparing different objective function values as in established methods such as the simplex method and simulated annealing. It can otherwise be compared to annealing with state-dependent temperature.

Model Change Active Learning in Graph-Based Semi-supervised Learning

Kevin S. Miller, Andrea L. Bertozzi

2024, 6(2): 1270-1298. doi:10.1007/s42967-023-00328-z

摘要 ( 279 )

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Active learning in semi-supervised classification involves introducing additional labels for unlabelled data to improve the accuracy of the underlying classifier. A challenge is to identify which points to label to best improve performance while limiting the number of new labels. “Model Change” active learning quantifies the resulting change incurred in the classifier by introducing the additional label(s). We pair this idea with graph-based semi-supervised learning (SSL) methods, that use the spectrum of the graph Laplacian matrix, which can be truncated to avoid prohibitively large computational and storage costs. We consider a family of convex loss functions for which the acquisition function can be efficiently approximated using the Laplace approximation of the posterior distribution. We show a variety of multiclass examples that illustrate improved performance over prior state-of-art.

Anderson Acceleration of Gradient Methods with Energy for Optimization Problems

Hailiang Liu, Jia-Hao He, Xuping Tian

2024, 6(2): 1299-1318. doi:10.1007/s42967-023-00327-0

摘要 ( 289 )

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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 experimental results show that the proposed AA-AEGD algorithm requires little tuning of hyperparameters and exhibits superior fast convergence.

On the Use of Monotonicity-Preserving Interpolatory Techniques in Multilevel Schemes for Balance Laws

Antonio Baeza, Rosa Donat, Anna Martínez-Gavara

2024, 6(2): 1319-1341. doi:10.1007/s42967-023-00332-3

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Cost-effective multilevel techniques for homogeneous hyperbolic conservation laws are very successful in reducing the computational cost associated to high resolution shock capturing 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.

Convergent Data-Driven Regularizations for CT Reconstruction

Samira Kabri, Alexander Auras, Danilo Riccio, Hartmut Bauermeister, Martin Benning, Michael Moeller, Martin Burger

2024, 6(2): 1342-1368. doi:10.1007/s42967-023-00333-2

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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 computerized tomography (CT). As the (naïve) solution does not depend on the measured data continuously, regularization is needed to reestablish a continuous dependence. In this work, we investigate simple, but yet still provably convergent approaches to learning linear regularization methods from data. More specifically, we analyze two approaches: one generic linear regularization that learns how to manipulate the singular values of the linear operator 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 typically 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 different resolutions.

An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation

Kevin Bui, Yifei Lou, Fredrick Park, Jack Xin

2024, 6(2): 1369-1405. doi:10.1007/s42967-023-00339-w

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In this paper, we design an efficient, multi-stage image segmentation framework that incorporates a weighted difference of anisotropic and isotropic total variation (AITV). The segmentation 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 efficiently by the alternating direction method of multipliers (ADMMs) with a closed-form solution of a proximal operator of the $\ell_1-\alpha \ell_2$ 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, efficient 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 TV^p(0<p<1) counterparts, showcasing the qualitative and quantitative advantages of our proposed method.

A Second-Order Image Denoising Model for Contrast Preservation

Wei Zhu

2024, 6(2): 1406-1427. doi:10.1007/s42967-023-00344-z

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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 Lagrangian 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.

Lax-Oleinik-Type Formulas and Efficient Algorithms for Certain High-Dimensional Optimal Control Problems

Paula Chen, Jér?me Darbon, Tingwei Meng

2024, 6(2): 1428-1471. doi:10.1007/s42967-024-00371-4

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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 control 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. Additionally, 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 demonstrates 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.

A Non-parametric Gradient-Based Shape Optimization Approach for Solving Inverse Problems in Directed Self-Assembly of Block Copolymers

Daniil Bochkov, Frederic Gibou

2024, 6(2): 1472-1489. doi:10.1007/s42967-024-00394-x

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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 interconnect accesses (VIA).

Optimization of Random Feature Method in the High-Precision Regime

Jingrun Chen, Weinan E, Yifei Sun

2024, 6(2): 1490-1517. doi:10.1007/s42967-024-00389-8

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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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