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Yuwen Li

Publications and source records attributed to Yuwen Li.

At least 19 recordsLinked to original sources
arXiv2026

$H^2$-based equilibrated residual error estimates

We develop a framework for equilibrated residual a posteriori error analysis of finite element methods, based on decomposing vector- and tensor-valued Sobolev spaces into $H^2$ and $H^1$ spaces. To estimate the residual functional acting on $H^2$ component, the method reconstructs symmetric and piecewise polynomial tensors on vertex patches in broken $H(\operatorname{div}\operatorname{div})$ spaces. We apply the proposed framework to conforming finite element discretizations of biharmonic equation, Poisson's equation, and linear elasticity in mixed form. As a result, we obtain several novel equilibrated residual a posteriori error estimates, providing guaranteed error control for energy error of biharmonic equation, $L^2$ norm error of Poisson's equation, and stress error of mixed linear elasticity. They also serve as the first polynomial-degree-robust a posteriori lower bounds of discretization errors. Numerical experiments illustrate the effectiveness of the $H^2$-based error estimators.

math.NA↗
arXiv2026

Sparse-grid discretizations of BGG sequences

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-dimensional ingredients of BGG sparse-grid finite elements. We construct a bounded commuting sparse-grid projection that yields stable discrete potentials for Hessian, elasticity, and div-div complexes. Under mixed Sobolev regularity, we prove sparse-grid approximation estimates with dimension-independent convergence rates in the number of degrees of freedom. Experiments for a three-dimensional incompatibility problem and a two-dimensional biharmonic problem on an annulus illustrate the efficiency of sparse grids relative to full grids.

math.NA↗
arXiv2026

Nearly sharp $W^{1,\infty}$ stability of polynomial spectral Ritz projections

We establish nearly sharp $W^{1,\infty}$ stability estimates for the Legendre spectral Ritz projection associated with Laplacian on a square domain. Its norm grows proportionally to the square root of the polynomial degree, up to a cubic logarithmic factor. The upper bound uses an integral representation of the discrete gradient projection in terms of one-dimensional semigroup kernels and a rank-two Duhamel comparison between the mass-matrix semigroup and its infinite-dimensional counterpart. The lower bound follows from the endpoint concentration of the Legendre kernel. As an application, we derive gradient maximum-norm error estimates for the Poisson and heat equations, with spatial factors optimal up to logarithmic terms.

math.NA↗
arXiv2026

An adaptive localized orthogonal decomposition method for Helmholtz problems

We develop an adaptive localized orthogonal decomposition (ALOD) method for Helmholtz problems with parametrized loads. Under standard resolution, oversampling, and discrete-stability conditions, we establish, to our knowledge, the first wavenumber-robust residual a posteriori error estimate for a Helmholtz LOD discretization. The estimate is two-sided relative to the fine-grid solution and includes a computable bound for the corrector-localization error. In addition, we design a novel family-aware adaptive algorithm using additive Schwarz corrections to construct a shared regional enrichment space for Helmholtz problems with multiple loads. Energy-weighted proper orthogonal decomposition compresses this space. We establish stability and quasi-optimality of the enriched formulation, with bounds independent of the enrichment space. Numerical experiments demonstrate the effectiveness of ALOD for multiple loads and large wavenumbers.

math.NA↗
arXiv2026

Improving accuracy of finite element solutions by smoothing

This paper develops a smoothing-based postprocessing method for improving the accuracy of finite element solutions. The method applies a few smoothing iterations, such as damped Jacobi, symmetrized Gauss-Seidel, or preconditioned conjugate gradient, with the initial guess being the current finite element solution embedded in an augmented finite element space. The resulting procedure is algebraic, easy to implement, and applicable to high-order and three-dimensional discretizations. For symmetric and positive-definite problems, we show that the accuracy of the smoothed solutions is improved by a smoothing factor under additive and multiplicative smoothers. This smoothing factor decays exponentially during the first few smoothing steps. The effectiveness of the proposed method is numerically demonstrated using the Poisson, Maxwell, biharmonic and Helmholtz equations.

math.NA↗
arXiv2026

Shallow neural network approximation in mixed Sobolev spaces

We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $ρ$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{α,ρ\}$ for target functions of mixed smoothness $α$, up to explicit logarithmic factors. To verify this property for concrete activations, we introduce a structured univariate approximation condition that implies the Fourier-block property with explicit parameters. For $\mathrm{ReLU}^k$, a matching algebraic lower bound identifies $\min\{α,k+1\}$ as the optimal algebraic approximation exponent in any dimension, up to logarithmic factors in the upper bound. The framework also yields the exponent $\min\{α,k+1\}$ for cardinal B-splines and soft-$\mathrm{ReLU}^k$, and the full mixed-smoothness exponent $α$ for ELU and cosine activations, again up to logarithmic~factors.

math.NA↗
arXiv2026

Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces

Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $Ω\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0 k+d/p$ for $1<q\le\infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \[ \mathcal L_1(\mathbb{D})\hookrightarrow B^{k+1}_{p,2}(Ω), \] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow $\operatorname{ReLU}^k$ networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.

math.FA↗
arXiv2026

Nearly optimal Kolmogorov widths under holomorphic mappings

This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces. Given a compact parameter set whose Kolmogorov widths decay algebraically with rate s, we prove that the widths of its image under a holomorphic mapping decay algebraically with every rate t<s, thereby answering an open question raised by Cohen and DeVore. As an application, we obtain a sharp characterization of the approximability of solution manifolds associated with inf-sup stable parametrized PDEs. We also construct an explicit example showing that the arbitrarily small loss in the algebraic decay exponent is unavoidable. Finally, we provide similar characterization of asymptotic bounds for Kolmogorov widths in the exponentially decaying regime. Our analysis involves multilinear Taylor expansion in Banach spaces and a novel block dyadic expansion-truncation technique.

math.FA↗
arXiv2026

Locally computable error estimators for conforming approximations of interface problems cannot be robust

We prove an impossibility result for finite-range locally computable a posteriori error estimators for the conforming finite element discretization of an elliptic interface problem. For a class of interface problems with a checkerboard cross-point and $\{1,M\}$-valued coefficients, we construct two problem instances on the same interface-fitted mesh. The two instances share a piecewise-constant load for which the finite element solution and the data oscillation both vanish. The ratio of their exact energy errors grows at least proportionally to $M^{1/4}$. Locality together with efficiency forces identical estimator values for these instances. The product of reliability and efficiency constants is therefore bounded below by a constant multiple of $M^{1/4}$. Consequently, any locally computable error estimator for interface problems, whether of residual, equilibrated, or recovery type, cannot be simultaneously reliable and efficient with contrast-independent constants.

math.NA↗
arXiv2026

Localized maximum-norm error estimates for the Hellan-Herrmann-Johnson method

We derive localized maximum-norm bounds for bending moments computed by the Hellan-Herrmann-Johnson method for the clamped Kirchhoff plate problem. The main difficulty is to localize the discrete stress without leaving the HHJ space or violating its kernel constraint. Using symmetric-curl potentials, we construct a kernel-preserving localization and connect local Green stresses with a global discrete Green stress. This argument separates the local interpolation error from a weaker global pollution term. Under explicit regularity assumptions on the auxiliary problems, the resulting estimates give optimal pointwise convergence for bending-moment values and elementwise first derivatives. No logarithmic loss occurs for positive polynomial degrees, whereas the lowest-order value estimate retains a logarithmic factor. Under additional reflection symmetry, symmetric recovery improves bending-moment values for even polynomial degrees and first derivatives for odd degrees. The proved gains are one third and one half of an order, respectively. Numerical experiments confirm this parity dependence and exhibit gains close to one full order, exceeding those established theoretically.

math.NA↗
arXiv2026

Checkpoint Selection and Evaluation in EEG Emotion Recognition

Checkpoint selection can improve an electroencephalography (EEG) emotion-recognition score without improving performance on other trials. We compared selection and scoring on disjoint trial pools along fixed training trajectories. A same-session SEED study comprised 300 trajectories from 15 participants, two models and five trial-role rotations. Another 276 trajectories extended the comparison to separate training, validation and target sessions in SEED, SEED-IV and SEED-V, with 15, 15 and 16 participants, respectively. Increasing the candidate set from five to 80 checkpoints raised the same-session dynamical graph convolutional neural network (DGCNN) selection-pool score by 6.24 percentage points, while the other-pool change was -1.24 (descriptive 95% participant-bootstrap interval -2.67 to 0.26). Under the cross-session design, DGCNN retained gains of 4.47 [1.79, 7.25], 4.27 [2.00, 6.53] and 2.75 [0.79, 4.78] points across the three datasets. Multilayer perceptron results varied across datasets. Common-time-range analysis and trial-level scoring supported the respective patterns. Both target pools supplied labels for symmetric checkpoint selection; these results concern retention across trials, not evaluation without target labels. Complete policy curves and executable reconstruction support reporting selected-score improvements alongside performance on other trials, rather than assuming a universal penalty for checkpoint search.

cs.LG↗
arXiv2026

Chronological stability for nonstationary matrix refinement

A matrix cascade is an iterative refinement process in which vector-valued data are repeatedly filtered by matrix-valued masks across successively finer scales. Such cascades arise in multichannel subdivision, Hermite refinement, multiresolution synthesis, and exponential-spline constructions, and in nonstationary schemes the masks may vary with scale. We develop a chronological stability framework for this setting, allowing both the transition operators and the associated matching spaces to change from level to level. If the chronological radius satisfies $ρ_{\rm ch}<1$ and the matching defects are summable, then the cascade converges uniformly and, for every $q\in(ρ_{\rm ch},1)$, \[ \|F_n-Φ\|_\infty \lesssim q^n+\sum_{j=1}^nδ_j q^{\,n-j} +\sum_{j>n}δ_j . \] The time-weighted defect term records the level at which each perturbation is inserted, and the matching spaces need not converge; they may even alternate indefinitely. For compact piecewise-linear seeds, every finite ordered matrix cascade admits an exact fixed-width ReLU realization with $O(n)$ depth and parameters, and this linear depth order is worst-case optimal at fixed width. Consequently, exponentially decaying defects yield certified approximants of size $O(\log\varepsilon^{-1})$. We further construct stable two-channel classes in which the chronological radius and the leading metric-entropy coefficient are independent invariants, showing that stability does not determine finite-accuracy description complexity. The framework therefore provides tolerance-controlled finite representations for nonstationary multiresolution and exponential-spline refinement.

math.NA↗
arXiv2026

Closure complexity of longest-edge bisection for triangular meshes

On triangular meshes, we analyze local mesh refinement based on longest-edge bisection equipped with the serial longest-edge propagation-path closure. Ties are resolved by terminal priority: if the incoming shared edge is a longest edge of the neighboring triangle, the pair is declared terminal and that edge is bisected. For every adaptive mesh sequence $\mathcal{T}_0, \mathcal{T}_1, \ldots, \mathcal{T}_L$ with a sequence of marked sets $\mathcal{M}_0, \mathcal{M}_1, \ldots, \mathcal{M}_{L-1}$, we prove the cumulative closure estimate $$\#\mathcal{T}_L-\#\mathcal{T}_0 \lesssim\sum_{\ell=0}^{L-1}\#\mathcal{M}_\ell.$$ The proof derives single-mark locality from the uniform multiplicative gap between possible descendant diameters implied by finite similarity classes, and converts this locality into the cumulative estimate through a Binev--Dahmen--DeVore-type charging argument.

math.NA↗
arXiv2026

Closure complexity of Bänsch-type algorithms for tetrahedral mesh refinement

We prove, to our knowledge, the first unconditional cumulative closure estimates for the Arnold--Mukherjee--Pouly (AMP) refinement algorithm and the original face-marked tetrahedral algorithm of Bänsch on arbitrary conforming initial tetrahedral meshes. Let $T_0,\ldots,T_L$ be an adaptive mesh sequence generated by either algorithm, with $M_\ell\subseteq T_\ell$ denoting the marking set at step $\ell$. Then $$\#T_L-\#T_0\leq C_{\mathrm{clos}}(T_0)\sum_{\ell=0}^{L-1}\#M_\ell.$$ The proof is intrinsic to the physical three-dimensional mesh and requires neither an initial compatibility condition nor a higher-dimensional embedding. It organizes conformity refinements into a causal forest and combines a uniform horizontal-propagation estimate with a weighted packing argument to obtain an explicit closure constant. For the original Bänsch algorithm, every history-dependent resolution of the initial two-edge ambiguity is represented by one of finitely many AMP histories. The estimate therefore holds uniformly for arbitrary deterministic or nondeterministic choices. This resolves a long-standing complexity question for the Bänsch--AMP family.

math.NA↗
arXiv2026

Localized pointwise a posteriori error estimates for nonconforming finite element methods

This paper establishes the first localized pointwise a posteriori error estimates for nonconforming finite element discretizations of the Poisson and biharmonic equations. For the Poisson problem, we derive localized estimates for the function-value and broken gradient errors of the Crouzeix--Raviart method. For the Morley discretization of the biharmonic equation, we derive a localized a posteriori estimate that controls the local Hessian error. This provides the first pointwise a posteriori error analysis for the biharmonic equation, for either conforming or nonconforming finite element methods. The key ingredient, absent from pointwise analysis of second order PDEs, is the design of two novel weight functions that facilitate sharp estimates of the $L^1$ norms of derivatives of a regularized Green's function for the biharmonic operator.

math.NA↗
arXiv2026

Higher-order finite element de Rham complexes on sparse grids

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 coordinate direction. Their hierarchical decompositions combine Alpert multiwavelets with their integrated counterparts. We establish commuting canonical interpolation operators and corresponding approximation error bounds under mixed Sobolev regularity. For the sparse-grid de Rham complex of arbitrary polynomial degree on the unit cube in arbitrary dimension, we also prove exactness, polynomial-degree-robust stable discrete potentials, and an H(d)-bounded commuting projection by developing a novel stable homotopy operator. Numerical experiments for curl-curl source problems on a cube and a Maxwell eigenproblem on a square-annulus illustrate the effectiveness of the proposed higher-order sparse-grid method.

math.NA↗
arXiv2026

Diagnosing as Cardiologists Do: ECG Agents with Doctor-Grounded Priors for Clinical Reasoning Across Diseases and Populations

Cardiologists interpret electrocardiograms by localizing waveform components, measuring rhythm and interval patterns, and translating these structured observations into diagnostic evidence. Whether this expert reading process can serve as an effective prior for ECG agents remains unclear. To address this question, we introduce LuminaECG, a clinically structured ECG reasoning framework that reformulates ECG interpretation as measurement-grounded visual reading. ECG signals are rendered on standard electrocardiographic grid paper to preserve the spatial and scale cues used in clinical reading. P-wave, QRS-complex, and T-wave boundaries are explicitly delineated, and color-coded segmentation decomposes the waveform into discrete visual measurement primitives. A general 2B vision-language backbone is then trained with low-rank supervised fine-tuning to associate these primitives with diagnostic reasoning, without architectural modification. Across open, proprietary, and ECG-specialist zero-shot baselines, LuminaECG improves both waveform measurement and diagnostic recovery. It reaches a clinically meaningful reader tier on the CODE-test benchmark, transfers across geographically diverse ECG datasets without retraining, and generates reports whose structure contains an emergent prognostic signal. These findings suggest that effective ECG agents require not only larger models, but supervision that preserves the alignment between measurable waveform evidence and clinical knowledge.

eess.SP↗
arXiv2026

Knowledge-Guided Cross-Modal Fusion for Adult-to-Pediatric ECG Transfer via Label-Conditioned Contrastive Alignment

Adult and pediatric electrocardiogram (ECG) interpretation relies on age-sensitive criteria, and models pretrained mainly on adult ECGs often transfer poorly to pediatric populations when pediatric labels are scarce. Existing multimodal ECG--text methods typically align waveforms and text at the global sample level, entangling evidence from co-occurring diagnoses and limiting transfer under this gap. We propose Pediatric-Adult ECG Alignment via Cross-modal Enhancement (PEACE), a knowledge-guided framework pretrained on the largely adult MIMIC-IV ECG corpus. PEACE describes each diagnosis along rhythm, morphology, and ST--T axes and, per recording, composes only positive-label descriptors into three axis tokens and a fused embedding. A label query network (LQN) uses diagnostic labels as queries to cross-attend over ECG tokens and axis tokens, while label set aware bidirectional contrastive learning (LSBC) aligns pooled ECG features with the fused embedding when recordings share diagnoses. Curriculum adaptive fusion (CAF) gates alignment strength according to smoothed classification loss and training progress, limiting disruption during early optimization. The knowledge branch is used only for training supervision; inference uses ECG signals alone. On ZZU-pECG, PEACE reaches macro average AUCs of 59.39%, 81.74%, and 91.56% under zero-shot, 50-shot, and full fine-tuning, with the clearest gains over foundation and knowledge-pretraining baselines under limited supervision; versus domain adaptation initializations, zero-shot improves substantially while 50-shot AUC is comparable to DANN. After fine-tuning on PTB-XL, PEACE reaches 96.90% macro average AUC over nine harmonized labels. Ablations confirm that label-conditioned knowledge alignment, rather than global text fusion, is the key driver of pediatric transfer gains.

cs.LG↗