math.NA
127 postsarXiv:2501.00902v1 Announce Type: new Abstract: Potential theory for rational approximation is reviewed by means of examples computed with the AAA algorithm.
arXiv:2501.00143v1 Announce Type: cross Abstract: This paper presents the application of the Shifted Boundary Method (SBM) to thermal flow simulations, utilizing incomplete octree meshes (Octree-SBM) to perform multiphysics simulations that couple flow and heat transfer. By employing a linearized form of the Navier-Stokes equations, we accelerate the simulations while maintaining accuracy. SBM enables precise enforcement of field and derivative boundary conditions on intercepted elements, allowing for accurate flux calculations near complex geometries, when using non-boundary fitted meshes. Both Dirichlet and Neumann boundary conditions are implemented within the SBM framework, with results demonstrating that SBM ensures precise enforcement of Neumann boundary conditions on octree-based meshes. We illustrate this approach by simulating flows across different regimes, benchmarking results over several orders of magnitude variation in Rayleigh numbers ($Ra \sim 10^3$ to $10^9$) and Reynolds numbers ($Re \sim 10^0$ to $10^4$), covering laminar, transitional, and turbulent regimes. Coupled thermal-flow phenomena as well as summary statistics across all these regimes are accurately captured without any additional numerical treatments, beyond a Residual-based Variational Multiscale formulation (RB-VMS). This approach offers a reliable and efficient solution for complex geometries, boundary conditions and flow regimes in computational multiphysics simulations.
arXiv:2501.00783v1 Announce Type: new Abstract: In this work,we consider the solid-state dewetting of an axisymmetric thin film on a curved-surface substrate,with the assumption that the substrate morphology is also axisymmetric.Under the assumptions of axisymmetry,the surface evolution problem on a curved-surface substrate can be reduced to a curve evolution problem on a static curved substrate.Based on the thermodynamic variation of the anisotropic surface energy,we thoroughly derive a sharp-interface model that is governed by anisotropic surface diffusion,along with appropriate boundary conditions.The continuum system satisfies the laws of energy decay and volume conservation,which motivates the design of a structure-preserving numerical algorithm for simulating the mathematical model.By introducing a symmetrized surface energy matrix, we derive a novel symmetrized variational formulation. Then, by carefully discretizing the boundary terms of the variational formulation, we establish an unconditionally energy-stable parametric finite element approximation of the axisymmetric system. By applying an ingenious correction method, we further develop another structure-preserving method that can preserve both the energy stability and volume conservation properties. Finally, we present extensive numerical examples to demonstrate the convergence and structure-preserving properties of our proposed numerical scheme. Additionally, several interesting phenomena are explored, including the migration of 'small' particles on a curved-surface substrate generated by curves with positive or negative curvature, pinch-off events, and edge retraction.
arXiv:2501.00898v1 Announce Type: new Abstract: An analytic function can be continued across an analytic arc $\Gamma$ with the help of the Schwarz function $S(z)$, the analytic function satisfying $S(z) = \bar z$ for $z\in \Gamma$. We show how $S(z)$ can be computed with the AAA algorithm of rational approximation, an operation that is the basis of the AAALS method for solution of Laplace and related PDE problems in the plane. We discuss the challenge of computing $S(z)$ further away from from $\Gamma$, where it becomes multi-valued.
arXiv:2501.01167v1 Announce Type: new Abstract: We propose novel methods for approximate sampling recovery and integration of functions in the Freud-weighted Sobolev space $W^r_{p,w}(\mathbb{R})$ with the approximation error measured in the norm of the Freud-weighted Lebesgue space $L_{q,w}(\mathbb{R})$. Namely, we construct equidistant compact-supported B-spline quasi-interpolation and interpolation sampling algorithms $Q_{\rho,m}$ and $P_{\rho,m}$ which are asymptotically optimal in terms of the sampling $n$-widths $\varrho_n(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R}))$ for every pair $p,q \in [1,\infty]$, and prove the right convergence rate of these sampling $n$-widths, where $\boldsymbol{W}^r_{p,w}(\mathbb{R})$ denotes the unit ball in $W^r_{p,w}(\mathbb{R})$. The algorithms $Q_{\rho,m}$ and $P_{\rho,m}$ are based on truncated scaled B-spline quasi-interpolation and interpolation, respectively. We also prove the asymptotical optimality and right convergence rate of the equidistant quadratures generated from $Q_{\rho,m}$ and $P_{\rho,m}$, for Freud-weighted numerical integration of functions in $W^r_{p,w}(\mathbb{R})$.
arXiv:2501.00037v1 Announce Type: cross Abstract: Several numerical simulations of a co-axial particle-laden swirling air flow in a vertical circular pipe were performed. The air flow was modeled using the unsteady Favre-averaged Navier-Stokes equations. A Lagrangian model was used for the particle motion. The gas and particles are coupled through two-way momentum exchange. The results of the simulations using three versions of the k-epsilon turbulence model (standard, re-normalization group (RNG), and realizable) are compared with experimental mean velocity profiles. The standard model achieved the best overall performance. The realizable model was unable to satisfactorily predict the radial velocity; it is also the most computationally-expensive model. The simulations using the RNG model predicted additional recirculation zones. We also compared the particle and parcel approaches in solving the particle motion. In the latter, multiple similar particles are grouped in a single parcel, thereby reducing the amount of computation.
arXiv:2501.00677v1 Announce Type: new Abstract: Robust matrix completion (RMC) is a widely used machine learning tool that simultaneously tackles two critical issues in low-rank data analysis: missing data entries and extreme outliers. This paper proposes a novel scalable and learnable non-convex approach, coined Learned Robust Matrix Completion (LRMC), for large-scale RMC problems. LRMC enjoys low computational complexity with linear convergence. Motivated by the proposed theorem, the free parameters of LRMC can be effectively learned via deep unfolding to achieve optimum performance. Furthermore, this paper proposes a flexible feedforward-recurrent-mixed neural network framework that extends deep unfolding from fix-number iterations to infinite iterations. The superior empirical performance of LRMC is verified with extensive experiments against state-of-the-art on synthetic datasets and real applications, including video background subtraction, ultrasound imaging, face modeling, and cloud removal from satellite imagery.
arXiv:2501.00780v1 Announce Type: new Abstract: We introduce a novel meshless simulation method for the McKean-Vlasov Stochastic Differential Equation (MV-SDE) utilizing deep learning, applicable to both self-interaction and interaction scenarios. Traditionally, numerical methods for this equation rely on the interacting particle method combined with techniques based on the It\^o-Taylor expansion. The convergence rate of this approach is determined by two parameters: the number of particles $N$ and the time step size $h$ for each Euler iteration. However, for extended time horizons or equations with larger Lipschitz coefficients, this method is often limited, as it requires a significant increase in Euler iterations to achieve the desired precision $\epsilon$. To overcome the challenges posed by the difficulty of parallelizing the simulation of continuous interacting particle systems, which involve solving high-dimensional coupled SDEs, we propose a meshless MV-SDE solver grounded in Physics-Informed Neural Networks (PINNs) that does not rely on the propagation of chaos result. Our method constructs a pseudo MV-SDE using It\^o calculus, then quantifies the discrepancy between this equation and the original MV-SDE, with the error minimized through a loss function. This loss is controlled via an optimization algorithm, independent of the time step size, and we provide an error estimate for the loss function. The advantages of our approach are demonstrated through corresponding simulations.
arXiv:2501.00849v1 Announce Type: new Abstract: In this paper, we examine a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Stokes equations ($i.e.$, $p(\cdot,\cdot)$ is time- and space-dependent), employing a backward Euler step in time and conforming, discretely inf-sup stable finite elements in space. More precisely, we derive error decay rates for the vector-valued velocity field imposing fractional regularity assumptions on the velocity and the kinematic pressure. In addition, we carry out numerical experiments that confirm the optimality of the derived error decay rates in the case $p(\cdot,\cdot)\ge 2$.
arXiv:2501.00887v1 Announce Type: new Abstract: In this work, we develop a fast and accurate method for the scattering of flexural-gravity waves by a thin plate of varying thickness overlying a fluid of infinite depth. This problem commonly arises in the study of sea ice and ice shelves, which can have complicated heterogeneities that include ridges and rolls. With certain natural assumptions on the thickness, we present an integral equation formulation for solving this class of problems and analyze its mathematical properties. The integral equation is then discretized and solved using a high-order-accurate, FFT-accelerated algorithm. The speed, accuracy, and scalability of this approach are demonstrated through a variety of illustrative examples.
arXiv:2501.00938v1 Announce Type: new Abstract: We analyze overlapping multiplicative Schwarz methods as smoothers in the geometric multigrid solution of two-dimensional anisotropic diffusion problems. For diffusion equations, it is well known that the smoothing properties of point-wise smoothers, such as Gauss--Seidel, rapidly deteriorate as the strength of anisotropy increases. On the other hand, global smoothers based on line smoothing are known to generally provide good smoothing for diffusion problems, independent of the anisotropy strength. A natural question is whether global methods are really necessary to achieve good smoothing in such problems, or whether it can be obtained with locally overlapping block smoothers using sufficiently large blocks and overlap. Through local Fourier analysis and careful numerical experimentation, we show that global methods are indeed necessary to achieve anisotropy-robust smoothing. Specifically, for any fixed block size bounded sufficiently far away from the global domain size, we find that the smoothing properties of overlapping multiplicative Schwarz rapidly deteriorate with increasing anisotropy, irrespective of the amount of overlap between blocks. Moreover, our results indicate that anisotropy-robust smoothing requires blocks of diameter ${\cal O}(\epsilon^{-1/2})$ for anisotropy ratio $\epsilon \in (0,1]$.
arXiv:2501.00997v1 Announce Type: new Abstract: These lecture notes are intended to cover some introductory topics in stochastic simulation for scientific computing courses offered by the IT department at Uppsala University, as taught by the author. Basic concepts in probability theory are provided in the Appendix A, which you may review before starting the upcoming sections or refer to as needed throughout the text.
arXiv:2501.01199v1 Announce Type: new Abstract: Spectral differentiations are basic ingredients of spectral methods. In this work, we analyze the pointwise rate of convergence of spectral differentiations for functions containing singularities and show that the deteriorations of the convergence rate at the endpoints, singularities and other points in the smooth region exhibit different patterns. As the order of differentiation increases by one, we show for functions with an algebraic singularity that the convergence rate of spectral differentiation by Jacobi projection deteriorates two orders at both endpoints and only one order at each point in the smooth region. The situation at the singularity is more complicated and the convergence rate either deteriorates two orders or does not deteriorate, depending on the parity of the order of differentiation, when the singularity locates in the interior of the interval and deteriorates two orders when the singularity locates at the endpoint. Extensions to some related problems, such as the spectral differentiation using Chebyshev interpolation, are also discussed. Our findings justify the error localization property of Jacobi approximation and differentiation and provide some new insight into the convergence behavior of Jacobi spectral methods.
arXiv:2501.01299v1 Announce Type: new Abstract: Traditional linear approximation methods, such as proper orthogonal decomposition and the reduced basis method, are ineffective for transport-dominated problems due to the slow decay of the Kolmogorov $n$-width. This results in reduced-order models that are both inefficient and inaccurate. In this work, we present an approach for the model reduction of transport-dominated problems by employing cross-correlation based snapshot registration, accelerating the Kolmogorov $n$-width decay, and enabling the construction of efficient reduced-order models using linear methods. We propose a complete framework comprising offline-online stages for the development of reduced-order models using the cross-correlation based snapshots registration. The effectiveness of the proposed approach is demonstrated using two test cases: 1D travelling waves and the higher-order methods benchmark test case, 2D isentropic convective vortex.
arXiv:2501.00150v1 Announce Type: new Abstract: Quasi-Monte Carlo sampling can attain far better accuracy than plain Monte Carlo sampling. However, with plain Monte Carlo sampling it is much easier to estimate the attained accuracy. This article describes methods old and new to quantify the error in quasi-Monte Carlo estimates. An important challenge in this setting is that the goal of getting accuracy conflicts with that of estimating the attained accuracy. A related challenge is that rigorous uncertainty quantifications can be extremely conservative. A recent surprise is that some RQMC estimates have nearly symmetric distributions and that has the potential to allow confidence intervals that do not require either a central limit theorem or a consistent variance estimate.
arXiv:2501.00245v1 Announce Type: new Abstract: This paper proposes a theoretical framework for analyzing Modified Incomplete LU (MILU) preconditioners. Considering a generalized MILU preconditioner on a weighted undirected graph with self-loops, we extend its applicability beyond matrices derived by Poisson equation solvers on uniform grids with compact stencils. A major contribution is, a novel measure, the \textit{Localized Estimator of Condition Number (LECN)}, which quantifies the condition number locally at each vertex of the graph. We prove that the maximum value of the LECN provides an upper bound for the condition number of the MILU preconditioned system, offering estimation of the condition number using only local measurements. This localized approach significantly simplifies the condition number estimation and provides a powerful tool or analyzing the MILU preconditioner applied to previously unexplored matrix structures. To demonstrate the usability of LECN analysis, we present three cases: (1) revisit to existing results of MILU preconditioners on uniform grids, (2) analysis of high-order implicit finite difference schemes on wide stencils, and (3) analysis of variable coefficient Poisson equations on hierarchical adaptive grids such as quadtree and octree. For the third case, we also validate LECN analysis numerically on a quadtree.
arXiv:2501.00263v1 Announce Type: new Abstract: In this paper, we propose and implement a structure-preserving stochastic particle method for the Landau equation. The method is based on a particle system for the Landau equation, where pairwise grazing collisions are modeled as diffusion processes. By exploiting the unique structure of the particle system and a spherical Brownian motion sampling, the method avoids additional temporal discretization of the particle system, ensuring that the discrete-time particle distributions exactly match their continuous-time counterparts. The method achieves $O(N)$ complexity per time step and preserves fundamental physical properties, including the conservation of mass, momentum and energy, as well as entropy dissipation. It demonstrates strong long-time accuracy and stability in numerical experiments. Furthermore, we also apply the method to the spatially non-homogeneous equations through a case study of the Vlasov--Poisson--Landau equation.
arXiv:2501.00288v1 Announce Type: new Abstract: Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method. In this paper, we introduce a random feature based framework toward efficiently solving PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a shallow neural network as well. We provide an error analysis for our proposed method along with comprehensive numerical results on several PDE benchmarks. In contrast to the state-of-the-art solvers that face challenges with a large number of collocation points, our proposed method reduces the computational complexity. Moreover, the implementation of our method is simple and does not require additional computational resources. Due to the theoretical guarantee and advantages in computation, our approach is proven to be efficient for solving PDEs.
arXiv:2501.00345v1 Announce Type: new Abstract: In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data.
arXiv:2501.00389v1 Announce Type: cross Abstract: We study the momentum-based minimization of a diffuse perimeter functional on Euclidean spaces and on graphs with applications to semi-supervised classification tasks in machine learning. While the gradient flow in the task at hand is a parabolic partial differential equation, the momentum-method corresponds to a damped hyperbolic PDE, leading to qualitatively and quantitatively different trajectories. Using a convex-concave splitting-based FISTA-type time discretization, we demonstrate empirically that momentum can lead to faster convergence if the time step size is large but not too large. With large time steps, the PDE analysis offers only limited insight into the geometric behavior of solutions and typical hyperbolic phenomena like loss of regularity are not be observed in sample simulations.