A Convergence Framework for Energy Minimization of Linear Self-Adjoint Elliptic PDEs in Nonlinear Approximation Spaces

Abstract. Recent years have seen growing interest in solving partial differential equations (PDEs) using nonlinear approximation spaces, where tunable parameters control the basis functions and allow the discretization to adapt dynamically to the solution. While these approaches often perform well in practice, convergence guarantees for the underlying optimization problem remain scarce. This work develops a general optimization framework for energy minimization problems arising from linear self-adjoint elliptic PDEs, formulated over nonlinear approximation spaces admitting a natural split between linear and nonlinear parameters. Linear variables are updated via linear solves or steepest descent, while nonlinear variables are handled using projected gradient descent. We establish local convergence in gradients and in values under modular structural assumptions, including differentiability, boundedness, regularity, and directional convexity, and provide explicit convergence rates in terms of the structural constants of the problem. These assumptions are verified for a broad class of nonlinear approximation spaces, including adaptive finite elements, free-knot splines, kernel methods, and weight-constrained neural networks. Numerical experiments illustrate the benefits of the parameter split and several key aspects of the framework. In a recent work [Magueresse, Badia (2025, arXiv:2508.17705)], we apply the framework to overlapping free-knot tensor-product B-splines, obtaining a geometrically adaptive solver with rigorous convergence guarantees.