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In the present paper, we are concerned with the study of the spectral distribution of matrix-sequences showing a non-Hermitian block structure with Toeplitz blocks. We use the notion of geometric mean of matrices and the theory of Generalized Locally Toeplitz (GLT) sequences to perform our analysis and produce some numerical tests and visualizations to confirm our theoretical derivations.
Real-time tsunami early warning relies on distributed sensor networks to infer seismic sources and seafloor motion. Optimizing these networks via Bayesian optimal experimental design (OED) is exceptionally challenging for systems governed by hyperbolic partial differential equations, which lack the spectral decay required by standard low-rank approximations. We present a scalable Bayesian OED framework for linear time-invariant systems. By reformulating the inverse problem in the data space, we transform OED into dense matrix subset selection. We propose a multi-GPU, Schur-complement-update-based, greedy algorithm that solves the OED problem using a pipelined approach that fully overlaps I/O with GPU computations. Our framework achieves near-perfect weak and strong scaling across hundreds of GPUs on Perlmutter and Frontier. Applied to the 2025 Gordon Bell Prize-winning digital twin for tsunami forecasting in the Cascadia Subduction Zone, we optimize a 175-sensor network, minimizing the uncertainty of a parameter field with over one billion degrees of freedom.
We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field affects the flow through concentration-dependent density and viscosity. The model serves as a proxy for temperature-triggered condensation-like phase separation; humidity, latent heat, vapor pressure, and capillary forcing are absorbed into the choice of the threshold temperature $Θ_S$. We motivate the chemical potential through a temperature-dependent Landau free energy and use a regularized auxiliary formulation to prove local-in-time existence of weak solutions. For the numerical analysis, we employ a first-order sequential finite-element discretization of a simplified quasi-static formulation. The heat equation is advanced by implicit diffusion, the variable-coefficient Stokes problem is treated by a Taylor-Hood discretization, and the Cahn-Hilliard bulk derivative is evaluated at the previous time level, so each algebraic subproblem is linear. An isothermal diffusive test confirms mass conservation to roundoff and exhibits monotone discrete-energy decay for the tested parameters. Time-step and mesh-refinement studies show first-order temporal and approximately second-order spatial behavior. The remaining computations provide qualitative, parameter-specific illustrations; no global discrete energy law is claimed for the non-isothermal sequential scheme.
We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. Building on ideas underpinning the recently introduced subspace conjugate gradient algorithm, we derive low-rank short recurrences that generalize one-dimensional subspace projection methods to the nonsymmetric matrix equation setting. To limit memory consumption and maximize computational efficiency, rank truncation strategies and modern randomization procedures are incorporated into the proposed algorithms. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.
We introduce a Grid-Based Multinode Shepard Collocation Method (GBMSC) for two-dimensional elliptic boundary value problems on rectangular domains. The method combines Shepard-type partition functions with tensor-product Lagrange interpolation on overlapping local Cartesian subgrids. The Cartesian structure avoids the local unisolvency search required by multinode Shepard constructions on scattered data, while the local character of the approximation yields sparse collocation matrices and moderate conditioning. We establish the main structural properties of the method, including the cardinal property, tensor-product polynomial reproduction, and local interpolation estimates. Numerical experiments for Poisson problems with Dirichlet and mixed boundary conditions confirm the polynomial reproduction property up to round-off accuracy and show regular convergence for smooth non-polynomial solutions. Comparisons with Multinode Shepard Collocation Method (MSC) and Kansa's collocation methods indicate that the proposed discretization preserves high accuracy with significantly better conditioning than the RBF-based schemes considered.
We propose a novel regularity-informed filtering framework for data assimilation in the context of hyperbolic conservation laws and other time-dependent partial differential equations. We focus on systems whose states exhibit steep gradients and jump discontinuities. While filtering is widely used to improve numerical simulations by incorporating observational data, traditional filtering methods lack awareness of the spatial regularity of states produced in these systems. As a result, data assimilation often produces unphysical state estimates, introducing spurious oscillations in smooth regions and smearing sharp features. To address this limitation, we introduce a filtering framework that incorporates edge-preserving regularization into the filter's analysis step; this framework balances simulation forecasts, observational data, and structural prior knowledge. We formalize this approach using the ensemble Kalman filter (EnKF) and a class of hierarchical generalized sparse Bayesian learning (GSBL) priors, which adaptively infer spatially varying hyperparameters to promote non-oscillatory behavior in smooth regions while preserving discontinuities. We demonstrate the effectiveness of the resulting GSBL-EnKF method on challenging benchmark problems governed by hyperbolic conservation laws. Our results show that enforcing regularity in the analysis step yields sharper, less oscillatory state estimates and lower errors of the ensemble mean. This sometimes comes at the cost of ensemble spread, which we quantify and discuss.
The Fokker-Planck equation admits both a macroscopic Eulerian description through probability densities and a microscopic Lagrangian description through stochastic trajectories. Consequently, optimization problems constrained by the Fokker-Planck equation can be formulated from either perspective. Surprisingly, the corresponding adjoint equations appear to be fundamentally different: the macroscopic adjoint is governed by the backward Kolmogorov equation, whereas the microscopic adjoint evolves pathwise along stochastic trajectories. In this note, we reconcile these two formulations by establishing their correspondence in the continuum setting. We further show that, although their discrete gradients no longer coincide after discretization, both provide consistent numerical approximations of the continuum gradient. Explicit convergence rates are established for both discretization strategies.
The Johnson-Lindenstrauss (JL) lemma guarantees that a random projection of $n$ points to $m=O(\varepsilon^{-2}\log n)$ dimensions preserves pairwise squared distances within relative error $\varepsilon$ with high probability, and this dimension order is asymptotically optimal. In high dimensions, however, distances concentrate around a baseline while key geometric information lies in much smaller fluctuations. We show that the JL bound can therefore be uninformative about retained geometry: an independent Gaussian replacement map can satisfy it even though the replacement cloud is independent of the original data. We then ask how well any decoder can recover a feature $f(D)$ of a squared distance $D$ from a linear sketch. Under squared-error loss, the optimal decoder is conditional expectation, so recovery defines a linear operator whose singular values quantify feature recovery. For isotropic Gaussian data ($Σ=σ^2 I_d$), we diagonalize this operator in closed form. For fixed $k$ with $m,d-m\to\infty$, its $k$th singular value satisfies $\ell_k\approx(m/ d)^{k/2}$. This yields three sharp consequences. A rank-$m$ sketch retains at most an $m/d$ fraction of the variance of any feature of one squared distance. If $m\to\infty$ and $m/d\to0$, the expected Kendall correlation is $\frac{2}π\sqrt{m/d}(1+o(1))$; for fixed $q$, nearest- neighbor agreement tends to $1/q$. Yet one projection can satisfy the JL bound while mean Kendall correlation vanishes when $\log n\ll m\ll d$. After removing scale, Haar-averaged retained covariance-shape information is $(m/d)^2$. Thus JL distance preservation does not quantify the geometry available for comparison or inference.
Numerical experiments reported by Quan and Wu [SIAM J Numer Anal 61 (2023) 2106-2132] show that the observed temporal convergence rates of nonuniform L2 methods for subdiffusion models are not consistent with the theoretically predicted order $3-α$. This discrepancy suggests that a more refined analysis is needed and motivates the development of a nonuniform BDF3-L2 method for the subdiffusion equation. To account for the initial solution singularity, we employ the multiscale solution decomposition to decompose the original solution and approximate a smoother unknown variable that satisfies the subdiffusion model with a smoother source term. The resulting formulation, however, involves restrictive high-order boundary conditions on the source term and initial data. To overcome this difficulty, we introduce a spectral truncation technique that requires only slightly stronger regularity of the data and a controllable truncation error. We establish high-order regularity estimates of the solution to the truncated problem and develop a nonuniform BDF3-L2 method for its numerical approximation, based on which we derive a rigorous error estimate of temporal convergence order $2+α$. Numerical experiments are carried out to substantiate the theoretical findings.
We introduce a compactly supported multinode Shepard operator for the interpolation of scattered data on the torus embedded in $\mathbb{R}^3$. The method combines local polynomial interpolation of total degree $d\in \mathbb{N}$ with compactly supported Shepard-type weights, so that the approximation at each evaluation point depends only on neighbouring stencils of nodes. The torus is treated as an algebraic surface defined by a quartic polynomial, and Gröbner bases are used to construct reduced polynomial spaces on the surface by removing the redundancy induced by the defining equation. This yields local Vandermonde systems adapted to the toroidal geometry. We discuss the metric structure of the torus, show the local equivalence between the periodic parameter distance and the Euclidean distance inherited from $\mathbb{R}^3$, and use this equivalence to motivate the compact support construction. We establish a uniform error estimate in terms of the maximal support radius and the local Lebesgue constants. Under uniform locality and stability assumptions, the method converges with order $d+1$ with respect to the fill distance. Numerical experiments on analytical test functions confirm polynomial reproduction and exhibit an error decay consistent with the theoretical analysis. The approach is also tested on Computational Fluid Dynamics data mapped onto the torus, including the interpolation of the velocity components, the reconstruction of the velocity magnitude from the interpolated components, and the reconstruction of a tangent velocity field through an orthonormal lifting of the interpolated components.
We introduce a stream-function formulation for the Surface Stokes-Helfrich model and study an Arbitrary Lagrangian-Eulerian (ALE) Surface Finite Element (SFEM) discretization in space and a semi-implicit discretization in time of these equations. In contrast with special cases of a surface Stokes model on stationary or prescribed evolving surfaces the surface pressure remains as an unknown and needs to be reconstructed in each time step. As for the surface Stokes model on a prescribed evolving surface the formulation also requires an additional gradient potential. A detailed computational comparison with an ALE-SFEM discretization in space and a similar semi-implicit discretization in time of the corresponding velocity-pressure formulation is explored on a representative problem formulation for simply-connected surfaces, demonstrating the validity of the approach.
We consider reaction-diffusion eigenproblems with oscillatory diffusion and reaction coefficients. The reaction coefficient magnitude is large: the corrector equation identified by periodic homogenization involves both the diffusion and the reaction operators. We study the numerical approximation of this problem using the Multiscale Finite Element Method (MsFEM). This now classical method is a finite element type method that performs a Galerkin approximation of the oscillatory problem on a specific, problem dependent, basis set. The basis functions are precomputed in an offline stage. Inspired by homogenization theory and using some filtering ideas, we show how to define these basis functions in order to obtain an efficient method. The comprehensive set of numerical experiments that we present, in periodic and non-periodic cases, for the scalar-valued version of the problem (which is then self-adjoint) and for the vector-valued version of the problem (which is then in general non self-adjoint), demonstrates the performance of the approach. Some theoretical arguments complement the numerical observations.
Determining the dimension of the $C^1$ cubic spline space $S_3^1(\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker's lower bound includes a local correction $σ$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\in I\setminus\{1/5,3/83\}$, the lower bound is attained and $\dim S_3^1(\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $σ=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $σ=0$, yet $\dim S_3^1(\mathcal{T}(1/5))=34>P_{\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein--Bézier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\dim S_3^1(\mathcal{T})$; genuinely global compatibility must also be taken into account.
In this paper, we consider the Smagorinsky model for the Navier-Stokes equations within a virtual element framework. Under the standard assumption of small data, we prove the existence and uniqueness of a solution. Assuming more regularity to the solutions, we derive the known convergence rates $h$ for the a priori error estimates of the Smagorinsky model in two dimensional domains. We additionally prove that divergence-free virtual discretizations provide improved convergence orders, with weaker regularity assumptions than in the finite element literature. We conclude the paper with numerical results that corroborate the theory.
In many systems, the interaction between a deformable surface or interface and the surrounding bulk fluid is coupled with intrinsic spatiotemporal dynamics within the moving surface. Examples include tumor growth, biological tissue morphogenesis, cardiac mechanics, multi-phase surfactant chemistry, additive manufacturing, clothing wear-and-tear, and reactive combustion flows. Solving such problems requires both geometric computing algorithms to track and resolve the surface and numerical methods to solve the coupled governing equations in the surface and the surrounding bulk phase. Here, we present a fully meshfree numerical solver for such coupled bulk-surface problems with deformable interfaces. The presented solver tracks the surface implicitly, solving for the dynamic surface geometry based on stress balance coupled to surrounding fluid phases. We show convergence for a mass-conserving case on a growing sphere and solve bulk-surface problems with incompressible Navier-Stokes fluids coupled to in-surface nonlinear reaction-diffusion dynamics. Finally, we show a model of biological morphogenesis, solving simultaneously for the dynamic surface shape and the fields on the curved surface with two-way coupling.
The Boltzmann equation plays an important role in modeling mesoscopic behavior in a wide range of scientific and engineering applications. However, its numerical solution is computationally expensive due to the high dimensionality of the model and the nonlinear nonlocal collision operator, especially for steady-state problems that require iterative solvers. This cost becomes prohibitive for inverse problems, where the induced optimization problem requires repeated forward solves. In this work, we propose a reduced-order model (ROM) for the parametric Boltzmann equation to address this computational challenge. The ROM constructs a low-dimensional approximation space for the parameter-induced solution manifold through a residual-based greedy strategy, and the reduced solution is then obtained via residual minimization over the reduced space, subject to mass conservation. The overall efficiency of the ROM is achieved by exploiting the quadratic structure of the collision operator and a precomputed separable approximation of the collision kernel. The resulting ROM is further applied to a thermally-driven inverse problem for reconstructing collision parameters from the observed macroscopic temperature data. This is accomplished either by directly replacing the PDE constraint with the ROM, leading to a bilevel optimization formulation, or by reformulating the task as a single-level optimization problem through the Karush--Kuhn--Tucker (KKT) conditions. Numerical experiments in both collision-dominated and transport-dominated cases are performed to demonstrate the efficiency and accuracy of the proposed ROM and its effectiveness in inverse problems. In particular, the resulting inverse problem is computationally much more tractable, achieving speedups of several orders of magnitude over that based on the full-order model while maintaining comparable accuracy.
This paper studies the computational implementation of proper orthogonal decomposition reduced-order models (POD-ROMs) for evolutionary parametric time-dependent partial differential equations (PDEs). For a one-parameter model, many papers in the literature build the correlation matrix by projecting onto $L^2(Ω)$ even though optimal pointwise error estimates are proved when projecting onto $H^1_0(Ω)$. We compare both scenarios and observe the similar performance in practice. Additionally, to get the POD approximation it is necessary to compute the nonlinear term in the reduced equations. This requires a high computational cost that increases when the problem gets more complex. Numerical results in this paper show different approaches to address this issue. Discrete Empirical Interpolation Method (DEIM) is the most efficient approach. It reduces computational time by approximately half compared to computing the whole FEM formulation by a tensor construction while maintaining the same accuracy. For a multiparameter model, we analyze the new and standard method proposed in \cite{newmethod} using a two-dimensional Brusselator model to complement the results and support the error analysis with a more complex system.
How small can the positivity loss be for an $N$-bandlimited spectral operator that exactly reproduces all modes up to degree $L$? For spherical polynomial approximation on $\mathbb S^d$, we prove that the smallest possible excess of the uniform operator norm above 1, equivalently the least positivity loss, is of sharp order $\left({L}/{(N+1)}\right)^2$ when $1\le L< N$. The lower bound follows from a Fejér peak test and a concentration estimate for bandlimited kernels, while a matching upper bound is obtained by correcting a positive Jackson operator with a smooth filter. We illustrate the result in three settings. On the circle, taking $N=sL-1$, this determines the sharp order of the generalized-projection constant above 1 and identifies the gap between the $s^{-1}$ excess of delayed de la Vallée--Poussin means and the optimal $s^{-2}$ order. For filtered hyperinterpolation, whose operator norm has long been known to be uniformly bounded, we give a quantitative lower bound on its separation from the positivity threshold 1. Finally, we identify an operator-level obstruction to maximum principles.