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We develop the first tensor-product sparse-grid finite element discretizations of Bernstein--Gelfand--Gelfand (BGG) complexes. Alpert wavelets and their first and second primitives yield compatible hierarchies of discontinuous, continuous, and continuously differentiable piecewise polynomials, that serve as one-dimensi...
Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0<p\leq 1$. Specifically, \[ B^s_{p,q}(\Ome...
We construct, for the first time, a family of higher-order finite element differential forms on tensor-product sparse grids. The construction starts from compatible one-dimensional spaces of continuous piecewise polynomials and discontinuous piecewise polynomials of one degree lower, linked by differentiation in each c...
We develop an adaptive localized orthogonal decomposition (ALOD) method for Helmholtz problems with localized oscillations and singularities. The main result is the first wavenumber-robust residual a posteriori error estimate for a Helmholtz LOD discretization within the standard resolution, oversampling, and discrete-...
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