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This paper focuses on computing spectral solutions for high-order eigenvalue problems using an efficient discretization method based on Chebfun spectral discretization algorithms and domain truncation. We solve several numerical eigenvalue problems, demonstrating both the accuracy and computational efficiency of the proposed approach.
Cubature rules, which approximate definite integrals as a linear combination of a set of function values, are ubiquitous and necessary for computational methods in the physical sciences. A superconvergence result for cubature rules on polygonal domains is developed, whereby a rule that is exact for all bivariate polynomials of a fixed even degree realize an extra order of convergence under a decrease in the spacing between nodes.
We extend the two-dimensional singular Zienkiewicz element to three dimensions, leading to a novel $H^2$-conforming finite element on tetrahedral meshes based on rational shape functions. Besides the finite element construction and its conformity analysis, we develop an exact iterative integration procedure for the associated class of rational functions. The resulting formulae allow for the exact integration of the basis functions, their derivatives, and the products occurring in finite element assembly.
We establish the global convergence of the (non-homogeneous) Fitness Centrality algorithm for general graphs, deriving an explicit convergence bound for the corresponding fixed-point iteration. Furthermore, we show how the convergence can be dramatically improved by Anderson acceleration and by switching to Newton's method once a sufficiently good approximation to the fixed point has been found. The efficacy of this strategy is illustrated by numerical experiments on different types of graphs.
In this paper, we propose an efficient preconditioner for solving indefinite complex symmetric linear systems within a block preconditioning framework. We analyze the convergence of the corresponding iterative method and investigate several spectral properties of the preconditioned matrix, including eigenvalue distributions and eigenvector structures. The new preconditioner is used to accelerate the convergence of the flexible version of GMRES. Numerical experiments are presented to illustrate the effectiveness of the proposed preconditioner, and comparisons with existing block preconditioners demonstrate its superior performance.
We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds
This paper investigates the sparse phase retrieval problem, which aims to recover a sparse signal from a system of quadratic measurements. In this work, we propose a novel non-convex algorithm, termed Gradient Hard Thresholding Pursuit (GraHTP), for sparse phase retrieval with complex sensing vectors. GraHTP is theoretically provable and exhibits high efficiency, achieving a quadratic convergence rate after a finite number of iterations, while maintaining low computational complexity per iteration. Numerical experiments further demonstrate GraHTP's superior performance compared to state-of-the-art algorithms.
We introduce a fully discrete numerical scheme for semilinear SPDEs with additive space-time white noise that overcomes the previous order barrier for the spatial convergence rate. The scheme achieves a strong convergence rate of $M^{-1+ε}$ in time and $N^{-3/2+ε}$ in space for any $ε>0$, where $M^{-1}$ and $N^{-1}$ are the temporal, respectively the spatial, meshsizes. This substantially improves the standard spatial error bounds of order $N^{-1/2}$ in the literature.
We develop an immersed interface finite difference method for a one-dimensional nonlinear parabolic interface problem with jump condition \[ [u]_α=λu^+u^-. \] The method combines a Crank--Nicolson immersed interface discretization with an \(s\)-parameter reduction of the nonlinear interface condition, thereby reducing the nonlinear coupling to a scalar quadratic equation. We also discuss two different viewpoints for combining Newton iteration with immersed interface discretization, namely the IIM--Newton and Newton--IIM formulations. Numerical experiments are presented to illustrate the behavior and accuracy of the method.
We first prove the spectral convergence of the random feature method (RFM) when used to solve second-order elliptic equations and eigenvalue problems in one dimension, provided that the solutions belong to Gevrey classes or Sobolev spaces. Second, we derive the convergence rate of RFM when integrated with the Partition of Unity Method (PUM) in terms of the patch size. Finally, we show that the singular values of the resulting random feature matrix decay exponentially, leading to exponential growth of the condition number. We also prove that PUM can mitigate this excessive singular-value decay.
The robustness of approximate Riemann solutions has been a crucial topic in computational magnetohydrodynamics, from both theoretical and practical perspectives. Recently, the widely used HLLD approximate Riemann solution was revised, becoming significantly more robust under strong magnetic fields. Yet, simplifications were needed as the compressible slow magnetoacoustic mode is not included in the HLLD scheme, and as a result non-physical density distribution has been found when having strong slow shocks. In this note, a simple fix is introduced for the estimated intermediate internal energy, providing robust results in several representative test cases, preserving pressure-positivity when the magnetic field is enhanced by a factor of 1000.
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 propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the uniformly elliptic Hamilton-Jacobi-Bellman (HJB) equation. For both linear elliptic PDE in non-divergence form and the HJB equation, we prove the well-posedness of strong solution in $W^{2,p}(Ω)$ and optimal convergence in discrete $W^{2,p}$-norm of the finite element approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a more restricted region. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.
This work presents a high-order isogeometric formulation for magnetoquasistatic eddy-current problems based on a decomposition into Biot-Savart-driven source fields and finite-element reaction fields. Building upon a recently proposed surface-only Biot-Savart evaluation, we generalize the reduced magnetic vector potential framework to the quasistatic regime and introduce a consistent high-order spline discretization. The resulting method avoids coil meshing, supports arbitrary winding paths, and enables high-order field approximation within a reduced computational domain. Beyond establishing optimal convergence rates, the numerical investigation identifies the requirements necessary to recover high-order accuracy in practice, including geometric regularity of the enclosing interface, accurate kernel quadrature, and compatible trace spaces for the source-reaction coupling.
Independently weighted semidefinite penalties arise in augmented-Lagrangian and constrained formulations. This paper characterizes when an exact additive subspace-correction preconditioner remains uniformly effective over all nonnegative penalty weights on a fixed finite-dimensional space. Robustness holds precisely when the correction spaces decompose every joint kernel generated by a nonempty subset of penalties. If one condition fails, a computable constant determines the exact first-order decay of the smallest preconditioned eigenvalue along the associated parameter ray, and the condition number grows linearly; none of the subset conditions can be discarded in general. Filtered decompositions provide computable sufficient bounds on parameter-ordering cones, while distributive kernel lattices permit a single common splitting. Exact-additive computations confirm the characterization and predicted rates. Separate Scott-Vogelius experiments produce stable multilevel iteration counts over the tested weights and mesh levels. The analysis does not establish mesh-uniformity.
This paper investigates a stochastic multiscale subdiffusion model driven by fractional Gaussian noise, where the multiscale Abel kernel with variable exponent $α(t)\in(0,1)$ is used to capture multiscale and crossover behavior in anomalous diffusion. The main difficulties of this model lie in the complexity of the multiscale Abel kernel (e.g. non-monotonicity and non-coercivity) and the low regularity caused by the noise. Concerning these issues, we prove the well-posedness and regularity of the mild solutions by means of solution operator approach and a perturbation technique for multiscale Abel kernel. Then both the semidiscrete-in-time and fully-discrete numerical schemes are proposed and analyzed under the low-regularity numerical analysis framework, with proved temporal and spatial convergence rates. Numerical experiments are presented to substantiate the theoretical results.
We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold $U(n)$ of unitary matrices of size n or on the flag manifold $U (n)/U (1)^n$. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.
Convenient, easy to implement stochastic integration methods are developed on the basis of abstract one-step deterministic order $p$ integration techniques. The abstraction as an arbitrary one step map allows the inspection of easy to implement stochastic exponential time differencing Runge-Kutta (SETDRK), stochastic integrating factor Runge-Kutta (SIFRK) and stochastic RK (SRK) schemes. Such schemes require minimal modifications to existing deterministic schemes and converging to the Stratonovich SDE. These schemes capture all symmetric terms in the Stratonovich-Taylor expansion, are order $p$ in the limit of vanishing noise, can attain at least strong order $p/2$ or $p/2-1/2$ (parity dependent) for drift commutative noise, strong order $1$ for commutative noise, and strong order $1/2$ for multidimensional non-commutative noise. Numerical convergence is demonstrated using different bases of noise for 2nd, 3rd and 4th order SETDRK, SIFRK and SRK schemes.