Adaptive BEM for elliptic PDE systems, Part I: Abstract framework for weakly-singular integral equations

由非残差估计器驱动的二阶线性椭圆型偏微分方程自适应有限元方法的最优复杂度，第一部分：对称偏微分方程

Optimal complexity of adaptive FEM for second-order linear elliptic PDEs driven by non-residual estimators, Part I: Symmetric PDEs

研究对称二阶线性椭圆型偏微分方程的自适应有限元方法，在特定假设下证明算法能无条件全R线性收敛，对足够小参数保证最优复杂度，重点分析如基于平均的非残差估计器。

关于最优目标导向自适应性的抽象分析

An abstract analysis of optimal goal-oriented adaptivity

本文提出一个抽象框架，用于有限元和边界元方法的最优目标导向自适应性分析，扩展了现有结果至一般二阶线性椭圆方程。

基于状态嵌套迭代求解偏微分方程约束的最优控制问题

State-based nested iteration solution of optimal control problems with PDE constraints

本文提出一个抽象框架，用时空有限元离散化求解受偏微分方程约束的最优控制问题，推导正则化与网格尺寸的最优关系，并设计基于状态的嵌套迭代以实现渐近最优复杂度求解。

自适应各向异性复合求积法用于神经PDE近似中的残差最小化

Adaptive anisotropic composite quadratures for residual minimisation in neural PDE approximations

本文研究了数值求积在神经网络求解偏微分方程残差最小化方法中的作用，提出了一种自适应各向异性复合求积策略，通过更丰富的参考求积和基于二分法的细化控制求积误差，提升求解精度与效率。

非线性逼近空间中线性自伴椭圆型偏微分方程能量极小化的收敛性框架

A convergence framework for energy minimisation of linear self-adjoint elliptic PDEs in nonlinear approximation spaces

针对线性自伴椭圆型PDEs能量极小化问题，提出通用优化框架，支持线性与非线性参数混合优化策略，建立算法收敛性，可应用于广泛非线性逼近流形，配套论文提供满足假设的具体空间实例。

Gevrey 类隐映射定理及其在半线性椭圆型 PDE 的 UQ 中的应用

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

本文建立 Gevrey 类隐映射定理，证明残差的 $s$-Gevrey 正则性可推出局部数据到解映射 Fréchet 导数的 $s$-Gevrey 界，并用于半线性椭圆型 PDE 的不确定性量化。

On parameter identification problems for elliptic boundary value problems in divergence form, Part I: An abstract framework

Parameter identification problems for partial differential equations are an important subclass of inverse problems. The parameter-to-state map, which maps the parameter of interest to the respective solution of the PDE or state of the system, plays the central role in the (usually nonlinear) forward operator. Consequently, one is interested in well-definedness and further analytic properties such as continuity and differentiability of this operator w.r.t. the parameter in order to make sure that techniques from inverse problems theory may be successfully applied to solve the inverse problem. I…

Generalised multilevel Picard approximations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numerical approximation schemes studied in the scientific literature suffer under the curse of dimensionality in the sense that the number of computational operations needed to compute an approximation with an error of size at most $ \varepsilon > 0 $ grows at least exponentially in the PDE dimension $ d \in \mathbb{N} $ or in the reciprocal of $ \varepsilon $. Recently, so-called full-history recursive multilevel Picard (…

伴随方法与物理信息神经网络在PDE约束逆问题中的比较

Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems

针对PDE约束逆问题，公平比较伴随优化与PINN，发现未知参数表示决定方法选择：网格场适合伴随，神经表示适合PINN；PINN在时间依赖问题中成本更低，且可预热启动伴随。

A short note on plain convergence of adaptive least-squares finite element methods

We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract f…

并行子空间校正方法用于半 coercive 和近似半 coercive 凸优化问题及其在非线性 PDEs 中的应用

Parallel subspace correction methods for semicoercive and nearly semicoercive convex optimization with applications to nonlinear PDEs

本文提出并行子空间校正方法在无约束半 coercive 和近似半 coercive 凸优化问题中的收敛性分析，扩展了线性奇异和近似奇异问题的理论到非线性问题，并展示了其在非线性偏微分方程中的应用。

在等变演化方程中波的计算与稳定性

Computation and stability of waves in equivariant evolution equations

本文研究了在等变演化方程中行波和旋转波的计算与稳定性问题，探讨了通过冻结方法将PDE的演化分解为底层李群和某些约简相空间的动力学，并分析了线性稳定性与非线性稳定性、离散化下的稳定性保持、谱结构分析、一阶与二阶演化系统、偏微分代数方程的适定性、波形的空间衰减、截断到有界域、波相互作用的分析与数值处理，以及与动力系统连接轨道的关系。