Numerical Analysis
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Showing new listings for Friday, 15 November 2024
- [1] arXiv:2411.08986 [pdf, html, other]
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Title: A Priori Error Bounds and Parameter Scalings for the Time Relaxation Reduced Order ModelSubjects: Numerical Analysis (math.NA)
The a priori error analysis of reduced order models (ROMs) for fluids is relatively scarce. In this paper, we take a step in this direction and conduct numerical analysis of the recently introduced time relaxation ROM (TR-ROM), which uses spatial filtering to stabilize ROMs for convection-dominated flows. Specifically, we prove stability, an a priori error bound, and parameter scalings for the TR-ROM. Our numerical investigation shows that the theoretical convergence rate and the parameter scalings with respect to ROM dimension and filter radius are recovered numerically. In addition, the parameter scaling can be used to extrapolate the time relaxation parameter to other ROM dimensions and filter radii. Moreover, the parameter scaling with respect to filter radius is also observed in the predictive regime.
- [2] arXiv:2411.09029 [pdf, html, other]
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Title: Newton's Method Applied to Nonlinear Boundary Value Problems: A Numerical ApproachComments: 9 pages, 2 figuresSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling contexts, where an exact solution is often unfeasible due to the intrinsic complexity of these equations. Thus, a numerical approach is employed, using Newton's method to solve the system resulting from the discretization of the original problem. The procedure involves the iterative formulation of the method, which enables the approximation of solutions and the evaluation of convergence with respect to the problem parameters. The results demonstrate that Newton's method provides a robust and efficient solution, highlighting its applicability to complex boundary value problems and reinforcing its relevance for the numerical analysis of nonlinear systems. It is concluded that the methodology discussed is suitable for solving a wide range of boundary value problems, ensuring precision and stability in the results.
- [3] arXiv:2411.09090 [pdf, html, other]
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Title: A Vectorial Envelope Maxwell Formulation for Electromagnetic Waveguides with Application to Nonlinear Fiber OpticsSubjects: Numerical Analysis (math.NA)
This article presents an ultraweak discontinuous Petrov-Galerkin (DPG) formulation of the time-harmonic Maxwell equations for the vectorial envelope of the electromagnetic field in a weakly-guiding multi-mode fiber waveguide. This formulation is derived using an envelope ansatz for the vector-valued electric and magnetic field components, factoring out an oscillatory term of $exp(-i \mathsf{k}z)$ with a user-defined wavenumber $\mathsf{k}$, where $z$ is the longitudinal fiber axis and field propagation direction. The resulting formulation is a modified system of the time-harmonic Maxwell equations for the vectorial envelope of the propagating field. This envelope is less oscillatory in the $z$-direction than the original field, so that it can be more efficiently discretized and computed, enabling solution of the vectorial DPG Maxwell system for $1000\times$ longer fibers than previously possible. Different approaches for incorporating a perfectly matched layer for absorbing the outgoing wave modes at the fiber end are derived and compared numerically. The resulting formulation is used to solve a 3D Maxwell model of an ytterbium-doped active gain fiber amplifier, coupled with the heat equation for including thermal effects. The nonlinear model is then used to simulate thermally-induced transverse mode instability (TMI). The numerical experiments demonstrate that it is computationally feasible to perform simulations and analysis of real-length optical fiber laser amplifiers using discretizations of the full vectorial time-harmonic Maxwell equations. The approach promises a new high-fidelity methodology for analyzing TMI in high-power fiber laser systems and is extendable to including other nonlinearities.
- [4] arXiv:2411.09113 [pdf, html, other]
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Title: Convergence rates of Landweber-type methods for inverse problems in Banach spacesSubjects: Numerical Analysis (math.NA)
Landweber-type methods are prominent for solving ill-posed inverse problems in Banach spaces and their convergence has been well-understood. However, how to derive their convergence rates remains a challenging open question. In this paper, we tackle the challenge of deriving convergence rates for Landweber-type methods applied to ill-posed inverse problems, where forward operators map from a Banach space to a Hilbert space. Under a benchmark source condition, we introduce a novel strategy to derive convergence rates when the method is terminated by either an {\it a priori} stopping rule or the discrepancy principle. Our results offer substantial flexibility regarding step sizes, by allowing the use of variable step sizes. By extending the strategy to deal with the stochastic mirror descent method for solving nonlinear ill-posed systems with exact data, under a benchmark source condition we also obtain an almost sure convergence rate in terms of the number of iterations.
- [5] arXiv:2411.09162 [pdf, html, other]
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Title: An Asymptotic-Preserving Scheme for Isentropic Flow in Pipe NetworksSubjects: Numerical Analysis (math.NA)
We consider the simulation of isentropic flow in pipelines and pipe networks. Standard operating conditions in pipe networks suggest an emphasis to simulate low Mach and high friction regimes -- however, the system is stiff in these regimes and conventional explicit approximation techniques prove quite costly and often impractical. To combat these inefficiencies, we develop a novel asymptotic-preserving scheme that is uniformly consistent and stable for all Mach regimes. The proposed method for a single pipeline follows the flux splitting suggested in [Haack et al., Commun. Comput. Phys., 12 (2012), pp. 955--980], in which the flux is separated into stiff and non-stiff portions then discretized in time using an implicit-explicit approach. The non-stiff part is advanced in time by an explicit hyperbolic solver; we opt for the second-order central-upwind finite volume scheme. The stiff portion is advanced in time implicitly using an approach based on Rosenbrock-type Runge-Kutta methods, which ultimately reduces this implicit stage to a discretization of a linear elliptic equation.
To extend to full pipe networks, the scheme on a single pipeline is paired with coupling conditions defined at pipe-to-pipe intersections to ensure a mathematically well-posed problem. We show that the coupling conditions remain well-posed in the low Mach/high friction limit -- which, when used to define the ghost cells of each pipeline, results in a method that is accurate across these intersections in all regimes. The proposed method is tested on several numerical examples and produces accurate, non-oscillatory results with run times independent of the Mach number. - [6] arXiv:2411.09244 [pdf, html, other]
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Title: Parallel in time partially explicit splitting scheme for high contrast multiscale problemsSubjects: Numerical Analysis (math.NA)
Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting scheme is proposed. By appropriately constructing multiscale spaces, the spatial multiscale property is effectively managed, and it has been demonstrated that the temporal step size is independent of the contrast. To enhance simulation speed, we propose a parallel algorithm for the multiscale flow problem that leverages the partially explicit temporal splitting scheme. The idea is first to evolve the partially explicit system using a coarse time step size, then correct the solution on each coarse time interval with a fine propagator, for which we consider both the sequential solver and all-at-once solver. This procedure is then performed iteratively till convergence. We analyze the stability and convergence of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm achieves high numerical accuracy for high-contrast problems and converges in a relatively small number of iterations. The number of iterations stays stable as the number of coarse intervals increases, thus significantly improving computational efficiency through parallel processing.
- [7] arXiv:2411.09285 [pdf, other]
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Title: Existence of solutions to numerical schemes using regularization: application to two-phase flow in porous media schemesThomas Crozon (LMJL, Nantes Univ - ECN)Subjects: Numerical Analysis (math.NA)
The present document corresponds to the 4 th chapter of my thesis, the problem setting is not definitive, what matters most here are the mathematical results and the methodology of the existence proofs. In this work, we propose a framework and some tools for establishing the existence of solutions to numerical schemes in the case of the two-phase flow model. These schemes are sharing some key a priori mathematical properties. It applies to a large variety of continuous models. We propose the definition of a regularized scheme and show that if solutions exist to this regularization, then the existence of the initial one is ensured. This perturbation of the scheme facilitates the regularized existence. The main aim is to handle degenerate systems such as the two-phase Darcy flows in porous media. We illustrate the strength of our framework on two practical schemes, a finite volume one using the DDFV framework, and the other based on a Control Volume Finite Element (CVFE) method.
- [8] arXiv:2411.09314 [pdf, other]
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Title: Theory of the lattice Boltzmann method: discrete effects due to advectionSubjects: Numerical Analysis (math.NA)
Lattice Boltzmann models are briefly introduced together with references to methods used to predict their ability for simulations of systems described by partial differential equations that are first order in time and low order in space derivatives. Several previous works have been devoted to analyzing the accuracy of these models with special emphasis on deviations from pure Newtonian viscous behaviour, related to higher order space derivatives of even order. The presentcontribution concentrates on possible inaccuracies of the advection behaviour linked to space derivatives of odd order. Detailed properties of advection-diffusion and athermal fluids are presented for two-dimensional situations allowing to propose situations that are accurate to third order in space derivatives. Simulations of the advection of a gaussian dot or vortex are presented. Similar results are discussed in appendices for three-dimensional advection-diffusion.
- [9] arXiv:2411.09329 [pdf, html, other]
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Title: Improving hp-Variational Physics-Informed Neural Networks for Steady-State Convection-Dominated ProblemsComments: 25 pages, 11 figures, 8 tablesSubjects: Numerical Analysis (math.NA); Computational Engineering, Finance, and Science (cs.CE); Machine Learning (cs.LG)
This paper proposes and studies two extensions of applying hp-variational physics-informed neural networks, more precisely the FastVPINNs framework, to convection-dominated convection-diffusion-reaction problems. First, a term in the spirit of a SUPG stabilization is included in the loss functional and a network architecture is proposed that predicts spatially varying stabilization parameters. Having observed that the selection of the indicator function in hard-constrained Dirichlet boundary conditions has a big impact on the accuracy of the computed solutions, the second novelty is the proposal of a network architecture that learns good parameters for a class of indicator functions. Numerical studies show that both proposals lead to noticeably more accurate results than approaches that can be found in the literature.
- [10] arXiv:2411.09444 [pdf, html, other]
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Title: Learning efficient and provably convergent splitting methodsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
Splitting methods are widely used for solving initial value problems (IVPs) due to their ability to simplify complicated evolutions into more manageable subproblems which can be solved efficiently and accurately. Traditionally, these methods are derived using analytic and algebraic techniques from numerical analysis, including truncated Taylor series and their Lie algebraic analogue, the Baker--Campbell--Hausdorff formula. These tools enable the development of high-order numerical methods that provide exceptional accuracy for small timesteps. Moreover, these methods often (nearly) conserve important physical invariants, such as mass, unitarity, and energy. However, in many practical applications the computational resources are limited. Thus, it is crucial to identify methods that achieve the best accuracy within a fixed computational budget, which might require taking relatively large timesteps. In this regime, high-order methods derived with traditional methods often exhibit large errors since they are only designed to be asymptotically optimal. Machine Learning techniques offer a potential solution since they can be trained to efficiently solve a given IVP with less computational resources. However, they are often purely data-driven, come with limited convergence guarantees in the small-timestep regime and do not necessarily conserve physical invariants. In this work, we propose a framework for finding machine learned splitting methods that are computationally efficient for large timesteps and have provable convergence and conservation guarantees in the small-timestep limit. We demonstrate numerically that the learned methods, which by construction converge quadratically in the timestep size, can be significantly more efficient than established methods for the Schrödinger equation if the computational budget is limited.
- [11] arXiv:2411.09485 [pdf, other]
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Title: Exact Integration for singular Zienkiewicz and Guzman-Neilan Finite Elements with ImplementationSubjects: Numerical Analysis (math.NA)
We develop a recursive integration formula for a class of rational polynomials in 2D. Based on this, we present implementations of finite elements that have rational basis functions. Specifically, we provide simple Matlab implementations of the singular Zienkiewicz and the lowest-order Guzman-Neilan finite element in 2D.
- [12] arXiv:2411.09498 [pdf, html, other]
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Title: Analysis and discretization of the Ohta-Kawasaki equation with forcing and degenerate mobilitySubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
The Ohta-Kawasaki equation models the mesoscopic phase separation of immiscible polymer chains that form diblock copolymers, with applications in directed self-assembly for lithography. We perform a mathematical analysis of this model under degenerate mobility and an external force, proving the existence of weak solutions via an approximation scheme for the mobility function. Additionally, we propose a fully discrete scheme for the system and demonstrate the existence and uniqueness of its discrete solution, showing that it inherits essential structural-preserving properties. Finally, we conduct numerical experiments to compare the Ohta-Kawasaki system with the classical Cahn-Hilliard model, highlighting the impact of the repulsion parameter on the phase separation dynamics.
- [13] arXiv:2411.09504 [pdf, html, other]
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Title: Asymptotic Analysis of IMEX-RK Methods for ES-BGK Model at Navier-Stokes levelSubjects: Numerical Analysis (math.NA)
Implicit-explicit Runge-Kutta (IMEX-RK) time discretization methods are very popular when solving stiff kinetic equations. In [21], an asymptotic analysis shows that a specific class of high-order IMEX-RK schemes can accurately capture the Navier-Stokes limit without needing to resolve the small scales dictated by the Knudsen number. In this work, we extend the asymptotic analysis to general IMEX-RK schemes, known in literature as Type I and Type II. We further suggest some IMEX-RK methods developed in the literature to attain uniform accuracy in the wide range of Knudsen numbers. Several numerical examples are presented to verify the validity of the obtained theoretical results and the effectiveness of the methods.
- [14] arXiv:2411.09509 [pdf, html, other]
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Title: Enhanced HLLEM and HLL-CPS schemes for all Mach number flows based using anti-diffusion coefficientsSubjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
This paper compares the HLLEM and HLL-CPS schemes for Euler equations and proposes improvements for all Mach number flows. Enhancements to the HLLEM scheme involve adding anti-diffusion terms in the face normal direction and modifying anti-diffusion coefficients for linearly degenerate waves near shocks. The HLL-CPS scheme is improved by adjusting anti-diffusion coefficients for the face normal direction and linearly degenerate waves. Matrix stability, linear perturbation, and asymptotic analyses demonstrate the robustness of the proposed schemes and their ability to capture low Mach flow features. Numerical tests confirm that the schemes are free from shock instabilities at high speeds and accurately resolve low Mach number flow features.
- [15] arXiv:2411.09525 [pdf, html, other]
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Title: Data-driven parameterization refinement for the structural optimization of cruise ship hullsSubjects: Numerical Analysis (math.NA)
In this work, we focus on the early design phase of cruise ship hulls, where the designers are tasked with ensuring the structural resilience of the ship against extreme waves while reducing steel usage and respecting safety and manufacturing constraints. The ship's geometry is already finalized and the designer can choose the thickness of the primary structural elements, such as decks, bulkheads, and the shell. Reduced order modeling and black-box optimization techniques reduce the use of expensive finite element analysis to only validate the most promising configurations, thanks to the efficient exploration of the domain of decision variables. However, the quality of the results heavily relies on the problem formulation, and on how the structural elements are assigned to the decision variables. A parameterization that does not capture well the stress configuration of the model prevents the optimization procedure from achieving the most efficient allocation of the steel. To address this issue, we extended an existing pipeline for the structural optimization of cruise ships developed in collaboration with Fincantieri S.p.A. with a novel data-driven reparameterization procedure, based on the optimization of a series of sub-problems. Moreover, we implemented a multi-objective optimization module to provide the designers with insights into the efficient trade-offs between competing quantities of interest and enhanced the single-objective Bayesian optimization module. The new pipeline is tested on a simplified midship section and a full ship hull, comparing the automated reparameterization to a baseline model provided by the designers. The tests show that the iterative refinement outperforms the baseline on the more complex hull, proving that the pipeline streamlines the initial design phase, and helps the designers tackle more innovative projects.
- [16] arXiv:2411.09583 [pdf, other]
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Title: A Nonuniform Fast Hankel TransformSubjects: Numerical Analysis (math.NA)
We describe a fast algorithm for computing discrete Hankel transforms of moderate orders from $n$ nonuniform points to $m$ nonuniform frequencies in $O((m+n)\log\min(n,m))$ operations. Our approach combines local and asymptotic Bessel function expansions with nonuniform fast Fourier transforms. The order of each expansion is adjusted automatically according to error analysis to obtain any desired precision $\varepsilon$. Several numerical examples are provided which demonstrate the speed and accuracy of the algorithm in multiple regimes and applications.
- [17] arXiv:2411.09584 [pdf, html, other]
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Title: A Sylvester equation approach for the computation of zero-group-velocity points in waveguidesComments: 18 pagesSubjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
Eigenvalues of parameter-dependent quadratic eigenvalue problems form eigencurves. The critical points on these curves, where the derivative vanishes, are of practical interest.
A particular example is found in the dispersion curves of elastic waveguides, where such points are called zero-group-velocity (ZGV) points. Recently, it was revealed that the problem of computing ZGV points can be modeled as a multiparameter eigenvalue problem (MEP), and several numerical methods were devised. Due to their complexity, these methods are feasible only for problems involving small matrices. In this paper, we improve the efficiency of these methods by exploiting the link to the Sylvester equation. This approach enables the computation of ZGV points for problems with much larger matrices, such as multi-layered plates and three-dimensional structures of complex cross-sections. - [18] arXiv:2411.09605 [pdf, html, other]
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Title: An explicit, energy-conserving particle-in-cell schemeSubjects: Numerical Analysis (math.NA); Plasma Physics (physics.plasm-ph)
We present an explicit temporal discretization of particle-in-cell schemes for the Vlasov equation that results in exact energy conservation when combined with an appropriate spatial discretization. The scheme is inspired by a simple, second-order explicit scheme that conserves energy exactly in the Eulerian context. We show that direct translation to particle-in-cell does not result in strict conservation, but derive a simple correction based on an analytically solvable optimization problem that recovers conservation. While this optimization problem is not guaranteed to have a real solution for every particle, we provide a correction that makes imaginary values extremely rare and still admits $\mathcal{O}(10^{-12})$ fractional errors in energy for practical simulation parameters. We present the scheme in both electrostatic -- where we use the Ampère formulation -- and electromagnetic contexts. With an electromagnetic field solve, the field update is most naturally linearly implicit, but the more computationally intensive particle update remains fully explicit. We also show how the scheme can be extended to use the fully explicit leapfrog and pseudospectral analytic time-domain (PSATD) field solvers. The scheme is tested on standard kinetic plasma problems, confirming its conservation properties.
- [19] arXiv:2411.09617 [pdf, other]
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Title: Riemannian optimisation methods for ground states of multicomponent Bose-Einstein condensatesSubjects: Numerical Analysis (math.NA)
This paper addresses the computation of ground states of multicomponent Bose-Einstein condensates, defined as the global minimiser of an energy functional on an infinite-dimensional generalised oblique manifold. We establish the existence of the ground state, prove its uniqueness up to scaling, and characterise it as the solution to a coupled nonlinear eigenvector problem. By equipping the manifold with several Riemannian metrics, we introduce a suite of Riemannian gradient descent and Riemannian Newton methods. Metrics that incorporate first- or second-order information about the energy are particularly advantageous, effectively preconditioning the resulting methods. For a Riemannian gradient descent method with an energy-adaptive metric, we provide a qualitative global and quantitative local convergence analysis, confirming its reliability and robustness with respect to the choice of the spatial discretisation. Numerical experiments highlight the computational efficiency of both the Riemannian gradient descent and Newton methods.
New submissions (showing 19 of 19 entries)
- [20] arXiv:2411.08905 (cross-list from cs.CE) [pdf, html, other]
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Title: Synthesis Method for Obtaining Characteristic Modes of Multi-Structure SystemsSubjects: Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA)
This paper introduces an efficient method of characteristic mode decomposition for multi-structure systems. Our approach leverages the translation and rotation matrices associated with vector spherical wavefunctions, enabling the synthesis of a total system's characteristic modes through independent simulation of each constituent structure. We simplify the computationally demanding translation problem by dividing it into three manageable sub-tasks: rotation, z-axis translation, and inverse rotation, which collectively enhance computational efficiency. Furthermore, this method facilitates the exploration of structural orientation effects without incurring additional computational overhead. To demonstrate the effectiveness of our approach, we present a series of compelling numerical examples that not only validate the accuracy of the method but also highlight its significant advantages.
- [21] arXiv:2411.09038 (cross-list from quant-ph) [pdf, html, other]
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Title: An Implementation of the Finite Element Method in Hybrid Classical/Quantum ComputersSubjects: Quantum Physics (quant-ph); Numerical Analysis (math.NA)
This manuscript presents the Quantum Finite Element Method (Q-FEM) developed for use in noisy intermediate-scale quantum (NISQ) computers, and employs the variational quantum linear solver (VQLS) algorithm. The proposed method leverages the classical FEM procedure to perform the unitary decomposition of the stiffness matrix and employs generator functions to design explicit quantum circuits corresponding to the unitaries. Q-FEM keeps the structure of the finite element discretization intact allowing for the use of variable element lengths and material coefficients in FEM discretization. The proposed method is tested on a steady-state heat equation discretized using linear and quadratic shape functions. Numerical verification studies demonstrate that Q-FEM is effective in converging to the correct solution for a variety of problems and model discretizations, including with different element lengths, variable coefficients, and different boundary conditions. The formalism developed herein is general and can be extended to problems with higher dimensions. However, numerical examples also demonstrate that the number of parameters for the variational ansatz scale exponentially with the number of qubits to increase the odds of convergence, and deterioration of system conditioning with problem size results in barren plateaus, and hence convergence difficulties.
- [22] arXiv:2411.09511 (cross-list from math.AP) [pdf, html, other]
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Title: Structure-informed operator learning for parabolic Partial Differential EquationsComments: 19 pagesSubjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA); Probability (math.PR)
In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of functions. We utilize Fréchet space neural networks (Benth et al. (2023)) to address this operator learning problem. Our approach provides an alternative to Deep Operator Networks (DeepONets), using basis functions to span the relevant function spaces rather than relying on finite-dimensional approximations through censoring. With this method, structural information encoded in the basis coefficients is leveraged in the learning process. This results in a neural network designed to learn the mapping between infinite-dimensional function spaces. Our numerical proof-of-concept demonstrates the effectiveness of our method, highlighting some advantages over DeepONets.
- [23] arXiv:2411.09530 (cross-list from math-ph) [pdf, html, other]
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Title: Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanicsSubjects: Mathematical Physics (math-ph); Differential Geometry (math.DG); Dynamical Systems (math.DS); Numerical Analysis (math.NA)
In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically, we develop $(\pm)$-discrete induced Dirac structures as discrete analogues of the induced Dirac structure on the cotangent bundle over a configuration manifold, as described by Yoshimura and Marsden (2006). We demonstrate that $(\pm)$-discrete Lagrange--Dirac systems can be naturally formulated in conjunction with the $(\pm)$-induced Dirac structure on the cotangent bundle. Furthermore, we show that the resulting equations of motion are equivalent to the $(\pm)$-discrete Lagrange--d'Alembert equations proposed in Cortés and Martínez (2001) and McLachlan and Perlmutter (2006). We also clarify the variational structures of the discrete Lagrange--Dirac dynamical systems within the framework of the $(\pm)$-discrete Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed discrete Lagrange--Dirac systems with some illustrative examples of nonholonomic systems through numerical tests.
- [24] arXiv:2411.09555 (cross-list from cs.CV) [pdf, html, other]
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Title: Image Processing for Motion MagnificationNadaniela Egidi, Josephin Giacomini, Paolo Leonesi, Pierluigi Maponi, Federico Mearelli, Edin TrebovicSubjects: Computer Vision and Pattern Recognition (cs.CV); Numerical Analysis (math.NA)
Motion Magnification (MM) is a collection of relative recent techniques within the realm of Image Processing. The main motivation of introducing these techniques in to support the human visual system to capture relevant displacements of an object of interest; these motions can be in object color and in object location. In fact, the goal is to opportunely process a video sequence to obtain as output a new video in which motions are magnified and visible to the viewer. We propose a numerical technique using the Phase-Based Motion Magnification which analyses the video sequence in the Fourier Domain and rely on the Fourier Shifting Property. We describe the mathematical foundation of this method and the corresponding implementation in a numerical algorithm. We present preliminary experiments, focusing on some basic test made up using synthetic images.
- [25] arXiv:2411.09644 (cross-list from math.OC) [pdf, html, other]
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Title: Neural Operators Can Play Dynamic Stackelberg GamesSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Numerical Analysis (math.NA); Probability (math.PR); Computational Finance (q-fin.CP)
Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the \textit{follower's best-response operator} can be approximately implemented by an \textit{attention-based neural operator}, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.
Cross submissions (showing 6 of 6 entries)
- [26] arXiv:2306.06383 (replaced) [pdf, html, other]
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Title: Using orthogonally structured positive bases for constructing positive $k$-spanning sets with cosine measure guaranteesComments: V4 fixes a typo in the arXiv abstractSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such a set covers the space of interest. In this paper, we investigate the construction of positive $k$-spanning sets with geometrical guarantees. Our results build on recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures efficiently. We then focus our study on positive $k$-spanning sets, for which we provide a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets. By combining our results, we are able to use orthogonally structured positive bases to create positive $k$-spanning sets with guarantees on the value of their cosine measures.
- [27] arXiv:2311.00534 (replaced) [pdf, html, other]
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Title: Error analysis for a finite element approximation of the steady $p(\cdot)$-Navier-Stokes equationsComments: 37 pages, 4 figures, 4 tablesSubjects: Numerical Analysis (math.NA)
In this paper, we examine a finite element approximation of the steady $p(\cdot)$-Navier-Stokes equations ($p(\cdot)$ is variable dependent) and prove orders of convergence by assuming natural fractional regularity assumptions on the velocity vector field and the kinematic pressure. Compared to previous results, we treat the convective term and employ a more practicable discretization of the power-law index $p(\cdot)$. Numerical experiments confirm the quasi-optimality of the $\textit{a priori}$ error estimates (for the velocity) with respect to fractional regularity assumptions on the velocity vector field and the kinematic pressure.
- [28] arXiv:2312.11166 (replaced) [pdf, html, other]
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Title: Volume-Preserving Transformers for Learning Time Series Data with StructureComments: Will be published as part of "Cemracs Proceedings 2023" (status: accepted)Subjects: Numerical Analysis (math.NA); Machine Learning (cs.LG)
Two of the many trends in neural network research of the past few years have been (i) the learning of dynamical systems, especially with recurrent neural networks such as long short-term memory networks (LSTMs) and (ii) the introduction of transformer neural networks for natural language processing (NLP) tasks.
While some work has been performed on the intersection of these two trends, those efforts were largely limited to using the vanilla transformer directly without adjusting its architecture for the setting of a physical system.
In this work we develop a transformer-inspired neural network and use it to learn a dynamical system. We (for the first time) change the activation function of the attention layer to imbue the transformer with structure-preserving properties to improve long-term stability. This is shown to be of great advantage when applying the neural network to learning the trajectory of a rigid body. - [29] arXiv:2404.18069 (replaced) [pdf, other]
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Title: Local Discontinuous Galerkin method for fractional Korteweg-de Vries equationComments: There are some incomplete estimates and require to work on it. It may lead to incorrect justificationSubjects: Numerical Analysis (math.NA)
We propose a local discontinuous Galerkin (LDG) method for fractional Korteweg-de Vries equation involving the fractional Laplacian with exponent $\alpha\in (1,2)$ in one and two space dimensions. By decomposing the fractional Laplacian into a first order derivative and a fractional integral, we prove $L^2$-stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We analyze the error estimate by considering linear convection term and utilizing the estimate, we derive the error estimate for general nonlinear flux and demonstrate an order of convergence $\mathcal{O}(h^{k+1/2})$. Moreover, the stability and error analysis have been extended to multiple space dimensional case. Numerical illustrations are shown to demonstrate the efficiency of the scheme by obtaining an optimal order of convergence.
- [30] arXiv:2406.02458 (replaced) [pdf, other]
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Title: Deep Block Proximal Linearised Minimisation Algorithm for Non-convex Inverse ProblemsChaoyan Huang, Zhongming Wu, Yanqi Cheng, Tieyong Zeng, Carola-Bibiane Schönlieb, Angelica I. Aviles-RiveroComments: 6 figures, 3 tablesSubjects: Numerical Analysis (math.NA)
Image restoration is typically addressed through non-convex inverse problems, which are often solved using first-order block-wise splitting methods. In this paper, we consider a general type of non-convex optimisation model that captures many inverse image problems and present an inertial block proximal linearised minimisation (iBPLM) algorithm. Our new method unifies the Jacobi-type parallel and the Gauss-Seidel-type alternating update rules, and extends beyond these approaches. The inertial technique is also incorporated into each block-wise subproblem update, which can accelerate numerical convergence. Furthermore, we extend this framework with a plug-and-play variant (PnP-iBPLM) that integrates deep gradient denoisers, offering a flexible and robust solution for complex imaging tasks. We provide comprehensive theoretical analysis, demonstrating both subsequential and global convergence of the proposed algorithms. To validate our methods, we apply them to multi-block dictionary learning problems in image denoising and deblurring. Experimental results show that both iBPLM and PnP-iBPLM significantly enhance numerical performance and robustness in these applications.
- [31] arXiv:2408.07691 (replaced) [pdf, html, other]
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Title: A family of high-order accurate contour integral methods for strongly continuous semigroupsSubjects: Numerical Analysis (math.NA)
Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and lead to global-in-time approximations that can be efficiently evaluated anywhere within a finite time horizon. However, there are core theoretical challenges that restrict their use cases to analytic semigroups, e.g., parabolic equations. In this article, we use carefully regularized contour integral representations to construct a family of new high-order quadrature schemes for the larger, less regular, class of strongly continuous semigroups. Our algorithms are accompanied by explicit high-order error bounds and near-optimal parameter selection. We demonstrate key features of the schemes on singular first-order PDEs from Koopman operator theory.
- [32] arXiv:2410.23061 (replaced) [pdf, html, other]
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Title: Learned RESESOP for solving inverse problems with inexact forward operatorComments: 21 pages, 7 figures, 4 tablesSubjects: Numerical Analysis (math.NA)
When solving inverse problems, one has to deal with numerous potential sources of model inexactnesses, like object motion, calibration errors, or simplified data models. Regularized Sequential Subspace Optimization (ReSeSOp) allows to compensate for such inaccuracies within the reconstruction step by employing consecutive projections onto suitably defined subspaces. However, this approach relies on a priori estimates for the model inexactness levels which are typically unknown. In dynamic imaging applications, where inaccuracies arise from the unpredictable dynamics of the object, these estimates are particularly challenging to determine in advance. To overcome this limitation, we propose a learned version of ReSeSOp which allows to approximate inexactness levels on the fly. The proposed framework generalizes established unrolled iterative reconstruction schemes to inexact forward operators and is particularly tailored to the structure of dynamic problems. We also present a comprehensive mathematical analysis regarding the effect of dependencies within the forward problem, clarifying when and why dividing the overall problem into subproblems is essential. The proposed method is evaluated on various examples from dynamic imaging, including datasets from a rheological CT experiment, brain MRI, and real-time cardiac MRI. The respective results emphasize improvements in reconstruction quality while ensuring adequate data consistency.
- [33] arXiv:2411.08702 (replaced) [pdf, html, other]
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Title: A Deep Uzawa-Lagrange Multiplier Approach for Boundary Conditions in PINNs and Deep Ritz MethodsComments: 19 pages, 13 figures; updated address of authorSubjects: Numerical Analysis (math.NA)
We introduce a deep learning-based framework for weakly enforcing boundary conditions in the numerical approximation of partial differential equations. Building on existing physics-informed neural network and deep Ritz methods, we propose the Deep Uzawa algorithm, which incorporates Lagrange multipliers to handle boundary conditions effectively. This modification requires only a minor computational adjustment but ensures enhanced convergence properties and provably accurate enforcement of boundary conditions, even for singularly perturbed problems.
We provide a comprehensive mathematical analysis demonstrating the convergence of the scheme and validate the effectiveness of the Deep Uzawa algorithm through numerical experiments, including high-dimensional, singularly perturbed problems and those posed over non-convex domains.