使用分数阶伪牛顿法求解与不确定性下的投资相关的非线性系统

一种用于非线性参数变分不等式的超降阶神经网络增强半光滑牛顿法

Neural-Manifold Primal-Dual Model Order Reduction and Hyper-Reduction for Nonlinear Parametric Variational Inequalities

针对计算力学中非线性参数化变分不等式，提出模型降阶框架，通过POD构建降阶模型，引入神经网络增强，用稀疏求积超降阶，在二维障碍和三维摩擦接触问题上验证，大幅降成本且逼近精确。

Robust preconditioning for a mixed formulation of phase-field fracture problems

In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the…

一种适用于大变形接触问题的全局收敛滤波-信任区域方法

A globally convergent filter-trust-region method for large deformation contact problems

本文提出一种全局收敛方法，用于求解无摩擦大变形接触问题，采用mortar方法提高稳定性，并结合滤波-信任区域方案和TNNMG方法高效求解非凸约束优化问题。

Inexact Newton combined approximations in the topology optimization of geometrically nonlinear elastic structures and compliant mechanisms

This work blends the inexact Newton method with iterative combined approximations (ICA) for solving topology optimization problems under the assumption of geometric nonlinearity. The density-based problem formulation is solved using a sequential piecewise linear programming (SPLP) algorithm. Five distinct strategies have been proposed to control the frequency of the factorizations of the Jacobian matrices of the nonlinear equilibrium equations. Aiming at speeding up the overall iterative scheme while keeping the accuracy of the approximate solutions, three of the strategies also use an ICA sch…

具有混合边界条件的拟线性椭圆问题的最优有限元误差估计和牛顿收敛性

Optimal finite element error estimates and Newton convergence for a quasilinear elliptic problem with mixed boundary conditions

研究具有混合边界条件的拟线性椭圆热传导问题，用牛顿法求解伽辽金离散化产生的非线性代数系统，结合混合边界椭圆正则性与对偶论证，证明离散解满足最优收敛速率，数值实验证实收敛速率及准则可行性。

A BDF2-Semismooth Newton Algorithm for the Numerical Solution of the Bingham Flow with Temperature Dependent Parameters

This paper is devoted to the numerical solution of the non-isothermal instationary Bingham flow with temperature dependent parameters by semismooth Newton methods. We discuss the main theoretical aspects regarding this problem. Mainly, we focus on existence of solutions and a multiplier formulation which leads us to a coupled system of PDEs involving a Navier-Stokes type equation and a parabolic energy PDE. Further, we propose a Huber regularization for this coupled system of partial differential equations, and we briefly discuss the well posedness of these regularized problems. A detailed fin…

基于Newton迭代的非线性特征值问题的多重网格方法

Multigrid method for nonlinear eigenvalue problems based on Newton iteration

本文提出一种基于Newton迭代的多重网格方法，通过将特征对视为乘积空间中的单一元素，每层仅需求解一个线性边值问题，显著提高了求解非线性特征值问题的效率，并给出了严格的收敛性理论证明。

用于求解稳态玻尔兹曼方程的加速迭代方法

Accelerated iterative method for solving the steady-state Boltzmann equation

针对稳态玻尔兹曼方程，提出带宏观合成系统的修正牛顿法（Newton-MS）。用修正牛顿迭代为外层非线性求解器，内层源迭代求解牛顿修正方程，还推导宏观合成系统加速收敛，通过数值例子验证其高效。

Structure-Preserving and Efficient Numerical Methods for Ion Transport

Ion transport, often described by the Poisson--Nernst--Planck (PNP) equations, is ubiquitous in electrochemical devices and many biological processes of significance. In this work, we develop conservative, positivity-preserving, energy dissipating, and implicit finite difference schemes for solving the multi-dimensional PNP equations with multiple ionic species. A central-differencing discretization based on harmonic-mean approximations is employed for the Nernst--Planck (NP) equations. The backward Euler discretization in time is employed to derive a fully implicit nonlinear system, which is …

Parallel Newton-Krylov-BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac Bidomain equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A fin…

Cluster Gauss-Newton method for finding multiple approximate minimisers of nonlinear least squares problems with applications to parameter estimation of pharmacokinetic models

Parameter estimation problems of mathematical models can often be formulated as nonlinear least squares problems. Typically these problems are solved numerically using iterative methods. The local minimiserobtained using these iterative methods usually depends on the choice of the initial iterate. Thus, the estimated parameter and subsequent analyses using it depend on the choice of the initial iterate. One way to reduce the analysis bias due to the choice of the initial iterate is to repeat the algorithm from multiple initial iterates (i.e. use a multi-start method). However, the procedure ca…

迭代求解Born级数及缩减逆Born级数方案

Iterative inversion schemes for the Born series and the reduced inverse Born series

本文提出了一种快速迭代方案用于求解Born级数及缩减逆Born级数，以解决非线性反问题中的计算成本问题。