SIAM Journal on Scientific Computing最新文献_第9页

Spectral Deferred Correction Methods for Second-Order Problems 二阶问题的光谱延迟校正方法

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-14 DOI: 10.1137/23m1592596

Ikrom Akramov, Sebastian Götschel, Michael Minion, Daniel Ruprecht, Robert Speck

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1690-A1713, June 2024.
Abstract. Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit, or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge–Kutta–Nyström method. Reproducibility of computational results.This paper has been awarded the “SIAM Reproducibility Badge: code and data available,” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/Parallel-in-Time/pySDC/tree/master/pySDC/projects/Second_orderSDC.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 A1690-A1713 页，2024 年 6 月。摘要。谱延迟修正（SDC）是一类用于常微分方程数值解的迭代方法。SDC 可以解释为一种 Picard 迭代法，用于解决用低阶方法预处理的全隐式配位问题。对于一阶问题，使用显式、隐式或隐式-显式欧拉法和其他低阶方法作为预处理，SDC 已得到广泛研究。对于一阶问题，SDC 可达到任意精度阶数，并具有良好的稳定性。虽然已有将 SDC 应用于二阶洛伦兹方程的数值结果，但还没有将 SDC 应用于二阶问题的理论结果。我们以速度-韦勒为基础方法，对一般二阶初值问题的 SDC 的收敛性和稳定性进行了分析。我们的分析证明，收敛阶数取决于系统中的力是否取决于速度。我们还证明了 SDC 迭代在某些条件下是稳定的。最后，我们证明了 SDC 比简单的 Picard 迭代或四阶 Runge-Kutta-Nyström 方法的计算效率更高。本文被授予 "SIAM 可重复性徽章：代码和数据可用性"，以表彰作者遵循了 SISC 和科学计算界重视的可重复性原则。读者可通过 https://github.com/Parallel-in-Time/pySDC/tree/master/pySDC/projects/Second_orderSDC 获取代码和数据，以重现本文中的结果。

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引用次数: 0

A Multiscale Hybrid Method 多尺度混合方法

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-13 DOI: 10.1137/22m1542556

Gabriel R. Barrenechea, Antonio Tadeu A. Gomes, Diego Paredes

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1628-A1657, June 2024.
Abstract. In this work we propose, analyze, and test a new multiscale finite element method called Multiscale Hybrid (MH) method. The method is built as a close relative to the Multiscale Hybrid Mixed (MHM) method, but with the fundamental difference that a novel definition of the Lagrange multiplier is introduced. The practical implication of this is that both the local problems to compute the basis functions, as well as the global problem, are elliptic, as opposed to the MHM method (and also other previous methods) where a mixed global problem is solved and constrained local problems are solved to compute the local basis functions. The error analysis of the method is based on a hybrid formulation, and a static condensation process is done at the discrete level, so the final global system only involves the Lagrange multipliers. We tested the performance of the method by means of numerical experiments for problems with multiscale coefficients, and we carried out comparisons with the MHM method in terms of performance, accuracy, and memory requirements.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 A1628-A1657 页，2024 年 6 月。摘要在这项工作中，我们提出、分析并测试了一种新的多尺度有限元方法，称为多尺度混合（MH）方法。该方法与多尺度混合（MHM）方法近似，但有一个根本区别，即引入了拉格朗日乘数的新定义。其实际意义在于，计算基函数的局部问题和全局问题都是椭圆问题，而 MHM 方法（以及之前的其他方法）则是解决混合全局问题，并解决受约束局部问题以计算局部基函数。该方法的误差分析基于混合表述，并在离散级完成了静态压缩过程，因此最终的全局系统只涉及拉格朗日乘数。我们通过多尺度系数问题的数值实验测试了该方法的性能，并在性能、精度和内存要求方面与 MHM 方法进行了比较。

引用次数: 0

Point Spread Function Approximation of High-Rank Hessians with Locally Supported Nonnegative Integral Kernels 利用局部支持的非负积分核的点展宽函数近似高方差赫赛因数

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-13 DOI: 10.1137/23m1584745

Nick Alger, Tucker Hartland, Noemi Petra, Omar Ghattas

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1658-A1689, June 2024.
Abstract. We present an efficient matrix-free point spread function (PSF) method for approximating operators that have locally supported nonnegative integral kernels. The PSF-based method computes impulse responses of the operator at scattered points and interpolates these impulse responses to approximate entries of the integral kernel. To compute impulse responses efficiently, we apply the operator to Dirac combs associated with batches of point sources, which are chosen by solving an ellipsoid packing problem. The ability to rapidly evaluate kernel entries allows us to construct a hierarchical matrix (H-matrix) approximation of the operator. Further matrix computations are then performed with fast H-matrix methods. This end-to-end procedure is illustrated on a blur problem. We demonstrate the PSF-based method’s effectiveness by using it to build preconditioners for the Hessian operator arising in two inverse problems governed by PDEs: inversion for the basal friction coefficient in an ice sheet flow problem and for the initial condition in an advective-diffusive transport problem. While for many ill-posed inverse problems the Hessian of the data misfit term exhibits a low-rank structure, and hence a low-rank approximation is suitable, for many problems of practical interest, the numerical rank of the Hessian is still large. The Hessian impulse responses, on the other hand, typically become more local as the numerical rank increases, which benefits the PSF-based method. Numerical results reveal that the preconditioner clusters the spectrum of the preconditioned Hessian near one, yielding roughly [math]–[math] reductions in the required number of PDE solves, as compared to classical regularization-based preconditioning and no preconditioning. We also present a comprehensive numerical study for the influence of various parameters (that control the shape of the impulse responses and the rank of the Hessian) on the effectiveness of the advection-diffusion Hessian approximation. The results show that the PSF-based method is able to form good approximations of high-rank Hessians using only a small number of operator applications.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 A1658-A1689 页，2024 年 6 月。摘要我们提出了一种高效的无矩阵点扩散函数（PSF）方法，用于逼近具有局部支持非负积分核的算子。基于 PSF 的方法计算算子在散点处的脉冲响应，并将这些脉冲响应插值为积分核的近似项。为了高效计算脉冲响应，我们将算子应用于与成批点源相关的狄拉克梳状体，这些点源是通过解决椭圆体打包问题选择的。快速评估核项的能力使我们能够构建算子的分层矩阵（H 矩阵）近似值。然后，再利用快速 H 矩阵方法进行进一步的矩阵计算。我们在一个模糊问题上演示了这一端到端的过程。我们用基于 PSF 的方法为两个受 PDEs 控制的逆问题中出现的 Hessian 算子建立预处理：冰原流动问题中的基底摩擦系数反演和平流扩散传输问题中的初始条件反演，从而证明了该方法的有效性。虽然对于许多问题严重的逆问题，数据失配项的 Hessian 具有低秩结构，因此适合采用低秩近似方法，但对于许多实际问题，Hessian 的数值秩仍然很大。另一方面，随着数值秩的增加，Hessian 脉冲响应通常会变得更加局部，这有利于基于 PSF 的方法。数值结果表明，与基于正则化的经典预处理方法和无预处理方法相比，预处理方法可将预处理后的 Hessian 频谱集中在 1 附近，从而使所需的 PDE 求解次数减少约 [math]-[math]。我们还对各种参数（控制脉冲响应的形状和 Hessian 的等级）对平流扩散 Hessian 近似有效性的影响进行了全面的数值研究。结果表明，基于 PSF 的方法只需应用少量算子，就能很好地逼近高阶 Hessian。

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引用次数: 0

Matrix-Free Monolithic Multigrid Methods for Stokes and Generalized Stokes Problems 斯托克斯和广义斯托克斯问题的无矩阵整体多网格方法

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-09 DOI: 10.1137/22m1504184

Daniel Jodlbauer, Ulrich Langer, Thomas Wick, Walter Zulehner

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1599-A1627, June 2024.
Abstract. We consider the widely used continuous [math]-[math] quadrilateral or hexahedral Taylor–Hood elements for the finite element discretization of the Stokes and generalized Stokes systems in two and three spatial dimensions. For the fast solution of the corresponding symmetric, but indefinite system of finite element equations, we propose and analyze matrix-free monolithic geometric multigrid solvers that are based on appropriately scaled Chebyshev–Jacobi smoothers. The analysis is based on results by Schöberl and Zulehner (2003). We present and discuss several numerical results for typical benchmark problems.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 A1599-A1627 页，2024 年 6 月。摘要。我们考虑将广泛使用的连续[math]-[math]四边形或六面体 Taylor-Hood 元素用于二维和三维斯托克斯和广义斯托克斯系统的有限元离散化。为了快速求解相应的对称但不确定的有限元方程组，我们提出并分析了基于适当比例的切比雪夫-雅可比平滑器的无矩阵单片几何多网格求解器。分析以 Schöberl 和 Zulehner (2003) 的结果为基础。我们介绍并讨论了几个典型基准问题的数值结果。

引用次数: 0

Observability and Effective Region of Partial Differential Equations with Application to Data Assimilation 偏微分方程的可观测性和有效区域及其在数据同化中的应用

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-09 DOI: 10.1137/23m1586690

Wei Kang, Liang Xu, Hong Zhou

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page C249-C271, June 2024.
Abstract. In this work, we introduce a new definition of observability for dynamical systems, formulated on the principles of dynamic optimization. This definition gives rise to the concept of an effective region, specifically designed for partial differential equations (PDEs). The usefulness of these concepts is demonstrated through examples of state estimation using observational information for PDEs in a limited area. The findings empower a more efficient analysis of PDE observability. By confining computations to an effective region significantly smaller than the overall region in which the PDE is defined, we demonstrate a substantial reduction in computational demand of evaluating observability. As an application of observability and effective region, we propose a learning-based surrogate data assimilation (DA) model for efficient state estimation in a limited area. Our model employs a feedforward neural network for online computation, eliminating the need for integrating high-dimensional limited-area models. This approach offers significant computational advantages over traditional DA algorithms. Furthermore, our method avoids the requirement of lateral boundary conditions for the limited-area model in both online and offline computations.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 C249-C271 页，2024 年 6 月。摘要在这项工作中，我们根据动态优化原理，为动态系统引入了一个新的可观测性定义。该定义产生了专门针对偏微分方程 (PDE) 的有效区域的概念。这些概念的实用性通过在有限区域内利用偏微分方程的观测信息进行状态估计的例子得到了证明。这些发现使我们能够更有效地分析偏微分方程的可观测性。通过将计算限制在一个明显小于定义 PDE 的整个区域的有效区域内，我们证明了评估可观测性的计算需求大幅减少。作为可观测性和有效区域的一种应用，我们提出了一种基于学习的代用数据同化（DA）模型，用于在有限区域内进行高效的状态估计。我们的模型采用前馈神经网络进行在线计算，无需整合高维有限区域模型。与传统的数据归纳算法相比，这种方法具有显著的计算优势。此外，我们的方法还避免了在线和离线计算中对有限区域模型横向边界条件的要求。

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引用次数: 0

Modified Lawson Methods for Vlasov Equations 弗拉索夫方程的修正劳森方法

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-08 DOI: 10.1137/22m154301x

Benjamin Boutin, Anaïs Crestetto, Nicolas Crouseilles, Josselin Massot

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1574-A1598, June 2024.
Abstract. In this work, Lawson type numerical methods are studied to solve Vlasov type equations on a phase space grid. These time integrators are known to satisfy enhanced stability properties in this context since they do not suffer from the stability condition induced from the linear part. We introduce here a class of modified Lawson integrators in which the linear part is approximated in such a way that some geometric properties of the underlying model are preserved, which has important consequences for the analysis of the scheme. Several Vlasov–Maxwell examples are presented to illustrate the good behavior of the approach.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 A1574-A1598 页，2024 年 6 月。摘要本文研究了求解相空间网格上 Vlasov 型方程的 Lawson 型数值方法。众所周知，这些时间积分器在这种情况下满足增强的稳定性能，因为它们不受线性部分引起的稳定性条件的影响。我们在此介绍一类修正的 Lawson 积分器，其中线性部分的近似方式保留了基础模型的某些几何特性，这对方案分析具有重要影响。我们列举了几个 Vlasov-Maxwell 例子来说明这种方法的良好性能。

引用次数: 0

Multigrid Solvers for the de Rham Complex with Optimal Complexity in Polynomial Degree 多项式度最佳复杂度的德拉姆复数多网格求解器

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-07 DOI: 10.1137/22m1537370

Pablo D. Brubeck, Patrick E. Farrell

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page A1549-A1573, June 2024.
Abstract. The Riesz maps of the [math] de Rham complex frequently arise as subproblems in the construction of fast preconditioners for more complicated problems. In this work, we present multigrid solvers for high-order finite-element discretizations of these Riesz maps with the same time and space complexity as sum-factorized operator application, i.e., with optimal complexity in polynomial degree in the context of Krylov methods. The key idea of our approach is to build new finite elements for each space in the de Rham complex with orthogonality properties in both the [math]- and [math]-inner products ([math] on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold–Falk–Winther, and Hiptmair space decompositions in the separable case. In the nonseparable case, the method can be applied to an auxiliary operator that is sparse by construction. With exact Cholesky factorizations of the sparse patch problems, the application complexity is optimal, but the setup costs and storage are not. We overcome this with the finer Hiptmair space decomposition and the use of incomplete Cholesky factorizations imposing the sparsity pattern arising from static condensation, which applies whether static condensation is used for the solver or not. This yields multigrid relaxations with time and space complexity that are both optimal in the polynomial degree. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://doi.org/10.5281/zenodo.7358044 and in the supplementary materials (pmg_de_rham.zip [61.2KB]).

SIAM 科学计算期刊》，第 46 卷第 3 期，第 A1549-A1573 页，2024 年 6 月。摘要。在构建更复杂问题的快速预处理时，[math] de Rham 复数的 Riesz 映射经常作为子问题出现。在这项工作中，我们提出了对这些里兹图进行高阶有限元离散化的多网格求解器，其时间和空间复杂度与和因式化算子应用相同，即在克雷洛夫方法背景下具有多项式度的最佳复杂度。我们方法的关键思路是为 de Rham 复数中的每个空间建立新的有限元，这些有限元在[math]-和[math]-内积（参考六面体上的[math]）中都具有正交性。由此产生的稀疏性使得在可分情况下，帕瓦里诺、阿诺德-福尔克-温特和希普迈空间分解中出现的补丁问题得以快速解决。在不可分离的情况下，该方法可用于构造稀疏的辅助算子。通过对稀疏补丁问题进行精确的 Cholesky 因子分解，应用复杂度达到最佳，但设置成本和存储成本却不尽人意。我们通过更精细的 Hiptmair 空间分解和使用不完全的 Cholesky 因子分解来克服这一问题，不完全的 Cholesky 因子分解施加了由静态压缩产生的稀疏性模式，无论求解器是否使用静态压缩，这种模式都适用。这就产生了多网格松弛，其时间和空间复杂度在多项式程度上都是最优的。计算结果的可重复性。本文被授予 "SIAM 可重现徽章"：代码和数据可用"，以表彰作者遵循了 SISC 和科学计算界重视的可重现性原则。读者可以通过 https://doi.org/10.5281/zenodo.7358044 和补充材料（pmg_de_rham.zip [61.2KB]）中的代码和数据重现本文的结果。

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引用次数: 0

Adaptive Space-Time Domain Decomposition for Multiphase Flow in Porous Media with Bound Constraints 多孔介质中具有约束条件的多相流的自适应时空域分解

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-06 DOI: 10.1137/23m1578139

Tianpei Cheng, Haijian Yang, Jizu Huang, Chao Yang

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B306-B330, June 2024.
Abstract. This paper proposes an adaptive space-time algorithm based on domain decomposition for the large-scale simulation of a recently developed thermodynamically consistent reservoir problem. In the approach, the bound constraints are represented by means of a minimum-type complementarity function to enforce the positivity of the reservoir model, and a space-time mixed finite element method is applied for the parallel-in-time monolithic discretization. In particular, we propose a time-adaptive strategy using the improved backward differencing formula of second order, to take full advantage of the high degree of space-time parallelism. Moreover, the complicated dynamics with higher nonlinearity of space-time discretization require some innovative nonlinear and linear solution strategies. Therefore, we present a class of modified semismooth Newton algorithms to enhance the convergence rate of nonlinear iterations. Multilevel space-time restricted additive Schwarz algorithms, whose subdomains cover both space and time variables, are also studied for domain decomposition-based preconditioning. Numerical experiments demonstrate the robustness and parallel scalability of the proposed adaptive space-time algorithm on a supercomputer with tens of thousands of processor cores.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 B306-B330 页，2024 年 6 月。摘要本文提出了一种基于域分解的自适应时空算法，用于大规模模拟最近开发的热力学一致储层问题。在该方法中，约束条件通过最小型互补函数来表示，以强制执行储层模型的实在性，并采用时空混合有限元法进行并行时整体离散化。特别是，我们提出了一种使用改进的二阶后向差分公式的时间自适应策略，以充分利用高度的时空并行性。此外，时空离散化的复杂动力学具有更高的非线性，需要一些创新的非线性和线性求解策略。因此，我们提出了一类改进的半滑牛顿算法，以提高非线性迭代的收敛速度。我们还研究了子域同时涵盖空间和时间变量的多级时空受限加法施瓦茨算法，用于基于域分解的预处理。数值实验证明了所提出的自适应时空算法在拥有数万个处理器内核的超级计算机上的鲁棒性和并行可扩展性。

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引用次数: 0

A Framework for Implementing General Virtual Element Spaces 实现通用虚拟元素空间的框架

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-03 DOI: 10.1137/23m1573653

Andreas Dedner, Alice Hodson

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B229-B253, June 2024.
Abstract. In this paper we present a framework for the construction and implementation of general virtual element spaces based on projections built from constrained least squares problems. Building on the triples used for finite element spaces, we introduce the concept of a virtual element method (VEM) tuple which encodes the necessary building blocks to construct these projections. Using this approach, a wide range of virtual element spaces can be defined. We discuss [math]-conforming spaces for [math] as well as divergence and curl free spaces. This general framework has the advantage of being easily integrated into any existing finite element package, and we demonstrate this within the open source software package Dune. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://gitlab.dune-project.org/dune-fem/dune-vem-paper and in the supplementary materials (128492_2_supp_546442_s3hsrj.zip [22KB]).

SIAM 科学计算期刊》，第 46 卷第 3 期，第 B229-B253 页，2024 年 6 月。摘要在本文中，我们提出了一个基于受约束最小二乘法问题建立的投影来构建和实现通用虚拟元素空间的框架。在用于有限元空间的三元组的基础上，我们引入了虚拟元素方法（VEM）元组的概念，它编码了构建这些投影的必要构件。利用这种方法，可以定义多种虚拟元素空间。我们讨论了[math]的[math]符合空间以及无发散和无卷曲空间。这种通用框架的优点是可以轻松集成到任何现有的有限元软件包中，我们在开源软件包 Dune 中演示了这一点。计算结果的可重复性。本文被授予 "SIAM 可再现性徽章"：代码和数据可用"，以表彰作者遵循了 SISC 和科学计算界重视的可重现性原则。读者可以通过 https://gitlab.dune-project.org/dune-fem/dune-vem-paper 和补充材料（128492_2_supp_546442_s3hsrj.zip [22KB]）中的代码和数据重现本文的结果。

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引用次数: 0

Finite Time Horizon Mixed Control of Vibrational Systems 振动系统的有限时域混合控制

IF 3.1 2区数学 Q1 Mathematics

SIAM Journal on Scientific Computing

Pub Date : 2024-05-03 DOI: 10.1137/22m1488648

Ivica Nakić, Marinela Pilj Vidaković, Zoran Tomljanović

SIAM Journal on Scientific Computing, Volume 46, Issue 3, Page B280-B305, June 2024.
Abstract. We consider a vibrational system control problem over a finite time horizon. The performance measure of the system is taken to be a [math]-mixed [math] norm which generalizes the standard [math] norm. We present an algorithm for efficient calculation of this norm in the case when the system is parameter dependent and the number of inputs or outputs of the system is significantly smaller than the order of the system. Our approach is based on a novel procedure which is not based on solving Lyapunov equations and which takes into account the structure of the system. We use a characterization of the [math] norm given in terms of integrals which we solve using adaptive quadrature rules. This enables us to use recycling strategies as well as parallelization. The efficiency of the new algorithm allows for an analysis of the influence of various system parameters and different finite time horizons on the value of the [math]-mixed [math] norm. We illustrate our approach by numerical examples concerning an [math]-mass oscillator with one damper.

SIAM 科学计算期刊》，第 46 卷第 3 期，第 B280-B305 页，2024 年 6 月。摘要我们考虑有限时间范围内的振动系统控制问题。系统的性能指标是一个[math]混合[math]规范，它是标准[math]规范的一般化。我们提出了一种算法，用于在系统与参数相关、系统的输入或输出数量明显小于系统阶数的情况下高效计算该准则。我们的方法基于一个新颖的程序，它不基于求解李亚普诺夫方程，而是考虑到了系统的结构。我们使用自适应正交规则求解的积分来描述 [math] 准则。这使我们能够使用循环策略和并行化。新算法的高效性使得我们可以分析各种系统参数和不同有限时间范围对[math]混合[math]准则值的影响。我们通过一个带有一个阻尼器的[math]质量振荡器的数值示例来说明我们的方法。

引用次数: 0