检索结果-内蒙古大学图书馆

The effect of smooth parametrizations on nonconvex optimization landscapes

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Levin, Eitan Kileel, Joe Boumal, Nicolas Dept. of Computing and Mathematical Sciences California Institute of Technology United States Dept. of Mathematics Oden Institute for Computational Engineering and Sciences University of Texas at Austin United States Switzerland

We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good guarantees) or for theoretical purposes (e.g., to reveal that the landscape satisfies a strict saddle property). In both cases, the central question is: how do the landscapes of the two problems relate? More precisely: how do desirable points such as local minima and critical points in one problem relate to those in the other problem? A key finding in this paper is that these relations are often determined by the parametrization itself, and are almost entirely independent of the cost function. Accordingly, we introduce a general framework to study parametrizations by their effect on landscapes. The framework enables us to obtain new guarantees for an array of problems, some of which were previously treated on a case-by-case basis in the literature. Applications include: optimizing low-rank matrices and tensors through factorizations;solving semidefinite programs via the Burer-Monteiro approach;training neural networks by optimizing their weights and biases;and quotienting out *** Codes 53Z50, 49J53, 90C26, 90C46 Copyright © 2022, The Authors. All rights reserved.

关键词： Optimization

A Partial Exact Penalty Function Approach for Constrained Optimization

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Xiao, Nachuan Liu, Xin Toh, Kim-Chuan The Institute of Operations Research and Analytics National University of Singapore Singapore State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China Department of Mathematics Institute of Operations Research and Analytics National University of Singapore Singapore119076 Singapore

In this paper, we focus on a class of constrained nonlinear optimization problems (NLP), where some of its equality constraints define a closed embedded submanifold M in Rn. Although NLP can be solved directly by various existing approaches for constrained optimization in Euclidean space, these approaches usually fail to recognize the manifold structure of M. To achieve better efficiency by utilizing the manifold structure of M in directly applying these existing optimization approaches, we propose a partial penalty function approach for NLP. In our proposed penalty function approach, we transform NLP into the corresponding constraint dissolving problem (CDP) in the Euclidean space, where the constraints that define M are eliminated through exact penalization. We establish the relationships on the constraint qualifications between NLP and CDP, and prove that NLP and CDP have the same stationary points and KKT points in a neighborhood of the feasible region under mild conditions. Therefore, various existing optimization approaches developed for constrained optimization in the Euclidean space can be directly applied to solve NLP through CDP. Preliminary numerical experiments demonstrate that by dissolving the constraints that define M, CDP gains superior computational efficiency when compared to directly applying existing optimization approaches to solve NLP, especially in high dimensional scenarios. Copyright © 2023, The Authors. All rights reserved.

关键词： Constrained optimization

Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Liao, Hong-Lin Tang, Tao Zhou, Tao School of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing211106 China MIIT Nanjing211106 China Division of Science and Technology BNU-HKBU United International College Guangdong Province Zhuhai China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see e.g. [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the L2 norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical *** Codes 65M06, 65M12 © 2022, CC BY-NC-SA.

关键词： Partial differential equations

A Reissner-Mindlin plate formulation using symmetric Hu-Zhang elements via polytopal transformations

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Sky, Adam Neunteufel, Michael Hale, Jack S. Zilian, Andreas Institute of Computational Engineering and Sciences Department of Engineering Faculty of Science Technology and Medicine University of Luxembourg 6 Avenue de la Fonte Esch-sur-AlzetteL-4362 Luxembourg Institute of Analysis and Scientific Computing Technische Universität Wien Wiedner Hauptstr. 8-10 Wien1040 Austria

In this work we develop new finite element discretisations of the shear-deformable Reissner-Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence M ∈ HZ ⊂ Hsym(Div). The latter results in highly accurate approximations of the bending moments M and in the rotation field being in the discontinuous Lebesgue space φ ∈ [L2]2, such that the Kirchhoff-Love constraint can be satisfied for t → 0. In order to preserve optimal convergence rates across all variables for the case t → 0, we present an extension of the formulation using Raviart-Thomas elements for the shear stress q ∈ RT ⊂ H(div). We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting. This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries. The robustness of the formulations and the construction of the Hu-Zhang element are tested for shearlocking, curved geometries and an L-shaped domain with a singularity in the bending moments M. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods. © 2023, CC BY-NC-ND.

关键词： Mindlin plates

ON FEEDBACK STABILISATION FOR THE CAHN-HILLIARD EQUATION AND ITS NUMERICAL APPROXIMATION

学校读者我要写书评

暂无评论

arXiv 2025年

作者： Egger, H. Fritz, M. Kunisch, K. Rodrigues, S.S. Johann Radon Institute for Computational and Applied Mathematics Linz Austria Institute for Numerical Mathematics Johannes Kepler University Linz Austria Institute for Mathematics and Scientific Computing Karl–Franzens University Graz Austria

We consider the stabilisation of solutions to the Cahn-Hilliard equation towards a given trajectory by means of a finite-dimensional static output feedback mechanism. Exponential stabilisation of the controlled state around the target trajectory is proven using careful energy estimates and a spectral condition which characterizes the strength of the feedback. The analysis is general enough to allow for pointwise and distributed measurements and actuation. The main results are derived via arguments that carry over to appropriate discretisation schemes which allows us to establish corresponding exponential stabilisation results also on the discrete level. The validity of our results and the importance of some of our assumptions are illustrated by numerical tests. Copyright © 2025, The Authors. All rights reserved.

关键词： Stabilization

A consistent diffuse-interface model for two-phase flow problems with rapid evaporation

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Schreter-Fleischhacker, Magdalena Munch, Peter Much, Nils Kronbichler, Martin Wall, Wolfgang A. Meier, Christoph Institute for Computational Mechanics Technical University of Munich Boltzmannstrasse 15 Garching85748 Germany Institute for High-Performance Scientific Computing University of Augsburg Universitätsstrase 12a Augsburg86159 Germany Department of Information Technology Uppsala University Box 337 Uppsala75105 Sweden Faculty of Mathematics Ruhr University Bochum Universitätsstrase 150 Bochum44780 Germany

We present accurate and mathematically consistent formulations of a diffuseinterface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several orders of magnitude, leading to high velocity and pressure jumps across the liquid-vapor interface, along with dynamically changing interface topologies. To this end, we integrate an incompressible Navier-Stokes solver combined with a conservative level-set formulation and a regularized, i.e., diffuse, representation of discontinuities into a matrix-free adaptive finite element framework. The achievements are three-fold: First, we propose mathematically consistent definitions for the level-set transport velocity in the diffuse interface region by extrapolating the velocity from the liquid or gas phase. They exhibit superior prediction accuracy for the evaporated mass and the resulting interface dynamics compared to a local velocity evaluation, especially for strongly curved interfaces. Second, we show that accurate prediction of the evaporation-induced pressure jump requires a consistent, namely a reciprocal, density interpolation across the interface, which satisfies local mass conservation. Third, the combination of diffuse interface models for evaporation with standard Stokes-type constitutive relations for viscous flows leads to significant pressure artifacts in the diffuse interface region. To mitigate these, we propose to introduce a correction term for such constitutive model types. Through selected analytical and numerical examples, the aforementioned properties are validated. The presented model promises new insights in simulation-based prediction of melt-vapor interactions in thermal multiphase flows such as in laser-based powder bed fusion of metals. Copyright © 2024, The Authors. All rights reserved.

关键词： Two phase flow

A FOUR-FIELD MIXED FORMULATION FOR INCOMPRESSIBLE FINITE ELASTICITY

学校读者我要写书评

暂无评论

arXiv 2025年

作者： Fu, Guosheng Neunteufel, Michael Schöberl, Joachim Zdunek, Adam Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Fariborz Maseeh Department of Mathematics and Statistics Portland State University 1825 SW Broadway PortlandOR97201 United States Institute of Analysis and Scientific Computing TU Wien Wiedner Hauptstr. 8-10 WienA-1040 Austria HB Berrit Solhemsbackarna 73 Spånga SE-163 56 Sweden

In this work, we generalize the mass-conserving mixed stress (MCS) finite element method for Stokes equations [12], involving normal velocity and tangential-normal stress continuous fields, to incompressible finite elasticity. By means of the three-field Hu–Washizu principle, introducing the displacement gradient and 1st Piola–Kirchhoff stress tensor as additional fields, we circumvent the inversion of the constitutive law. We lift the arising distributional derivatives of the displacement gradient to a regular auxiliary displacement gradient field. Static condensation can be applied at the element level, providing a global pure displacement problem to be solved. We present a stabilization motivated by Hybrid Discontinuous Galerkin methods. A solving algorithm is discussed, which asserts the solvability of the arising linearized subproblems for problems with physically positive eigenvalues. The excellent performance of the proposed method is corroborated by several numerical *** Codes 74S05 (Primary) 74B20 (Secondary) Copyright © 2025, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

HOMOGENIZATION FOR POLYNOMIAL OPTIMIZATION WITH UNBOUNDED SETS

学校读者我要写书评

暂无评论

arXiv 2021年

作者： Huang, Lei Nie, Jiawang Yuan, Ya-Xiang Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100190 China Department of Mathematics University of California 9500 Gilman Drive La Jolla San DiegoCA92093 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper considers polynomial optimization with unbounded sets. We give a homogenization formulation and propose a hierarchy of Moment-SOS relaxations to solve it. Under the assumptions that the feasible set is closed at infinity and the ideal of homogenized equality constraining polynomials is real radical, we show that this hierarchy of Moment-SOS relaxations has finite convergence, if some optimality conditions (i.e., the linear independence constraint qualification, strict complementarity and second order sufficient conditions) hold at every minimizer, including the one at infinity. Moreover, we prove extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with denominators is also studied. In particular, we give a positive answer to a conjecture of Mai, Lasserre and Magron in their recent work. Polynomial optimization and Homogenization and Moment-SOS relaxations and Optimality conditions. Copyright © 2021, The Authors. All rights reserved.

关键词： Polynomials

A force-based gradient descent method for ab initio atomic structure relaxation

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Hu, Yukuan Gao, Xingyu Zhao, Yafan Liu, Xin Song, Haifeng State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China CAEP Software Center for High Performance Numerical Simulation Beijing100088 China

Force-based algorithms for ab initio atomic structure relaxation, such as conjugate gradient methods, usually get stuck in the line minimization processes along search directions, where expensive ab initio calculations are triggered frequently to test trial positions before locating the next iterate. We present a force-based gradient descent method, WANBB, that circumvents the deficiency. At each iteration, WANBB enters the line minimization process with a trial stepsize capturing the local curvature of the energy surface. The exit is controlled by an unrestrictive criterion that tends to accept early trials. These two ingredients streamline the line minimization process in WANBB. The numerical simulations on nearly 80 systems with good universality demonstrate the considerable compression of WANBB on the cost for the unaccepted trials compared with conjugate gradient methods. We also observe across the board significant and universal speedups as well as the superior robustness of WANBB over several widely used methods. The latter point is theoretically established. The implementation of WANBB is pretty simple, in that no a priori physical knowledge is required and only two parameters are present without tuning. © 2022, CC BY.

关键词： Conjugate gradient method