TL;DR
This work introduces Schur–Riesz refinement, a variational framework for combining ordinary polynomial finite-element refinement with functions derived from known PDE structure, thereby bridging width-optimal spaces and stable, computable adaptive approximation.
The experiments demonstrate that Schur-Riesz refinement can construct smaller stable spaces with lower deployment memory and repeated-solve cost, while preserving a certified incumbent when richer functions do not help.
We introduce two derivative-free spectral projection methods for large-scale monotone equations with convex constraints. The first, SOPP (Spectral Optimal-Perry Projection), selects its Perry parameter by minimizing the condition number of a symmetrized Perry matrix over its positive definite range, in place of the eig...
We establish whole-sequence convergence of the primal iterates of the smoothing-based full-splitting proximal subgradient method (S-FSPS) of Bo\c{t}, Li, and Tao (SIAM J. Optim., 35 (2025), pp.~2623--2653) with a prescribed, nonsummable sequence of vanishing smoothing parameters. The challenge is that each iteration us...
Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approac...
We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our...
It is proved strong $L^2$ convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space.
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