Papers

Chirality plays an important role in many optical phenomena, and it is crucial to have efficient and accurate numerical algorithms to simulate solutions in this setting. In this paper, we discuss a High-Order Spectral method coupled to geometric regular perturbation theory, which results in just such a method. We view the chirality of the laterally periodic, constant permittivity/permeability structure as deviating from a background value, and demonstrate analyticity of the field scattered by a plane electromagnetic wave with respect to this deformation. This analyticity is not only with respect to chirality deformations of arbitrarily large real size, but also joint in the spatial variables. We also show how the High–Order Perturbation of Envelopes (HOPE) recursions, which we used to establish these results, can be implemented as a rapid, robust, and reliable numerical algorithm.

We propose a deep learning framework for image restoration from images degraded by both multiplicative Gamma noise and blur. Unlike conventional deep equilibrium (DEQ) models that rely on implicit neural regularization, the proposed method learns an explicit and interpretable regularizer parameterized by geometric priors associated with surface area and mean curvature. To minimize the resulting variational model, we develop a mirror descent algorithm tailored to the commonly used Gamma-noise fidelity terms. Leveraging the Kurdyka-Lojasiewicz property for functions defined in $o$-minimal structures, we establish the global convergence of the generated iterates to a critical point. Experimental results on both grayscale and color image restoration demonstrate that the proposed method consistently outperforms representative model-based approaches while achieving performance comparable to state-of-the-art DEQ models based on implicit regularization, despite requiring substantially fewer trainable parameters.

Learned spectral models fail at symmetry forced degenerate sectors for two distinct reasons. Where symmetry forces levels to coincide exactly, the per level observable one would normally fit is not well defined, since every unit vector of that shared space is an eigenvector; and near a symmetry protected crossing, the eigenvector observable gradient carries a factor 1/(lambda_i - lambda_j) that is genuinely singular as the gap closes. The usual response is to regularize the divergence or threshold the gap, and both carry a real cost: a fixed gap cannot both protect a forced multiplet and keep two genuinely distinct levels apart. We show a different fix, on synthetic operator families with a known symmetry answer key. A gate reads the symmetry structure directly from the observed operators, as the linear commutant of the family, one singular value decomposition nullspace, following the simultaneous block diagonalization of Maehara and Murota. Block identity is read from the centre of that commutant rather than from eigenvalue clustering, which makes the forced versus accidental distinction structural rather than metric, and the objective switches between a projector trace through forced blocks and a per level target elsewhere. Under operator estimation noise the gate classifies correctly to epsilon of about 0.3, where energy clustering already fails by 0.02. Gated fitting reaches the truth at machine precision in both the symmetric and the symmetry breaking regime, in the latter converging to a truth that is a singularity of the ungated objective, and observable bias sits at the noise floor. Validation off the regular representation, where multiplicity and dimension separate, eliminated two defective estimators that all regular representation tests had passed. The demonstrated object is a gated estimator; a full parametric matrix model, with the matrices learned, is the next experiment.

In optimization, one often encounters problems of the form $\min_x F(x)+E(x)+G(Kx)$. In this work, we combine primal-dual algorithms with multigrid techniques for their solution. To link the the fine-grid and coarse-grid problems problems, we introduce a nonsmooth primal-dual coherence condition, and an efficient partially linearized line search procedure. Our work is motivated by total variation regularized inverse imaging problems, on which we demonstrate the efficacy of the method, being able to solve problems not previously possible with forward-backward multigrid methods.

We study a parabolic-parabolic chemotaxis system motivated by microglial recruitment toward an amyloid-$\beta$-associated signal in Alzheimer's disease. The signal is subject to a nonnegative Radon measure-valued Neumann influx on a vascular portion of the boundary, while microglial cells respond to a nonlocal spatial average of the signal. For a fixed sensing length $\sigma>0$, the chemotactic velocity is $b_\sigma[v]=\nabla K_\sigma[v]$. For every fixed $\sigma>0$, the nonlocal operator maps finite signal mass into a bounded spatially Lipschitz velocity field. We introduce a weak-solution concept adapted to the low regularity induced by the boundary measure and construct a fully implicit upwind finite-volume approximation in which the boundary source is discretized through its exact mass on each boundary face-time cell. We establish existence and positivity of discrete solutions, together with uniform mass, energy, discrete-gradient, and compactness estimates. Finally, we prove subsequential convergence of the discrete solutions toward a nonnegative weak solution of the continuous problem.

We propose a phase field model of a macrocracked single crystal in which the crack and the geometrically necessary dislocations descend from a single energy functional. Energy minimization alone then decides dislocation nucleation, through an integral criterion evaluated in closed form along slip chords. The criterion yields a size effect inaccessible to point-wise strength conditions: a grain-size-dependent yield stress. With slip suppressed the model reproduces Griffith fracture; with fracture suppressed, the nucleation load measured by the full non-smooth solver agrees with the closed-form nucleation criterion to four percent. The coupled computations produce a two-stage response: at loads an order of magnitude below cleavage, dislocation bands emitted from the notch tip blunt and shield it, raising the initiation load; once the crack grows, the bands heal; a compact cluster of like-signed dislocations travels with the tip, its canceling partner walls pinned at the grain boundary, and the dissipated fracture resistance equals the elastic one. In the purely energetic, dissipationless limit, emission shields the crack but does not toughen it; toughening requires dissipation, incorporated in the sequel through the threshold resistance to dislocation motion.

In the present work, we derive functional upper bounds for the potential error arising from boundary element discretizations of the Laplace-Dirichlet problem. These bounds are based on local auxiliary problems on patches of boundary vertices and the resulting a posteriori error estimator is shown to be locally equivalent to the well-studied residual error estimator. This equivalence result allows us to prove R-linear convergence of the functional a posteriori error estimator and, together with a suitable mesh-refining strategy, to establish that the potential error as well as the functional error estimator converge with optimal rates with respect to the number of boundary elements. Numerical experiments affirm the theoretical findings and illustrate the practical performance of the related adaptive algorithm driven by the proposed functional error estimator.

Finite element methods are among the most widely used discretizations for flow in porous media, and their discrete fracture models (DFMs) for highly conductive fractures are well-established. Low-permeability barriers, by contrast, have long resisted this framework because they induce pressure discontinuities that continuous elements cannot represent directly. In this paper, we propose a simple extension of the linear finite element method for modeling low-permeability barriers through a closed-form modification of the local stiffness matrix of each barrier-cut element. The method retains exactly the same $H^1$-conforming $P^1$-finite element space, preserves the original sparsity pattern and symmetric positive definiteness, requires no mesh fitting, and coincides with the standard finite element method away from barriers. Moreover, an inexpensive, purely local post-processing step recovers the discontinuous pressure field with a sharp jump at the barrier interface, thereby removing the one-element-wide smearing present in the continuous solution. Convergence studies with manufactured solutions and two- and three-dimensional benchmark problems from the literature confirm the effectiveness of the method.

Filtering combines model predictions with measurements to estimate the probability density function (PDF) of a system state over time. The PDF often becomes highly asymmetric and even multimodal in nonlinear systems with oscillatory or chaotic dynamics. Such non-Gaussian features violate the single-Gaussian assumption underlying Kalman-type filters. To address this problem, Gaussian mixture filtering has been proposed. However, accurately propagating mixture components and adaptively adjusting their number and weights over time remain open challenges. Here, we develop an adaptive split-combine Gaussian mixture filter (AMF) that estimates the time evolution of asymmetric and multimodal PDFs by adaptively splitting and combining Gaussian particles without auxiliary online numerical optimization. Notably, the proposed splitting method guarantees a reduction in variance along a target level-set-point direction of a Gaussian particle. This enables accurate and efficient propagation of particles. We show that AMF consistently outperforms various baseline filters across diverse benchmarks, including single and coupled slow-fast Van der Pol oscillators and the Lorenz attractor. We also propose a parallel implementation of AMF, allowing high-fidelity PDF estimation with practical computational cost.

We study asynchronous optimization for finite-sum eigenspace computation in heterogeneous distributed systems. The theoretical foundations for asynchronous eigenspace computation remain scarce, with existing approaches offering limited coverage of dynamics directly on the Grassmannian under stale information. In this paper, we propose a Grassmannian incremental aggregation method that refreshes only arriving components and reuses cached gradients, retaining low per-update cost without global synchronization. The method employs an extrinsic polar update that preserves the intrinsic subspace geometry without requiring parallel transport of stale tangent vectors. Our analysis establishes a tight angle-dependent gradient-dominance characterization of the objective and a basin-invariance property for stale aggregated updates. These yield two-phase linear convergence, comprising an explicit broad-basin regime and a sharper local regime, with constants controlled by component spectral spreads. Experiments on serial and distributed PCA demonstrate improved sample efficiency and wall-clock convergence over representative baselines.

We make a convergence analysis on the joint bidiagonalization (JBD) method that computes several extreme generalized singular value decomposition (GSVD) components of a regular matrix pair $\{A,L\}$, and show that the right and left Ritz vectors obtained by it may converge erratically and even may fail to converge, while Ritz values converge. These convergence results hold for a class of general Rayleigh–Ritz projection methods for the GSVD problem under the hypothesis that the deviation of a desired right generalized singular vector from the right subspace tends to zero. We prove the interlacing property of Ritz values and generalized singular values, and extend it to the generalized singular values of $\{A,L\}$ and the matrix pairs consisting of subsets of its columns. To overcome the irregular convergence or possible non-convergence of the JBD method, we nontrivially extend the refined Rayleigh–Ritz projection for the eigenvalue problem to the GSVD problem, and propose a refined JBD (RJBD) method that replaces the right Ritz vectors by new approximations, called the right refined Ritz vectors, satisfying certain residual optimality; we define new approximate left generalized singular vectors, called the left refined Ritz vectors. We prove that the left and right refined Ritz vectors unconditionally converge under the same hypothesis. We extend the implicit restarting scheme to the RJBD method, and develop an implicitly restarted RJBD algorithm with the refined shifts proposed. Numerical experiments illustrate that the new algorithm is at least competitive and often considerably more efficient than the implicitly restarted JBD algorithm.

Matrix-valued kernels provide a flexible framework for approximating vector fields from scattered data, especially when structural constraints such as divergence-free or curl-free conditions must be preserved. Classical potential-based constructions enforce these constraints naturally, but they typically require the generating scalar function to possess relatively high smoothness. We develop an operator-based framework for constructing div-free and curl-free matrix-valued kernels using integral and differential operators, which substantially relaxes the regularity requirements of the potential approach. Using dimension-walking techniques, we show that the resulting native spaces are norm-equivalent to appropriate vector-valued Sobolev spaces. Another main contribution of the paper is a sharp error analysis for the corresponding kernel matrix-valued interpolation problem. We derive direct Sobolev error estimates that allow fractional regularity of the target field, and we establish Bernstein-type inequalities for the associated kernel trial spaces. These results lead to a complete inverse theorem. We also investigate stability by proving lower bounds for the smallest eigenvalues of the interpolation matrices. Numerical experiments are included to verify the theoretical results.

Near a planar fast–slow fold, a local maximal canard is selected by the parameter at which the attracting and repelling slow manifolds meet. We compare this threshold for a physical flow and a Runge–Kutta map, using actual invariant manifolds on a common fold section. An order-two Runge–Kutta method has two independent order-three rooted-tree defects. Both enter the pointwise one-step residual, but Gaussian fold transport acts on their leading contribution by $(\alpha,\beta)\mapsto-3\beta\Xi(J)/8$. Here $\Xi(J)$ is an explicit functional of the fold jet. Thus the singular passage filters the numerical defect space: it annihilates the bushy-tree direction and can retain only the chain-tree direction. For compact analytic classes of affinely normalizable folds and every fixed compact, uniformly finite-stage family of real Runge–Kutta methods of order at least two, the actual flow and map splittings obtained from independent continuations have unique roots whose displacement satisfies a uniform absolute estimate throughout the full small-step rectangle. Whenever the step-independent, exponentially small selection ambiguity is $o(h^2\varepsilon^2)$, the joint-fold law is $\lambda_{\mathrm{RK}}-\lambda_{\mathrm{flow}}=K_\theta(J)h^2\varepsilon^2+o(h^2\varepsilon^2)$. Fold-matched continuations additionally give ordinary second-order convergence as $h\to0$ with $\varepsilon$ fixed. The leading joint-fold bias therefore vanishes under $b^T A c=1/6$, without classical third order. This is cancellation in one nonlinear observable, not an increase in trajectory order or, in general, in fixed-$\varepsilon$ threshold order. Affine covariance transfers the coefficient to physical fold germs, and a van der Pol invariant-graph computation illustrates the sign change, cancellation, and fixed-$\varepsilon$ convergence.

A Strassen-type algorithm has many realizations with the same exact product and multiplication count yet different fp8 error because basis changes reshape coefficient geometry, posing the question of which to run. No current account settles this: classical stability controls worst-case $\ell_1$ growth, not the expected-error magnitude, and the Dumas–Pernet–Sedoglavic optimizer could only be called probably optimal, its global optimality unproved. To settle this, we attach to each realization a coefficient functional $\Phi$, a scalar summary of its coefficient geometry, which we minimize over the change-of-basis orbit. This Kempf–Ness problem on a Hadamard manifold lets us certify the global $\Phi$ optimum rather than merely search for it: an exact moment-map zero fixes $\Phi_{\min} = 200/9$, and de Groote's classification extends that optimality to every exact real rank-7 $2\times2$ decomposition. Every exact real rank-7 realization therefore has a $\Phi$-predicted RMS constant at least $5/3$ times that of the cubic algorithm, at fixed noise coefficient. We then introduce an explicit block-scaled e4m3 model in which $\Phi$ is the leading-order coefficient of relative expected mean-squared error, and we test the resulting $\Phi$-predicted ordering against realized fp8 error. Ordering and re-basing experiments support that prediction within tested fused block-scaled regimes, and on real matmul tiles from four architecture families the $\Phi$-optimal realization falls in the fp8 low-error region. Across two $\sim$70B models on real deep_gemm kernels, the same realization removes 10 to 55% of classic Strassen's excess NLL over the clean model. Algorithm realization thus becomes a mathematically certified design problem rather than a tuning choice: an independent low-precision axis with a global $\Phi$ optimum and measured fp8 relevance.

Physics-informed and hybrid models integrate data and physics, and read learned coefficients as physical. Yet a small prediction error guarantees neither that the postulated operator is right, nor that the design identifies the coefficient, nor that training reached its minimum. On one parabolic benchmark where a sensor aggregates several diffusion rates and the truth is known, a correct operator recovers the coefficient to machine precision. A wrong one leaves it $30\%$ off while the in-domain error is 1.36 times the noise, and drifts $27\%$ across windows. At $7\%$ noise the error falls below the noise; the coefficient stays $30\%$ off. Without the truth, a post-training coefficient diagnostic questions the trained model with three classical statistics, without oracle; the first two need no refit. Does it explain the data? Can the design resolve the named physics? Does the coefficient depend on the window? The specification test keeps its pre-registered false-alarm ceiling and rejects every replicate of both wrong operators tested; a network frozen at its minimum returns the same verdicts. Where the design is blind, a wrong operator confined to the unseen rate leaves the test mute; the design reading flags it beforehand. A network stopped short of its minimum mimics a wrong operator; an analytic second fit separates the two. The test has a notch against a missing reaction term, miscalibrates under correlated noise and loses power beyond $7\%$ noise. Each warning names its remedy (operator, experiment or training); silence certifies nothing.

Vector kinetic lattice Boltzmann (VKLB) methods have recently emerged as a promising framework for solving hyperbolic partial differential equation (PDE) systems. VKLB discretizes Boltzmann-type equations using a discrete set of lattice velocities, enforces discrete moment constraints, and carefully defines equilibrium distribution functions. In this work, we introduce novel upwinded equilibrium distribution functions constructed from conservation variables and numerical fluxes derived via continuous flux vector splitting based on the eigendecomposition of the flux Jacobian. This formulation enables the weighted-upwind VKLB equilibrium to be applied broadly to general hyperbolic systems. The method is verified on a set of challenging hyperbolic systems that includes the shallow water, Euler and ideal magnetohydrodynamics (MHD) equations. The proposed method demonstrates improved stability, reduced error norms, and sharper shock resolution across increasingly complex verification and benchmark problems.

We develop a concise error analysis for a linear-quadratic elliptic distributed optimal control problem with point tracking and pointwise control constraints, where a standard finite element discretization and a variational discretization are treated in a unified framework. The analysis is self-contained modulo standard techniques in finite element analysis and classical results for elliptic partial differential equations.

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

Nonlinear state estimation requires sequentially fusing model-based predictions with noisy measurements. Under imperfect dynamics and unknown, time-varying noise statistics, this fusion can degrade in both accuracy and statistical consistency. Existing learning-aided filters largely treat accuracy and uncertainty estimation separately, limiting their ability to correct model-mismatch-induced bias while retaining an explicit, calibrated posterior covariance. This paper introduces Unscented KalmanNet (UKN), a model-based deep learning architecture that extends the Unscented Kalman Filter (UKF) with learned mechanisms for these two sources of filtering error while preserving explicit posterior covariance propagation. NoiseNet learns time-varying process and measurement covariances as bounded multiplicative corrections to baseline covariances, guaranteeing positive definiteness, while GainNet learns a bounded residual correction to the analytical UKF gain to compensate for model-mismatch-induced bias. A calibration-aware training objective couples state error with posterior covariance and innovation consistency terms through adaptive weighting, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and real-flight UZH-FPV data. It achieves the lowest state-estimation error in all four examples and reduces RMSE by 26.4-49.7% compared with UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows 22.4% and 34.3% reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with dimension-normalized NEES and empirical coverage closest to nominal values among covariance-reporting filters. These results show that structured learned adaptation improves estimation accuracy while retaining calibrated uncertainty.

We study a physics-informed neural network (PINN) for the unsteady, two-dimensional incompressible Navier–Stokes equations in which the stiff divergence-free constraint is replaced by an artificial-compressibility (AC) relaxation governed by a single scalar parameter $\eps$. The relaxation reintroduces a pressure time derivative, converting a differential-algebraic constraint into an ordinary residual that a PINN can minimise directly. On the Taylor–Green vortex, which admits a closed-form unsteady solution, we quantify the effect of $\eps$: the residual divergence scales as $\eps\,|\partial_t p|$, so larger $\eps$ raises both the divergence and the velocity error, and both decrease monotonically and saturate as $\eps$ is reduced. On the $Re=100$ cylinder wake the plain forward AC-PINN collapses to the steady symmetric branch and does not reproduce von Kármán shedding; assimilating a few hundred sparse velocity sensors from a boundary-layer-resolved finite-element reference (whose Strouhal number, $0.176$, we bring close to the $0.164$–$0.172$ literature band by resolving the separating shear layer, though it remains just above it) recovers the unsteady vortex street to $7\%$ over the wake and its shedding frequency to within $3\%$ of that same reference — a bound set by the reference's own fidelity rather than an independent validation against the true flow.

We study reduced-order modeling for inverse problems in layered media, focusing on the recovery of impedance profiles from time-domain measurements. Using the Goupillaud structure, we formulate the forward problem as a discrete dynamical system and introduce a ROM-based objective defined at the level of reduced operators. Through numerical experiments on a 5-layer medium under various structured perturbations, we compare the ROM-based objective with a classical data misfit. The results show that the ROM-based inversion provides a more favorable reconstruction in the clean setting and in certain structured perturbation regimes, while remaining consistently competitive with the classical approach across all cases considered.

Tensor cross-concentrated sampling (t-CCS) bridges entrywise sampling and t-CUR slice-wise sampling by observing entries only within selected horizontal and lateral slices. Existing t-CCS completion methods, however, assume that the observations are free of gross corruption. In this work, we study robust recovery of a third-order low-tubal-rank tensor from partial t-CCS observations contaminated by sparse, arbitrarily large outliers. We propose Robust Iterative t-CUR (R-ItCUR), a tensor-native algorithm that partitions the sampled tensor cross into two exterior blocks and an intersection block, applies adaptive blockwise Welsch correction for outlier suppression, and updates the low-rank component through projected blockwise gradient descent. By operating directly on the sampled cross, R-ItCUR avoids reconstructing the full tensor throughout the iterations, resulting in substantial memory and computational savings. Experiments on synthetic tensors, cardiac MRI data, and three-dimensional seismic data demonstrate accurate recovery and strong robustness to sparse gross corruptions. The results further highlight the importance of explicitly exploiting the cross-concentrated sampling structure in robust tensor completion.

We present a novel method to recover the source intensity, $f : \mathbb{R}^n \to \mathbb{R}$, and attenuation coefficient, $\mu : \mathbb{R}^n \to \mathbb{R}$, in Compton camera imaging. We apply a non-linear model, which accounts for ray attenuation. We show that the data, $h$, can be modeled $h = \mathcal{R}(f,\mu) = R(fg)$, where $g = \exp(-G\mu)$ models attenuation, $G$ is a (linear) divergent beam transform, and $R$ is a linear operator which defines the integrals of $fg$ over cones. Commonly in the literature, $\mu$ is set to zero, and the data $h = Rf$ is linear. We address the case when $\mu \neq 0$ and the transform is non-linear. To simplify the analysis, we first transform the data into weighted line integrals, $\tilde{h} = \mathcal{D}_k(f,\mu) = D_k(fg)$, where $D_k$ is a weighted ray transform. Assuming practically reasonable geometric conditions, we show that $\tilde{h} = \exp(-X_{w_1}\mu)X_{w_2}f$, where $X_w$ is a weighted X-ray transform, and the $w_i$ are smooth weights. After which, we use the theory of conormal distributions to describe the singularities of $\tilde{h}$. We show that there are artifacts in the reconstruction, and we quantify their strength using Sobolev spaces. We combine this theory with a geometric argument to recover the edges of $f$ and ultimately prove that $f$ and $\mu$ are unique to $h$. The recovery of $f$ is notably more stable than that of $\mu$, which we also discuss. To validate our theory, we present simulated reconstructions of $f$ and $\mu$ using the proposed method.

Atmospheric temperature on Earth results from complex phenomena that climate models must account for. One important module is radiation, which has three sources: infrared radiation emitted by the Earth and the air, and visible light from the Sun. The latter arrives from an almost point-like source in the sky, producing a Dirac singularity in the boundary conditions known as collimated light. Following Siewert and Maiorino (1980), we propose a very fast numerical implementation to handle the singularity.

A classical result by Neubert, Caswell and Murray states that reactivity of a spatially homogeneous equilibrium is a necessary condition for diffusion-driven (Turing) instability. In this work, we investigate whether the same conclusion remains valid in the presence of chemotaxis. We consider a general reaction–diffusion system coupled with a chemotactic flux and establish necessary conditions for asymptotic instability. We show that the classical requirement of reactivity can be relaxed when the chemotactic contribution is sufficiently strong. In particular, while reactivity remains necessary for Turing instability, it is not a necessary condition for chemotaxis-driven instability. From a computational viewpoint, we extend the matrix-oriented formulation developed for reaction–diffusion systems to the more general class of reaction–diffusion–chemotaxis models. The chemotactic transport term is discretized in a form compatible with the matrix-oriented approximation of the diffusion operator, yielding an efficient numerical framework for the simulation of chemotaxis-driven pattern formation. A geometric interpretation of the instability region is presented, highlighting the distinct roles played by diffusion and chemotaxis. The theoretical and numerical developments are illustrated through two representative examples: a chemotaxis-extended Schnakenberg model, showing how chemotaxis modifies classical Turing patterns, and a predator–prey model, demonstrating that chemotaxis alone can induce pattern formation in the absence of both reactivity and diffusion-driven instability. These results reveal a fundamental difference between diffusion-driven and chemotaxis-driven mechanisms of spatial self-organization and provide new theoretical and computational insights into the role of non-symmetric transport processes in biological pattern formation.

This work introduces a robust deterministic framework for approximating solutions of the Boltzmann equation with binary collisions by discretizing their dependence on time, position, and velocity using Galerkin methods. By employing a family of parametric Galerkin closures based on $\varphi$-divergences in velocity space, we derive rigorous hierarchies of moment equations that govern fluid dynamic variables. Addressing the limitation that these closures alone do not guarantee dissipation of a $\varphi$-divergence entropy for the true binary collision operator, we restore this property by formulating a compatible approximate collision operator tailored to each closure. This constructed operator intrinsically retains fundamental physical properties essential for high-fidelity flow simulations, including Galilean invariance, exact conservation of mass, momentum, and energy, and strict dissipation of a $\varphi$-divergence entropy. Furthermore, we show that the resulting closed moment systems are symmetric-dissipative, yielding Cauchy problems that are well-posed locally in time. To translate this mathematical foundation into an efficient computational tool, we discretize the position and time variables with an entropy-stable discontinuous Galerkin (DG) finite element method. The fully implicit, entropy-stable space-time approach enables time steps far beyond typical CFL-limited step sizes and the direct computation of steady states. The robustness and accuracy of the methodology are verified and validated through numerical simulations on the supersonic nozzle flow of argon, mass flow through a channel, and heat transfer between parallel walls, demonstrating agreement with analytical benchmarks, experimental measurements, and stochastic particle simulations.

A three-dimensional (3-D) hybrid numerical method (HNM) is presented for electromagnetic scattering in structures with multiple inhomogeneous layered media coupled to an arbitrary 3-D non-layered scattering region. It integrates the 3-D numerical mode-matching (NMM) method with a tree-cotree-based mixed finite element method (MFEM). In the NMM, the fields in the 3-D layered media are reduced to a superposition of 2-D waveguide eigenmodes, while the MFEM discretizes the 3-D scattering region. The HNM thus inherits the dimensionality-reduction advantages of both conventional hybrid MM/FEM and pure NMM. Numerical experiments show that the HNM provides an effcient alternative for scattering problems involving non-layered regions embedded in layered media.

We study the iteratively regularized gradient method applied to the ill-posed problem of minimizing a general smooth functional on a Hilbert space. Under a source condition on the solution, we establish error estimates for this method when the objective functional is known exactly or only approximately. For these estimates we provide two proofs based on substantially different ideas and compare the assumptions and conclusions of the corresponding theorems. In addition, we present an improved proof of an error estimate for the Tikhonov method applied to ill-posed optimization problems with an exactly known functional. We also prove a theorem on the existence of a global minimizer of the Tikhonov functional without assuming weak lower semicontinuity of the original functional.

The paper proposes a novel proximal difference-of-convex (DC) algorithmic framework to solve general non-convex, non-smooth optimization problems. By combining Barzilai-Borwein (BB) step sizes with nonmonotone line search strategies, our approach effectively overcomes the conservative step sizes and stability issues inherent in standard proximal DC algorithms. Furthermore, we develop extrapolation mechanisms to accelerate convergence while ensuring global stability. The global convergence of the proposed algorithms is rigorously established under the Kurdyka-Łojasiewicz property. Numerical experiments on the SCAD-regularized least squares problem and graphic Ginzburg-Landau image segmentation models demonstrate that the proposed methods achieve highly competitive efficiency and accuracy compared to existing DC algorithms.

This article addresses adaptivity in the kernel multiscale method. Adaptively compressed kernel multiscale approximations have already been presented and analyzed in (LeGia & Wendland, 2014). The main contribution of this work is to attempt to avoid using function evaluations which will be deleted in the compression step anyways. In order to work with function values, we always work in the Lagrange representation. We still assume to have all function values available to compute error norms, but we do not include those values in our approximations. Finally, we also employ local Lagrange function to further reduce the numerical work.