Papers

This paper studies the finite element approximation of a one-dimensional nonlocal Poisson problem with heterogeneous localization and homogeneous local Dirichlet boundary conditions. These local boundary conditions induce localization kernels with spatially varying interaction neighborhoods, which lead to substantial numerical challenges for the assembly of the singular nonlocal stiffness matrix. An asymptotically compatible conforming finite element method is developed for the variational formulation, together with an exact geometric decomposition for the singular stiffness matrix assembly. Under additional smoothness assumptions on the localization profile, second-order operator consistency is established and error estimates are derived with the corresponding convergence orders. Numerical experiments confirm the theoretical convergence behavior and demonstrate the improved boundary behavior of the heterogeneous localization model.

Semiconductor quantum-device design still lacks an integrated Technology Computer-Aided Design (TCAD)-like environment that connects material geometry, quantum many-body simulation, and automated design. Here we introduce zenDot, a large-language model (LLM)-integrated quantum TCAD platform that links a material-labelled device state to a unified condensed-matter physics toolbox. The device and calculation components are integrated into a desktop workbench, Python API, and an embedded LLM agent, allowing electrostatics, charge and transport characterization, correlated-state calculations, and qubit modelling to be executed within one reproducible environment. We demonstrate zenDot on a Si/SiO2 double quantum dot, where a single device state reproduces the characterization workflow and supports hybrid, tunnel-charge, and singlet-triplet qubit analyses. A platform-level universal-control scan revises the singlet-triplet operating point and reduces the predicted worst-gate infidelity by nearly 30-fold. Beyond analysis, the LLM agent directly operates the same physics environment as human users, proposing design changes, executing registered simulations, and iterating on solver-returned metrics under physics-aware validation. Across three demonstration tasks it completes 18 validated design iterations, including geometry modification followed by a full re-solve from the material stack. zenDot establishes a machine-operable quantum TCAD workflow that connects device physics with LLM-driven design exploration.

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.

A compact two-stage fourth-order two-derivative IMEX method may retain its classical non-commuting fourth-order expansion and L-stable stiff damping while losing high-order accuracy when the relaxation time is comparable with the time step. We ask whether uniform fourth-order accuracy can be recovered without redesigning the original two-stage, two-implicit-solve core. For linear relaxation systems with a separated slow spectral subspace, we introduce a slow–fast response correction (SFRC): the exact slow projection is advanced by the unchanged compact IMEX step and projected back to the slow subspace, while the fast complement is advanced by its exact semigroup. For fixed nonzero relaxation time, the correction is $O(\dt^5)$, so the base mixed expansion through degree four is unchanged, and the pure-fast response remains L-stable. An abstract slow–fast argument reduces the full-state error to a slow-block defect and converts a uniform $O(\dt^5)$ local estimate into a uniform $O(\dt^4)$ global estimate. For each Jin–Xin Fourier mode, the required local estimate is proved by an exact projected-amplification factorization and a uniform bound in $x=\eps/\dt$. In particular, the unchanged compact core already has a uniform fifth-order one-step defect after restriction to the exact finite-$\eps$ slow eigenspace. A two-dimensional three-variable model yields the same mechanism on fixed Fourier grids. Symbolic verification and one- and two-dimensional experiments corroborate the analytical predictions. SFRC is therefore a rigorous linear benchmark identifying the finite-$\eps$ slow–fast information sufficient to reconcile compactness, mixed fourth-order accuracy, stiff damping, and uniform fourth-order time accuracy.

We present a computational framework for robust topology optimization that integrates polygonal finite-element discretizations, spatially correlated non-Gaussian material modeling, and non-intrusive polynomial-chaos surrogates. Spatial uncertainty in Young's modulus is represented as a homogeneous non-Gaussian random field obtained via a memoryless transformation of a truncated Karhunen-Loève expansion, ensuring physical admissibility through positivity of stiffness while preserving the prescribed autocovariance. Polygonal finite elements provide a stable discretization for density-based optimization on unstructured meshes and mitigate checkerboard artefacts and mesh bias, while the sparse polynomial-chaos expansion enables efficient estimation of low-order statistical moments required by the robust objective at a fraction of the cost of intrusive or Monte Carlo approaches. Numerical studies on a cantilever and a curved beam show that introducing non-Gaussian material variability leads to systematic load-path redistribution and a reallocation of 6-12% of the structural volume, together with a reduction in compliance scatter. The non-intrusive surrogate reproduces intrusive reference results within 3% using an order of magnitude fewer full finite-element analyses. These results demonstrate that the proposed framework offers a physically consistent and computationally efficient route to topology-optimized designs that remain reliable under realistic material uncertainty.

We propose a fully discrete method for linear fourth-order parabolic equations with Dirichlet boundary conditions, combining an implicit LBRFD multistep scheme in time with a mixed interior penalty discontinuous Galerkin (IPDG) method in space. The temporal discretization employs equispaced linear barycentric rational interpolants and incorporates a startup procedure. To facilitate the spatial discretization, the original problem is reformulated through an auxiliary variable. For certain parameter pairs $(n,d)$, the LBRFD method is shown to be $A(\alpha)$-stable and to possess a wider stability angle than the corresponding BDF$p$ method of the same order. Stability and a priori error estimates are established via a $G$-energy technique and the discrete Grönwall lemma. The theoretical analysis yields a total $L^2$ error estimate of order $h^{k-1}+\tau^p$, where $p=d$ if $n-d$ is even and $p=d+1$ if $n-d$ is odd. The reduced spatial convergence rate is attributed to boundary contributions on $\partial\Omega$. Despite this theoretical prediction, numerical experiments confirm the stability and demonstrate optimal convergence of order $h^{k+1}+\tau^p$.

We study optimization problems with noisy zeroth-order objective observations and deterministic nonlinear equality constraints with available derivatives. We propose a constrained variant of the adaptive-sampling trust-region derivative-free optimization algorithm—ASTRO-DF. The method builds quadratic local models from estimated objective values at interpolation points within a moving trust region and promotes feasibility through a Byrd–Omojokun composite-step based on linearized constraints, following an SQP-like framework. We prove almost sure convergence using a new constrained criticality test and present numerical results on an equality-constrained stochastic activity network problem.

We give a random-bit-efficient construction for the inverse star discrepancy. For every fixed $u\in(0,1)$, $k$-wise independent uniform points $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N$ with $k=O(d(1+\log(1+N/d)))$ satisfy the Monte Carlo bound $D_N^*(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N) =O(\sqrt{d/N})$ with probability at least $u$. Consequently, $N=O(d\varepsilon^{-2})$ and $k=O(d(1+\log\varepsilon^{-1}))$ suffice to attain discrepancy at most $\varepsilon$. The proof isolates the finitely many moments required by a chaining argument and gives explicit constants. A random vector-valued polynomial over a finite field realizes the required bounded independence on a grid using $O(d^2(1+\log(1+N/d))\log N)$ random bits, rather than the $\Theta(dN\log(dN))$ bits used by independent grid sampling.

We present a conservative, structure-preserving, finite-volume scheme for ideal MHD that ensures pressure positivity, remains applicable across all Mach and Alfven regimes, and handles general nonlinear equations of state. The scheme splits the MHD system into three sub-systems according to characteristic wave scales: an advective part for hydrodynamic transport, a magnetic part for velocity-field coupling, and a pressure part for pressure-velocity coupling. Nonlinear advective terms are explicit, while the other two sub-systems are implicit, yielding a mild, velocity-based CFL condition that is supported by extensive numerical evidence. This makes the scheme suitable for gas-pressure or magnetic-pressure dominated regimes and the incompressible limit. The implicit discretisation gives a pressure equation that reduces to an elliptic form in the low-Mach limit for ideal gases and general thermodynamics. Pressure positivity is ensured via a local conservation-preserving modification of the pressure-internal-energy relation, avoiding a posteriori clipping while preserving conservation. The divergence-free constraint is enforced exactly via constrained transport. Second-order accuracy is achieved with an IMEX Runge-Kutta time integration, TVD reconstruction for explicit fluxes, and central discretisation for implicit terms. The scheme is validated against numerous benchmarks, including high- and low-Mach regimes, strongly magnetised flows, and standard MHD shock problems in 1D and 2D, demonstrating accuracy, stability, and excellent shock-capturing.

Spatially varying coefficients are the primary source of circuit complexity in quantum simulation of linear advection-diffusion-reaction equations. This work presents a resource analysis of single-step quantum propagators constructed using sparse FABLE block encodings, Quantum Singular Value Transformation, and linear-combination-of-unitaries. We derive theoretical estimates for qubit count, gate complexity, and circuit depth in terms of the spatial discretization, sparsity, and polynomial degree, and compare these predictions with synthesized quantum circuits for one- and two-dimensional variable-coefficient problems. The resulting analysis quantifies the cost of encoding realistic transport operators and provides practical resource estimates for near-term implementations.

In this work, based on the discrete ordinate method, we propose a robust block preconditioning strategy for the 3D nonlinear steady-state radiation transport equation with heat diffusion term. The presence of the diffusive term of the temperature equation prevents its elimination into a single equation for the radiation intensity. To overcome this difficulty, all physical variables are assembled into a single monolithic linear system. The heat flux and temperature are treated as independent variables in a mixed $H(\mathrm{div})$-conforming finite element formulation. The equation for radiation intensity is discretised by a discontinuous Galerkin method with upwind flux, where a vectorial finite element space is used to couples the radiation intensity in different directions within each element. We then construct a Newton-Krylov iterative solver to solve the nonlinear equations, for which the core part is efficient preconditioning. To accelerate the convergence of Krylov's method, three block preconditioners are constructed, corresponding to different levels of approximation of the coupling between the temperature and radiation intensity. $P_{\mathrm{Schur}}$ retains the full coupling. $P_{\mathrm{Split}}$ drops the conductive contribution to the radiation block. $P_{\mathrm{BJ}}$ neglects the radiation-to-temperature coupling, retaining only the temperature-to-radiation coupling. Numerical experiments demonstrate the mesh independence and robustness of the proposed preconditioners.

This paper presents a numerical finite volume method for conservation law systems that are adapted to the principle of maximal dissipation. The general assumptions are the existence of a strictly convex entropy functional and the finite propagation speed property of a given system. The procedure is based on a numerical flux construction obtained by the minimization of the entropy functional in each time step. The scheme satisfies assumptions of the Lax-Wendroff theorem. A limiting solution obtained by this scheme is compared with the classical weak solutions obtained by the Glimm or the Wave Front Tracking algorithm for one-dimensional systems.

We study the dimension and dimensional stability of polynomial spline spaces of bi-degree $(m,m')$ with prescribed smoothness orders over planar T-meshes. The homological dimension formula writes the spline dimension as the sum of an Euler characteristic term and a correction term. For a fixed ordered bi-degree, the Euler characteristic term is determined by the mesh structure and the prescribed smoothness orders. The correction term can be written as a quotient of coefficient spaces attached to maximal interior segments (MISs). We prove a weighted deletion theorem: when the available vertex relations generate the coefficient space of an MIS, its summand can be removed from the quotient without changing the correction term. This operation changes neither the T-mesh nor its chain complexes. Repeating the deletion leaves a weighted completely non-diagonalizable component (CNDC). We prove that the weighted CNDC is independent of the order in which eligible MISs are removed and is the same for corresponding pairs in the structural class. The correction term can therefore be represented using only the MISs in the weighted CNDC, while all relations from the original T-mesh are retained. Dimensional stability is then equivalent to constancy of the dimension of the remaining relation space. In particular, an empty weighted CNDC is sufficient for stability. We also derive an upper bound for the remaining correction term and compare it with Mourrain's upper bound based on all MISs.

This paper investigates the uniform-in-time strong convergence rates of a fully discrete approximation for the stochastic Cahn–Hilliard equation driven by multiplicative noise in spatial dimensions $d\in\{1,2,3\}$. The proposed scheme combines a spectral Galerkin method in space with a backward Euler scheme in time. The main analytical difficulties arise from the state-dependent stochastic perturbation, the absence of a global monotonicity structure for the nonlinear term, and the fourth-order nature of the Cahn–Hilliard operator. In particular, these features make the derivation of uniform $L^{\infty}$-moment estimates highly nontrivial in three dimensions. For the continuous equation, by utilizing the Itô formula to $\|u\|^p$ and introducing the energy functional $\mathcal{E}(u(t))$, we derive the uniform moment boundedness of the solution. At the fully discrete level, we develop discrete energy estimates and close the required high-order moment bounds through an induction argument. Based on these regularity estimates, we deduce uniform-in-time strong convergence rates for the fully discrete scheme. Moreover, we prove the existence and uniqueness of invariant measures for both the exact dynamics and the fully discrete numerical dynamics. Numerical experiments are provided to confirm the theoretical findings.

We extend the scaled boundary cubature (SBC) scheme from planar regions to higher-dimensional regions described by oriented boundary patches. The resulting parametrization transforms integrals over compact regions in $\Re^d$ into sums of integrals over the boundary-patch parameter domains and a radial coordinate. In three dimensions, this yields a direct volume-integration rule for solids bounded by affine faces, triangular or tensor-product surface patches, B-spline patches, NURBS patches, and combinations of curved and affine boundary representations. For affine polytopes, recursive application of the scaled boundary map yields nested tensor-product rules over simplex sectors; in three dimensions, these reduce to tetrahedral-sector rules that apply equally to convex and nonconvex oriented polyhedra. We also develop transformations for weakly singular integrands. Placing the scaling center at a point singularity exposes the radial power cancelled by the Jacobian, while generalized radial scalings and Gauss–Jacobi quadrature handle fractional powers. A transverse scaled-boundary map provides the analogous construction for affine singular sets, with straight-line examples in three dimensions. Numerical examples verify polynomial exactness on affine polyhedra and a four-dimensional tesseract, rapid convergence on curved B-spline and NURBS solids, and the expected convergence improvements for point and line singularities. Near-boundary singularity tests also identify when additional patch subdivision or patch-parameter transformations are required.

Morphing flexible multibody aircraft require structural strain, aerodynamic geometry, surface velocity, and generalized loading to remain compatible as joints and flexible components change configuration. A compositional formulation is developed around an assumed material attachment between each lifting surface and a geometrically exact beam. Separating the component root pose from the section field shows that the body strain and elastic potential of a component depend on its own elastic coordinates, while upstream motion enters kinetic terms and external-load pullbacks. At element level, an exact relative logarithm $d$ supplies strain and potential energy, whereas a reference-anchored section coordinate $\sigma$ supplies deformed section geometry. Finite-order expansions retain the finite reference geometry exactly and truncate only endpoint perturbations. The attachment map then generates surface points, tangents, normals, velocities, and force Jacobians from common section kinematics. Euler–Poincare beam balance, moving-surface potential-flow relations, graph cotangent assembly, and the associated semidiscrete power identity are stated in a common twist–wrench convention. Collocation, pressure, equivalent-load, and structural-station sites are distinguished to expose their approximation errors. Verification gives the expected $N_d+1$ convergence order for degree-$N_d$ relative-log expansions. In a geometrically nonlinear cantilever comparison, a cubic static-manifold correction reduces mean full-record displacement error from $0.479$ to $0.255$ over four completed load cases. These results provide structural and interface-level evidence rather than validation of a complete aircraft aeroelastic prediction.

Inverse physics-informed neural networks (PINNs) can reconstruct a field accurately while returning an incorrect physical parameter. We introduce a two-axis post-training diagnosis that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric. The first axis repeatedly fits noisy observations with a matched forward estimator. At known synthetic truth, the second freezes the field and residual view and computes a local score displacement toward a nearby residual-profile minimum. Endpoint consistency then tests whether joint training delivers that preference under the same final view. Across three synthetic one-dimensional, scalar-parameter PDEs, matched-forward mean absolute relative error ranges from 2.34 percent to 17.46 percent. The displacement tracks frozen-profile minima across locked seeds, architectures, and fresh-noise retraining (r from .945 to .982), and it tracks delivered signed log-error in 240 fresh-noise RBA runs (r = .994; 237/240 correct directions). A coupled two-parameter Darcy check validates the full matrix calculation. The axes are complementary diagnostic coordinates, not additive error components or a deployable oracle-free estimator. Together, they route follow-up work toward observations, residual evidence, or endpoint delivery.

Trefftz functions satisfy the differential equation locally and exactly; quasi-Trefftz functions do so approximately to prescribed high order. This paper considers (quasi-)Trefftz approximations for two-dimensional scalar electromagnetic problems. Established discretizations include the Flexible Local Approximation MEthod (FLAME), Trefftz elements ($T$-elements), and Trefftz discontinuous Galerkin (Trefftz-DG) methods. New developments are gradient-enriched FLAME (GEFLAME), conforming Trefftz–FLAME finite elements (TFF), and a full-field Bloch-wavevector-vs-frequency solution with Schur–DtN reduction. Applications cover electrostatics, scattering, singular fields, and Bloch waves in periodic structures. These methods make different compromises between conformity and flexibility. FLAME and GEFLAME incorporate Trefftz functions directly into local difference schemes; TFF uses elementwise FLAME schemes to lift polynomial traces into element interiors; $T$-elements match elementwise Trefftz spaces weakly to such traces; and Trefftz-DG couples broken Trefftz spaces through fluxes and penalties. In the reported wave-scattering examples, GEFLAME gives field and gradient errors several orders of magnitude below those of the quadratic finite-element discretization at comparable algebraic cost. Localized singular-function enrichment removes the dominant reentrant-corner error. Conforming TFF and quasi-conforming $T$-elements admit standard finite-element assembly but expose polynomial edge traces as an accuracy bottleneck. For periodic media, the bilinear Trefftz-DG formulation gives an analytic polynomial wavenumber-vs-frequency eigenproblem without restricting the Bloch multiplier to the unit circle. Schur–DtN reduction yields compact boundary-response problems.

Arbitrary-order directional derivatives along multiple (possibly distinct) directions are computed through a generalized dual-number formulation for both scalar- and vector-valued functions. The proposed framework combines generalized dual evaluations with an inclusion–exclusion reconstruction of symmetric multilinear forms, allowing general multidirectional derivatives to be reconstructed from repeated-direction evaluations without explicitly constructing higher-order derivative tensors. Mixed directional and mixed partial derivatives arise naturally as particular cases of the formulation. The methodology further enables the systematic computation of arbitrary-order kinematic quantities and the construction of Taylor-series methods for systems of ordinary differential equations through automatically generated time derivatives. Numerical examples include the computation of high-order directional derivatives in high-dimensional functions, mixed partial derivatives, arbitrary-order kinematic quantities, and Taylor-series integration methods. The implementation is developed in modern Fortran within an open-source framework compatible with the Fortran Package Manager ecosystem.

The space-time Galerkin discretization of retarded layer potentials leads to a linear system where the coefficients are expressed in terms of integrals over the ansatz and test elements. They require carefully designed quadrature schemes because the kernel is singular in the origin and is discontinuous across the hyperbolicity cone. This paper introduces a new integration approach that leads to a scheme that converges exponentially with the number of quadrature points. The key here is to consider the integration domain as a convex polytope and to devise a decomposition into the convex hulls of simpler polytopes which are parameterized such that the singularity and discontinuity occurs in a single variable. The method is implemented for piecewise constant elements and tested on a scattering problem with known analytic solution.

The notion of Q-order convergence is arguably the most important tool for describing the asymptotic behavior of a convergent sequence. Loosely speaking, it captures the“speed”of convergence of an iterative method. The concept of Q-order convergence is not always well suited for sequences whose errors do not decrease monotonically at every step. In this paper, we introduce the notion of $p$-step Q-order convergence. It generalizes the classical notion of Q-order convergence by comparing errors that are $p$ iterations apart rather than errors of successive iterates. This definition recovers classical Q-order convergence as the special case $p=1$. We show that it extracts meaningful convergence information from certain non-monotonic sequences for which the classical Q-order either does not exist or assigns an overly pessimistic classification. We develop the basic theory of the new notion and locate it within the classical hierarchy by proving that $p$-step Q-order at least $\alpha$ implies R-order at least $\alpha$. Natural applications include iterative methods whose updates alternate or cycle over multiple steps.

Scientific machine learning for partial differential equations commonly targets solution fields, as in physics-informed neural networks, or solution maps, as in neural operators. We study a third target: the propagator itself, a phase and amplitude in phase space. The motivation is a gap in regularity. A transported discontinuity is nonsmooth in space and time, yet the rule that moves it can be a polynomial phase carrying unit amplitude, so the object that generates an evolution can be far smoother than the field it generates. The microlocal neural operator (MiNO) learns that object, using the eikonal equation for the phase and the transport equation for the amplitude, and recovers the solution by an oscillatory integral. Sharp fronts and caustics then belong to propagation geometry rather than to a field fitted pointwise. Small residuals certify more than the reconstructed field. They place the learned canonical relation, the geometry that carries singularities, close to the exact one, and they separate trainable error from the frequency-truncation tail. On a matched-budget discontinuous-advection benchmark, MiNO stops improving within 10,000 steps at the accuracy limit of its finite reconstruction window, a limit predicted in closed form, whereas a physics-informed neural network with neural-tangent-kernel loss balancing stays near its initial error. On smooth advection, the mean error is $3.84\times10^{-3}$ for MiNO and $3.12\times10^{-2}$ for a supervised Fourier neural operator. Single-branch MiNO is the smallest model compared, and one trained generator serves five unseen initial conditions without retraining.

When a linear differential problem is discretized by a linear numerical method characterized by a mesh fineness parameter $n$, the computation of the numerical solution reduces to solving a linear discrete problem identified by a matrix $A_n$ whose size grows with $n$. The sequence of discretization matrices $\{A_n\}_n$ often falls within the class of generalized locally Toeplitz (GLT) sequences, even when the numerical method belongs to the family of domain decomposition methods (DDMs), as illustrated herein through examples. Four widely used preconditioners for DDM discretization matrices are the block Jacobi (BJ), block Gauss–Seidel (BGS), additive Schwarz (AS), and multiplicative Schwarz (MS) preconditioners. In this paper, we provide formal definitions of the BJ/BGS/AS/MS preconditioners for arbitrary multilevel block matrices. These definitions and the associated notations are inspired by the theory of GLT sequences and are proposed as alternatives to those commonly used by the DDM community. We analyze the structure of the BJ/BGS/AS/MS preconditioners when applied to multilevel block matrices $A_n$ belonging to a GLT sequence $\{A_n\}_n$. Every GLT sequence $\{A_n\}_n$ is uniquely associated with a special function $\kappa$ called symbol. We prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $\kappa$, then the sequences of the BJ, BGS, and MS preconditioners are GLT sequences with symbol $\kappa$. For the AS preconditioner, we prove that $\{P_n^{AS}(A_n)\}_n$ is a GLT sequence with symbol $\kappa^{AS}\approx\kappa$, and $\kappa^{AS}=\kappa$ whenever the overlaps in the subdomains used for the construction of $P_n^{AS}(A_n)$ vanish as $n\to\infty$. A numerical validation of these results in the context of isogeometric DDMs is presented.

A recent interpolation-based Quantum Signal Processing (QSP) framework by Alase bypasses the phase-finding procedures required in conventional QSP, allowing for a direct encoding of the target polynomial into a quantum circuit. However, this approach assumes access to a diagonal block encoding of function values without providing an explicit circuit construction. In this work, we address this gap by developing an explicit circuit construction method for diagonal block encodings. The resulting algorithm achieves a computational cost of $\mathcal{O}(d\log d)$ for explicitly constructing block encodings of matrix polynomials, improving upon the best-known theoretical bounds of previous methods. Numerical results confirm this scaling, demonstrating that circuit parameters for polynomial degrees up to $10^7$ can be computed in about a minute on a standard CPU.

We study the full three-dimensional dynamics of the $L^2$-critical Zakharov-Kuznetsov equation with the fractional nonlinearity $|u|^{4/3}u$, equivalently $u^{7/3}$ for real-valued functions. This equation is a higher-dimensional extension of the generalized Korteweg-de Vries equation. In the critical setting solutions to this 3D ZK equation may blow up in finite time or exhibit global time dynamics. The novelties of this work is to treat a non-integer power and to study the dynamics of solutions in a higher dimension. We first review the finite time blow-up in 2D critical ZK, then do a formal analysis of the slightly mass-supercritical blow-up dynamics, deriving the corrections to the blow-up rate and profile for the critical ZK equation in any dimension. We then perform a computational study of solutions, utilizing full 3D numerical simulations. In particular, we use a Fourier pseudospectral discretization and an integrating factor fourth-order Runge-Kutta method on a full three-dimensional grid. A multi-GPU implementation makes it possible to follow blow-up solutions in a full 3D setting. We examine perturbations of the ground state, Gaussian data, and nonsymmetric two-bump configurations. The computations show dispersive and concentrating regimes, in both cases with radiation emitted in a conic-type region opposite to the direction of propagation and convergence of the concentrating core toward a rescaled ground-state profile. The two-bump experiments also demonstrate that total mass alone does not determine the blow-up dynamics. We discuss the numerical evidence for the predicted blow-up rate and identify the pre-asymptotic and resolution limitations that remain near the blow-up time.

Let $d\geq 2$ and let $\bf{f}_1,\ldots,\bf{f}_m\in\mathbb C^d$. We prove that if $m\leq 2d-1$, then the intensity measurement map \[ \bf{x}\longmapsto \bigl( |\langle \bf{x},\bf{f}_1\rangle|^2, \ldots, |\langle \bf{x},\bf{f}_m\rangle|^2 \bigr) \] fails to recover almost every signal in $\mathbb C^d$ uniquely up to a global phase factor. Combined with the known generic sufficiency of $2d$ measurements, our result establishes that the minimum number of measurements required for almost-everywhere phase retrieval in $\mathbb C^d$ is exactly $2d$. This resolves an open problem in phase retrieval by determining the exact measurement threshold for almost-everywhere phase retrieval in ${\mathbb C}^d$.

We study the spectral behavior and approximation properties of linearized shallow networks with analytic activation functions on the unit sphere $\mathbb S^d$. Motivated by earlier spherical approximation results for analytic zonal functions \cite{mhaskar1999approximation}, we first derive an asymptotic formula, expressed explicitly in terms of the residues, for the ultraspherical coefficients of a real-analytic function whose relevant singularities are a conjugate pair of simple poles on the imaginary axis. In particular, the formula gives a matching lower bound along the relevant high-degree subsequences and identifies the exponential decay scale and sign oscillation. It applies directly to $\tanh$, the logistic sigmoid, and the rational activations considered here, while the corresponding estimate for $\arctan$ is obtained through differentiation. For such activation functions and quasi-uniform points $\{\theta_j^\ast\}_{j=1}^n \subset \mathbb S^d$, functions $f \in \mathcal H^r(\mathbb S^d)$ satisfying the appropriate spectral compatibility condition can be approximated by a linear combination $$f_n(\eta) = \sum_{j=1}^n a_j \sigma(\theta_j^\ast \cdot \eta),$$ with the sharp Sobolev rate $$\|f - f_n\|_{\mathcal H^s(\mathbb S^d)} \lesssim n^{-\frac{r-s}{d}} \|f\|_{\mathcal H^r(\mathbb S^d)}.$$ The approximation theorem places the $\mathcal H^s$ norm on the left-hand side and simultaneously provides an explicit normalized $\ell^2$ bound on the coefficients. For spectrally compatible analytic target functions, the same construction further gives the stated analytic convergence rate. Thus, the analysis connects spectral asymptotics and lower bounds at the degrees where the ultraspherical coefficients do not vanish with Sobolev-norm approximation and coefficient control for fixed analytic activation functions.

Linear and nonlinear diffusion equations arise in a range of phenomena of mathematical interest, including slow and fast diffusion, sandpile dynamics, height-constrained transport, the two-dimensional Navier-Stokes equation, and dynamics for sampling probability measures. In recent years, blob methods have attracted significant interest as an approach for numerically simulating these types of PDEs. To address the $O(N^2)$ computational bottleneck of blob methods, we consider stochastic discretizations of space and time. We compare the random batch method with a new approach, which we call the random multirate method. Both of these methods build on classical stochastic methods in the optimization literature, with many similarities to stochastic gradient descent and random coordinate descent. We find that, for linear and nonlinear diffusion equations, in the tradeoff between computational complexity and accuracy, the random multirate method has the best performance. On one hand, we prove that the random multirate method converges to the underlying ODE system at a rate of $O(k \Delta t)$, matching forward Euler, and show by example that this rate is theoretically sharp. On the other hand, we observe even better rates of convergence for the random multirate method when applied to ODEs arising from blob methods for diffusive PDEs. Finally, due to its ability to simulate a wide range of nonlinear diffusion equations, including height-constrained transport and sandpile dynamics, our method succeeds in capturing key features of PDEs for which few numerical approaches exist.

This study presents a hybrid computational framework for simulating non-reacting inviscid high-speed flows of nitrogen gas (N$_2$) around a circular cylinder. Owing to the strongly convection-dominated nature of the compressible Euler equations, the compressible-flow streamline-upwind/Petrov–Galerkin (SUPG) formulation is combined with the YZ$\beta$ shock-capturing technique to stabilize the finite element discretization in the presence of strong discontinuities. Building upon the stabilized solution, a physics-informed neural network (PINN) is employed as a post-processing correction stage (\underline{P}INN-\underline{A}ugmented \underline{S}UPG with \underline{S}hock-\underline{C}apturing—PASSC). The network is anchored to the finite element solution through a shock-weighted data-consistency loss, while the governing equations are enforced in a conservative space–time control-volume form supplemented by macroscopic conservation windows, an entropy-admissibility penalty, and the boundary conditions of the underlying problem. Two-dimensional simulations are performed for free-stream Mach numbers ranging from $2.0$ to $12.0$, and the results are assessed against analytical normal-shock and stagnation relations, the semi-empirical Billig correlation, and reference solutions from the literature. The correction is designed to improve the numerical representation of shocks by reducing localized discretization-induced oscillations and mesh-scale serrations while preserving the agreement of the stabilized solution with the analytical and semi-empirical reference quantities.

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.