Special Session 154: Optimization methods and numerical methods for nonlinear PDEs

Virtual element methods for a class of fully nonlinear elliptic PDEs

Andrea Cangiani SISSA Italy Co-Author(s):    Guillaume Bonnet, Andreas Dedner, Ricardo H. Nochetto

Abstract: We propose and analyse $H^2$-conforming arbitrary order Virtual Element Methods (VEM) for the numerical solution of uniformly elliptic Isaacs equations, a general class of fully nonlinear equations which includes the Hamilton-Jacobi-Bellman as a special case. The approach builds upon our previous work on $H^2$-conforming VEM for linear elliptic problems in nondivergence form, extending the methodology to the fully nonlinear setting. A key feature of the proposed method is that the use of $H^2$-conforming spaces enables a direct, arbitrary order, discretisation of second-order operators in strong form. Moreover, the direct discretization of the strong form allows for a simple weak imposition of the boundary conditions. Assuming that the differential operator satisfies the Cordes conditions, we establish the wellposedness of the VEM and derive optimal a priori error bounds in the $H^2$-norm. Numerical experiments confirm the optimality of the method and its competitiveness with other approaches.

Unsupervised Phase Unwrapping via Tailed Nonconvex Optimization and Generalized Itoh Conditions

Huibin Chang Tianjin Normal University Peoples Rep of China Co-Author(s):    BIng Hu; Huibin Chang; Zhangling Chen.

Abstract: Phase Unwrapping (PU) is a fundamental task in MRI, InSAR, and fringe projection profilometry, serving as a critical step for recovering continuous physical quantities. Conventional PU algorithms rely heavily on the Itoh continuity condition, which often fails in experimental scenarios characterized by high noise, sharp discontinuities, or undersampling, leading to severe reconstruction artifacts. This talk presents two novel unsupervised frameworks addressing these challenges. First, we introduce a nonconvex optimization model utilizing L1-L2 regularization and tail minimization. By enhancing gradient sparsity and targeting Itoh condition residuals, this approach significantly improves reconstruction accuracy for complex phase maps. Second, we integrate Deep Image Prior (DIP) with generalized Itoh conditions to establish an unsupervised, high-fidelity unwrapping framework that requires no pre-training datasets. Numerical and experimental results demonstrate that our proposed methods achieve superior signal-to-noise ratios and detail preservation compared to classical and supervised algorithms. These works offer robust and efficient computational solutions for challenging phase retrieval tasks in modern imaging.

BDFs meet projection-free iterative schemes in non-convex constrained variational minimization

Guozhi Dong Central South University Peoples Rep of China Co-Author(s):    Zikang Gong, Ziqing Xie, Shuo Yang

Abstract: I will present a general framework for estimating constraint violations in projection-free iterative methods for minimizing energy functionals subject to pointwise constraints, which are non-convex and involve operator quadratic in the function and/or its derivatives. Our analysis proves to be universal to all projection-free methods that utilize tangent space update strategy. We show that the constraint error bounds are determined solely by the sum of finite differences (of various orders) of the iterates, which reflect (pseudo-)temporal regularities of their continuous counterparts. A new class of projection-free BDF-k second-order-flow methods are proposed achieving modified energy stability and ensuring higher order conditional/unconditional constraint consistency than existing projection-free gradient flow methods.

A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation

ZHAONAN DONG INRIA Paris France Co-Author(s):

Abstract: We establish rigorous a posteriori error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove a posteriori error bounds in the $L^\infty(L^2)$ norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically.

Numerical Analysis of PDE-Constrained Shape Optimization via Shape Gradient Flow

Wei Gong Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Chuwen Ma, Ziyi Zhang

Abstract: In this talk, we investigate the numerical approximation of shape gradient flows arising from PDE-constrained shape optimization problems. The shape gradient flow system comprises the state equation, adjoint state equation, velocity equation, and domain evolution map. For discretization, we discuss two strategies: evolving finite element method where the mesh adapts dynamically to the evolving domain using evolving finite elements, as well as unfitted finite element method by use of a fixed background mesh enhanced with cubic splines to implicitly track domain boundaries. For both approaches, we derive rigorous a priori error estimates. Numerical experiments are provided to validate these theoretical findings.

Entropy-consistent numerical methods and singular limit for nonlocal conservation laws

Kuang Huang The University of Hong Kong Hong Kong Co-Author(s):    Giuseppe Maria Coclite, Nicola De Nitti

Abstract: We consider a class of nonlocal conservation laws modeling traffic flow, where the velocity depends on a convolution of the density with a rescaled kernel that concentrates as a parameter $\epsilon$ tends to zero. A central question is whether solutions converge to the entropy-admissible solution of the corresponding local conservation law, and how to design numerical methods that faithfully capture this transition. We present numerical schemes that are asymptotically compatible: as the mesh size $h$ and $\epsilon$ vanish simultaneously, numerical solutions converge to the local entropy solution with an explicit rate of order $\sqrt{\epsilon}+\sqrt{h}$. We also discuss a complementary approach via compensated compactness that resolves the singular limit for rough initial data and kernels.

An edgewise iterative scheme for the discontinuous Galerkin method with Lagrange multiplier for Poisson`s equation

Mi Young Kim Inha University Korea Co-Author(s):    Dongwook Shin

Abstract: An edgewise iterative scheme is developed for large systems of equations resulting from the discretization by the discontinuous Galerkin method with Lagrange multiplier for the Poisson`s equation. The solution is computed element by element. Lagrange multiplier is edgewise updated, which is given as the average of the Robin type information on the elements sharing the edge. Analysis of the convergence of the scheme is given with the discrete maximum norm over all the edges. Several numerical experiments are presented.

Neural networks-based nonlinear preconditioning for Newton`s method

Yuhuang Meng Delft University of Technology Netherlands Co-Author(s):    Jing Zhao, Alexander Heinlein

Abstract: Many problems in computational science and engineering require solving nonlinear systems of equations arising from partial differential equations. The commonly used methods include Newton's methods and Picard`s iteration methods. However, the standard Newton's method is sensitive to the initial guess and may exhibit slow convergence or even stagnation due to unbalanced nonlinearities. To address this issue, nonlinear preconditioning techniques have been developed to improve the nonlinear convergence. Constructing an efficient preconditioner can be challenging. In addition to traditional numerical approaches, a promising direction is to leverage machine learning techniques, especially given the rapid development in the field of scientific machine learning. In this talk, we will explore how neural networks can be employed to build efficient preconditioners that accelerate the convergence of Newton's methods.

Adaptive multilevel Newton methods for nonlinear PDEs

Eun-Jae Park Yonsei University Korea Co-Author(s):    D. Kim and B. Seo

Abstract: The main part of the talk consists of an adaptive mesh-refining based on the multi-level algorithm and derive a unified a posteriori error estimate for a class of nonlinear problems in the abstract setting. The multi-level algorithm on adaptive meshes retains quadratic convergence of Newton`s method across different mesh levels both theoretically and numerically. As applications of our theory, we consider the pseudostress-velocity formulation of Navier-Stokes equations and the standard Galerkin formulation of semilinear elliptic equations. Reliable and efficient a posteriori error estimators for both approximations are derived. Several numerical examples are presented to test the performance of the algorithm and validity of the theory developed. Lastly, ongoing work on Darcy-Forchheimer flows is presented.

Randomized Neural Networks for PDEs with Applications in MHD

Fei Wang Xi`an Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: The incompressible magnetohydrodynamic (MHD) equations are fundamental in modeling many physical and engineering phenomena. Yet their strong nonlinearity and two divergence-free constraints make them notoriously difficult to solve with traditional numerical methods. Recently, neural networks have gained attention for solving PDEs, but deep learning approaches often struggle: they are slow to train, get trapped in local minima, and only approximate mass conservation. In this talk, I will present a Structure-Preserving Randomized Neural Network (SP-RaNN) designed to overcome these challenges. SP-RaNN automatically and exactly enforces divergence-free conditions, avoids costly nonlinear optimization, and works in a space-time framework that preserves both physical and mathematical structures. The method linearizes the governing equations, discretizes them at collocation points, and solves the resulting system efficiently with least-squares techniques. Through applications to Navier-Stokes, Maxwell, and MHD equations, I will show that SP-RaNN achieves higher accuracy and efficiency than both traditional solvers and deep neural network methods.

A class of stabilized nonconforming finite element methods for fourth-order PDEs on surfaces

Shuonan Wu Peking University Peoples Rep of China Co-Author(s):    Hao Zhou

Abstract: This talk presents the development and analysis of nonconforming finite element methods, specifically the Morley and New-Zienkiewicz-type (NZT) elements, for the surface biharmonic problem and the stream function formulation of the surface Stokes problem. By employing appropriate stabilization techniques and geometric approximations, we address the challenges arising from the lack of $H^2$-conformity on discrete surfaces. Theoretical analysis establishes optimal convergence rates for the proposed schemes in various norms. Furthermore, we investigate the necessity and impact of stabilization terms through numerical experiments, providing insights into the robustness and efficiency of these nonconforming approximations for complex surface geometries.

Machine learning method for solving high dimensional eigenvalue problems

Hehu Xie Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Yifan Wang, Qi Zhou, Teng Wu, Jianghao Liu, Qingyuan Sun, Zhenli Xu

Abstract: In this talk, we will consider the neural network-based machine learning method for solving eigenvalue problems of differential operators. Based on a new understanding of the error estimation for machine learning methods, we design a type of machine learning method with numerical integration to achieve high accuracy. As an example, we will design a tensor neural network-based machine learning method for solving high-dimensional eigenvalue problems, including the famous Schrodinger equations. Some numerical examples are provided to validate the high accuracy and efficiency of the proposed tensor neural network-based machine learning methods.

Hyperbolicity-Preserving Stochastic Galerkin Methods for Conservation Laws with Uncertainty via Associative Truncated Polynomial Products

Yulong Xing The Ohio State University USA Co-Author(s):    Haroun Meghaichi

Abstract: Stochastic Galerkin methods offer an attractive intrusive framework for uncertainty quantification in hyperbolic conservation laws, but for nonlinear systems the projected stochastic Galerkin system may fail to preserve the hyperbolicity of the underlying equations. In this talk, we present a new framework for constructing hyperbolicity-preserving stochastic Galerkin methods for general systems of conservation laws with uncertain data. The central idea is to discretize the multiplication operator arising in the stochastic Galerkin formulation through an associative truncated polynomial product. Within this framework, we establish approximation properties, compare several choices of truncated products, and highlight the mechanisms by which the construction supports hyperbolicity preservation. We then present numerical results for the isothermal Euler equations and the compressible Euler equations with uncertain data, which demonstrate the robustness and effectiveness of the proposed approach.

A Cartesian grid method for acoustic scattering

Wenjun Ying Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: We present a Cartesian grid method for homogeneous and inhomogeous scattering problems on complex domains. The method is a generalization of the traditional boundary integral method. It solves the scattering problem in the framework of boundary integral method but avoids direct evaluation of boundary and volume integrals. The evaluation is done by indirectly solving equivalent interface probelms on Cartesian grids with fast solvers. For (exterior) problems on unbounded domains, we introduce an artificial circle or sphere to accelerate the solution of interface problems, while preserving accuracy. In the talk, we shall also present numerical examples to demonstrate the method.