Numerical Analysis
New submissions
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New submissions for Fri, 3 May 24
- [1] arXiv:2405.00803 [pdf, other]
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Title: A perturbative analysis for noisy spectral estimationAuthors: Lexing YingSubjects: Numerical Analysis (math.NA)
Spectral estimation is a fundamental task in signal processing. Recent algorithms in quantum phase estimation are concerned with the large noise, large frequency regime of the spectral estimation problem. The recent work in Ding-Epperly-Lin-Zhang shows that the ESPRIT algorithm exhibits superconvergence behavior for the spike locations in terms of the maximum frequency. This note provides a perturbative analysis to explain this behavior. It also extends the discussion to the case where the noise grows with the sampling frequency.
- [2] arXiv:2405.00806 [pdf, other]
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Title: Strong convergence of the exponential Euler scheme for SDEs with superlinear growth coefficients and one-sided Lipschitz driftSubjects: Numerical Analysis (math.NA); Probability (math.PR)
We consider the problem of the discrete-time approximation of the solution of a one-dimensional SDE with piecewise locally Lipschitz drift and continuous diffusion coefficients with polynomial growth. In this paper, we study the strong convergence of a (semi-explicit) exponential-Euler scheme previously introduced in Bossy et al. (2021). We show the usual 1/2 rate of convergence for the exponential-Euler scheme when the drift is continuous. When the drift is discontinuous, the convergence rate is penalised by a factor {$\epsilon$} decreasing with the time-step. We examine the case of the diffusion coefficient vanishing at zero, which adds a positivity preservation condition and a convergence analysis that exploits the negative moments and exponential moments of the scheme with the help of change of time technique introduced in Berkaoui et al. (2008). Asymptotic behaviour and theoretical stability of the exponential scheme, as well as numerical experiments, are also presented.
- [3] arXiv:2405.00815 [pdf, other]
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Title: Extended Galerkin neural network approximation of singular variational problems with error controlSubjects: Numerical Analysis (math.NA)
We present extended Galerkin neural networks (xGNN), a variational framework for approximating general boundary value problems (BVPs) with error control. The main contributions of this work are (1) a rigorous theory guiding the construction of new weighted least squares variational formulations suitable for use in neural network approximation of general BVPs (2) an ``extended'' feedforward network architecture which incorporates and is even capable of learning singular solution structures, thus greatly improving approximability of singular solutions. Numerical results are presented for several problems including steady Stokes flow around re-entrant corners and in convex corners with Moffatt eddies in order to demonstrate efficacy of the method.
- [4] arXiv:2405.00904 [pdf, other]
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Title: Adjoint-based goal-oriented implicit shock tracking using full space mesh optimizationComments: 46 pages, 30 figuresSubjects: Numerical Analysis (math.NA)
Solutions to the governing partial differential equations obtained from a discrete numerical scheme can have significant errors, especially near shocks when the discrete representation of the solution cannot fully capture the discontinuity in the solution. A recent approach to shock tracking [1, 2] has been to implicitly align the faces of mesh elements with the shock, yielding accurate solutions on coarse meshes. In engineering applications, the solution field is often used to evaluate a scalar functional of interest, such as lift or drag over an airfoil. While functionals are sensitive to errors in the flow solution, certain regions in the domain are more important for accurate evaluation of the functional than the rest. Using this fact, we formulate a goal-oriented implicit shock tracking approach that captures a segment of the shock that is important for evaluating the functional. Shock tracking is achieved using Lagrange-Newton-Krylov-Schur (LNKS) full space optimizer, with the objective of minimizing the adjoint-weighted residual error indicator. We also present a method to evaluate the sensitivity and the Hessian of the functional error. Using available block preconditioners for LNKS [3, 4] makes the full space approach scalable. The method is applied to test cases of two-dimensional advection and inviscid compressible flows to demonstrate functional-dependent shock tracking. Tracking the entire shock without using artificial dissipation results in the error converging at the orders of $\mathcal{O}(h^{p+1})$.
- [5] arXiv:2405.00933 [pdf, ps, other]
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Title: Efficient Computation for Invertibility Sequence of Banded Toeplitz MatricesSubjects: Numerical Analysis (math.NA)
When solving systems of banded Toeplitz equations or calculating their inverses, it is necessary to determine the invertibility of the matrices beforehand. In this paper, we equate the invertibility of an $n$-order banded Toeplitz matrix with bandwidth $2k+1$ to that of a small $k*k$ matrix. By utilizing a specially designed algorithm, we compute the invertibility sequence of a class of banded Toeplitz matrices with a time complexity of $5k^2n/2+kn$ and a space complexity of $3k^2$ where $n$ is the size of the largest matrix. This enables efficient preprocessing when solving equation systems and inverses of banded Toeplitz matrices.
- [6] arXiv:2405.01079 [pdf, other]
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Title: Singular Value and Frame Decomposition-based Reconstruction for Atmospheric TomographyComments: 29 pages, 10 figuresSubjects: Numerical Analysis (math.NA)
Atmospheric tomography, the problem of reconstructing atmospheric turbulence profiles from wavefront sensor measurements, is an integral part of many adaptive optics systems used for enhancing the image quality of ground-based telescopes. Singular-value and frame decompositions of the underlying atmospheric tomography operator can reveal useful analytical information on this inverse problem, as well as serve as the basis of efficient numerical reconstruction algorithms. In this paper, we extend existing singular value decompositions to more realistic Sobolev settings including weighted inner products, and derive an explicit representation of a frame-based (approximate) solution operator. These investigations form the basis of efficient numerical solution methods, which we analyze via numerical simulations for the challenging, real-world Adaptive Optics system of the Extremely Large Telescope using the entirely MATLAB-based simulation tool MOST.
- [7] arXiv:2405.01082 [pdf, other]
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Title: A reduced scalar potential approach for magnetostatics avoiding the coenergySubjects: Numerical Analysis (math.NA)
The numerical solution of problems in nonlinear magnetostatics is typically based on a variational formulation in terms of magnetic potentials, the discretization by finite elements, and iterative solvers like the Newton method. The vector potential approach aims at minimizing a certain energy functional and, in three dimensions, requires the use of edge elements and appropriate gauging conditions. The scalar potential approach, on the other hand, seeks to maximize the negative coenergy and can be realized by standard Lagrange finite elements, thus reducing the number of degrees of freedom and simplifying the implementation. The number of Newton iterations required to solve the governing nonlinear system, however, has been observed to be usually higher than for the vector potential formulation. In this paper, we propose a method that combines the advantages of both approaches, i.e., it requires as few Newton iterations as the vector potential formulation while involving the magnetic scalar potential as the primary unknown. We discuss the variational background of the method, its well-posedness, and its efficient implementation. Numerical examples are presented for illustration of the accuracy and the gain in efficiency compared to other approaches.
- [8] arXiv:2405.01109 [pdf, other]
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Title: Hypergraph $p$-Laplacian regularization on point clouds for data interpolationComments: 33 pagesSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Analysis of PDEs (math.AP)
As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the $\varepsilon_n$-ball hypergraph and the $k_n$-nearest neighbor hypergraph on a point cloud and study the $p$-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph $p$-Laplacian regularization and the continuum $p$-Laplacian regularization in a semisupervised setting when the number of points $n$ goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of $\varepsilon_n$ and $k_n$. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal-dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph $p$-Laplacian regularization outperforms the graph $p$-Laplacian regularization in preventing the development of spikes at the labeled points.
- [9] arXiv:2405.01298 [pdf, other]
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Title: Reorthogonalized Pythagorean variants of block classical Gram-SchmidtSubjects: Numerical Analysis (math.NA)
Block classical Gram-Schmidt (BCGS) is commonly used for orthogonalizing a set of vectors $X$ in distributed computing environments due to its favorable communication properties relative to other orthogonalization approaches, such as modified Gram-Schmidt or Householder. However, it is known that BCGS (as well as recently developed low-synchronization variants of BCGS) can suffer from a significant loss of orthogonality in finite-precision arithmetic, which can contribute to instability and inaccurate solutions in downstream applications such as $s$-step Krylov subspace methods. A common solution to improve the orthogonality among the vectors is reorthogonalization. Focusing on the "Pythagorean" variant of BCGS, introduced in [E. Carson, K. Lund, & M. Rozlo\v{z}n\'{i}k. SIAM J. Matrix Anal. Appl. 42(3), pp. 1365--1380, 2021], which guarantees an $O(\varepsilon)\kappa^2(X)$ bound on the loss of orthogonality as long as $O(\varepsilon)\kappa^2(X)<1$, where $\varepsilon$ denotes the unit roundoff, we introduce and analyze two reorthogonalized Pythagorean BCGS variants. These variants feature favorable communication properties, with asymptotically two synchronization points per block column, as well as an improved $O(\varepsilon)$ bound on the loss of orthogonality. Our bounds are derived in a general fashion to additionally allow for the analysis of mixed-precision variants. We verify our theoretical results with a panel of test matrices and experiments from a new version of the \texttt{BlockStab} toolbox.
- [10] arXiv:2405.01346 [pdf, other]
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Title: Improved weak convergence for the long time simulation of Mean-field Langevin equationsComments: 68 pages, 1 FigureSubjects: Numerical Analysis (math.NA); Probability (math.PR)
We study the weak convergence behaviour of the Leimkuhler--Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean--Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption.
Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate $3/2$) than the standard Euler method (of weak order $1$). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests. - [11] arXiv:2405.01406 [pdf, other]
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Title: Reduced Order Modeling for Real-Time Monitoring of Structural Displacements due to Electromagnetic Forces in Large Scale TokamaksAuthors: Francesco Lucchini, Alessandro Frescura, Riccardo Torchio, Piergiorgio Alotto, Paolo BettiniSubjects: Numerical Analysis (math.NA)
The real-time monitoring of the structural displacement of the Vacuum Vessel (VV) of thermonuclear fusion devices caused by electromagnetic (EM) loads is of great interest. In this paper, Model Order Reduction (MOR) is applied to the Integral Equation Methods (IEM) and the Finite Elements Method (FEM) to develop Electromagnetic and Structural Reduced Order Models (ROMs) compatible with real-time execution which allows for the real-time monitoring of strain and displacement in critical positions of Tokamaks machines. Low-rank compression techniques based on hierarchical matrices are applied to reduce the computational cost during the offline stage when the ROMs are constructed. Numerical results show the accuracy of the approach and demonstrate the compatibility with real-time execution in standard hardware.
- [12] arXiv:2405.01443 [pdf, ps, other]
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Title: On the existence of approximate problems that preserve the type of a bifurcation point of a nonlinear problem. Application to the stationary Navier-Stokes equations. Part 1. The overdetermined extended systemAuthors: Cătălin - Liviu BichirSubjects: Numerical Analysis (math.NA); Functional Analysis (math.FA)
We consider a nonlinear problem $F(\lambda,u)=0$ on infinite-dimensional Banach spaces that correspond to the steady-state bifurcation case. In the literature, it is found again a bifurcation point of the approximate problem $F_{h}(\lambda_{h},u_{h})=0$ only in some cases. We prove that, in every situation, given $F_{h}$ that approximates $F$, there exists an approximate problem $F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$ that has a bifurcation point with the same properties as the bifurcation point of $F(\lambda,u)=0$. First, we formulate, for a function $\widehat{F}$ defined on general Banach spaces, some sufficient conditions for the existence of an equation that has a bifurcation point of certain type. For the proof of this result, we use some methods from variational analysis, Graves' theorem, one of its consequences and the contraction mapping principle for set-valued mappings. These techniques allow us to prove the existence of a solution with some desired components that equal zero of an overdetermined extended system. We then obtain the existence of a constant (or a function) $\widehat{\varrho}$ so that the equation $\widehat{F}(\lambda,u)-\widehat{\varrho} = 0$ has a bifurcation point of certain type. This equation has $\widehat{F}(\lambda,u) = 0$ as a perturbation. It is also made evident a class of maps $C^{p}$ - equivalent (right equivalent) at the bifurcation point to $\widehat{F}(\lambda,u)-\widehat{\varrho}$ at the bifurcation point. Then, for the study of the approximation of $F(\lambda,u)=0$, we give conditions that relate the exact and the approximate functions. As an application of the theorem on general Banach spaces, we formulate conditions in order to obtain the existence of the approximate equation $F_{h}(\lambda_{h},u_{h})-\varrho_{h} = 0$.
Cross-lists for Fri, 3 May 24
- [13] arXiv:2405.00782 (cross-list from math.DS) [pdf, other]
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Title: Rigged Dynamic Mode Decomposition: Data-Driven Generalized Eigenfunction Decompositions for Koopman OperatorsSubjects: Dynamical Systems (math.DS); Machine Learning (cs.LG); Numerical Analysis (math.NA); Optimization and Control (math.OC); Spectral Theory (math.SP)
We introduce the Rigged Dynamic Mode Decomposition (Rigged DMD) algorithm, which computes generalized eigenfunction decompositions of Koopman operators. By considering the evolution of observables, Koopman operators transform complex nonlinear dynamics into a linear framework suitable for spectral analysis. While powerful, traditional Dynamic Mode Decomposition (DMD) techniques often struggle with continuous spectra. Rigged DMD addresses these challenges with a data-driven methodology that approximates the Koopman operator's resolvent and its generalized eigenfunctions using snapshot data from the system's evolution. At its core, Rigged DMD builds wave-packet approximations for generalized Koopman eigenfunctions and modes by integrating Measure-Preserving Extended Dynamic Mode Decomposition with high-order kernels for smoothing. This provides a robust decomposition encompassing both discrete and continuous spectral elements. We derive explicit high-order convergence theorems for generalized eigenfunctions and spectral measures. Additionally, we propose a novel framework for constructing rigged Hilbert spaces using time-delay embedding, significantly extending the algorithm's applicability. We provide examples, including systems with a Lebesgue spectrum, integrable Hamiltonian systems, the Lorenz system, and a high-Reynolds number lid-driven flow in a two-dimensional square cavity, demonstrating Rigged DMD's convergence, efficiency, and versatility. This work paves the way for future research and applications of decompositions with continuous spectra.
- [14] arXiv:2405.00814 (cross-list from cs.CE) [pdf, other]
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Title: Solving Maxwell's equations with Non-Trainable Graph Neural Network Message PassingComments: 9 pages, 5 figuresSubjects: Computational Engineering, Finance, and Science (cs.CE); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)
Computational electromagnetics (CEM) is employed to numerically solve Maxwell's equations, and it has very important and practical applications across a broad range of disciplines, including biomedical engineering, nanophotonics, wireless communications, and electrodynamics. The main limitation of existing CEM methods is that they are computationally demanding. Our work introduces a leap forward in scientific computing and CEM by proposing an original solution of Maxwell's equations that is grounded on graph neural networks (GNNs) and enables the high-performance numerical resolution of these fundamental mathematical expressions. Specifically, we demonstrate that the update equations derived by discretizing Maxwell's partial differential equations can be innately expressed as a two-layer GNN with static and pre-determined edge weights. Given this intuition, a straightforward way to numerically solve Maxwell's equations entails simple message passing between such a GNN's nodes, yielding a significant computational time gain, while preserving the same accuracy as conventional transient CEM methods. Ultimately, our work supports the efficient and precise emulation of electromagnetic wave propagation with GNNs, and more importantly, we anticipate that applying a similar treatment to systems of partial differential equations arising in other scientific disciplines, e.g., computational fluid dynamics, can benefit computational sciences
- [15] arXiv:2405.00891 (cross-list from math.OC) [pdf, other]
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Title: An interacting particle consensus method for constrained global optimizationSubjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
This paper presents a particle-based optimization method designed for addressing minimization problems with equality constraints, particularly in cases where the loss function exhibits non-differentiability or non-convexity. The proposed method combines components from consensus-based optimization algorithm with a newly introduced forcing term directed at the constraint set. A rigorous mean-field limit of the particle system is derived, and the convergence of the mean-field limit to the constrained minimizer is established. Additionally, we introduce a stable discretized algorithm and conduct various numerical experiments to demonstrate the performance of the proposed method.
- [16] arXiv:2405.00929 (cross-list from quant-ph) [pdf, other]
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Title: Quantum Wave Packet Transforms with compact frequency supportSubjects: Quantum Physics (quant-ph); Numerical Analysis (math.NA)
Different kinds of wave packet transforms are widely used for extracting multi-scale structures in signal processing tasks. This paper introduces the quantum circuit implementation of a broad class of wave packets, including Gabor atoms and wavelets, with compact frequency support. Our approach operates in the frequency space, involving reallocation and reshuffling of signals tailored for manipulation on quantum computers. The resulting implementation is different from the existing quantum algorithms for spatially compactly supported wavelets and can be readily extended to quantum transforms of other wave packets with compact frequency support.
- [17] arXiv:2405.00951 (cross-list from cs.CV) [pdf, other]
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Title: Hyperspectral Band Selection based on Generalized 3DTV and Tensor CUR DecompositionSubjects: Computer Vision and Pattern Recognition (cs.CV); Numerical Analysis (math.NA); Optimization and Control (math.OC)
Hyperspectral Imaging (HSI) serves as an important technique in remote sensing. However, high dimensionality and data volume typically pose significant computational challenges. Band selection is essential for reducing spectral redundancy in hyperspectral imagery while retaining intrinsic critical information. In this work, we propose a novel hyperspectral band selection model by decomposing the data into a low-rank and smooth component and a sparse one. In particular, we develop a generalized 3D total variation (G3DTV) by applying the $\ell_1^p$-norm to derivatives to preserve spatial-spectral smoothness. By employing the alternating direction method of multipliers (ADMM), we derive an efficient algorithm, where the tensor low-rankness is implied by the tensor CUR decomposition. We demonstrate the effectiveness of the proposed approach through comparisons with various other state-of-the-art band selection techniques using two benchmark real-world datasets. In addition, we provide practical guidelines for parameter selection in both noise-free and noisy scenarios.
- [18] arXiv:2405.01038 (cross-list from math.AP) [pdf, other]
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Title: Evolution of multiple closed knotted curves in spaceComments: arXiv admin note: substantial text overlap with arXiv:2201.02895Subjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
We investigate a system of geometric evolution equations describing a curvature and torsion driven motion of a family of 3D curves in the normal and binormal directions. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semi-flows, we are able to prove local existence, uniqueness, and continuation of classical H\"older smooth solutions to the governing system of non-linear parabolic equations modelling $n$ evolving curves with mutual nonlocal interactions. We present several computational studies of the flow that combine the normal or binormal velocity and considering nonlocal interaction.
- [19] arXiv:2405.01098 (cross-list from quant-ph) [pdf, ps, other]
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Title: Multivariate trace estimation using quantum state space linear algebraSubjects: Quantum Physics (quant-ph); Machine Learning (cs.LG); Numerical Analysis (math.NA)
In this paper, we present a quantum algorithm for approximating multivariate traces, i.e. the traces of matrix products. Our research is motivated by the extensive utility of multivariate traces in elucidating spectral characteristics of matrices, as well as by recent advancements in leveraging quantum computing for faster numerical linear algebra. Central to our approach is a direct translation of a multivariate trace formula into a quantum circuit, achieved through a sequence of low-level circuit construction operations. To facilitate this translation, we introduce \emph{quantum Matrix States Linear Algebra} (qMSLA), a framework tailored for the efficient generation of state preparation circuits via primitive matrix algebra operations. Our algorithm relies on sets of state preparation circuits for input matrices as its primary inputs and yields two state preparation circuits encoding the multivariate trace as output. These circuits are constructed utilizing qMSLA operations, which enact the aforementioned multivariate trace formula. We emphasize that our algorithm's inputs consist solely of state preparation circuits, eschewing harder to synthesize constructs such as Block Encodings. Furthermore, our approach operates independently of the availability of specialized hardware like QRAM, underscoring its versatility and practicality.
- [20] arXiv:2405.01232 (cross-list from math.OC) [pdf, other]
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Title: Kinetic Theories for Metropolis Monte Carlo MethodsSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA); Probability (math.PR)
We consider generalizations of the classical inverse problem to Bayesien type estimators, where the result is not one optimal parameter but an optimal probability distribution in parameter space. The practical computational tool to compute these distributions is the Metropolis Monte Carlo algorithm. We derive kinetic theories for the Metropolis Monte Carlo method in different scaling regimes. The derived equations yield a different point of view on the classical algorithm. It further inspired modifications to exploit the difference scalings shown on an simulation example of the Lorenz system.
- [21] arXiv:2405.01372 (cross-list from stat.ME) [pdf, other]
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Title: Statistical algorithms for low-frequency diffusion data: A PDE approachComments: 50 pages, 11 figures, 5 tablesSubjects: Methodology (stat.ME); Numerical Analysis (math.NA); Statistics Theory (math.ST); Computation (stat.CO)
We consider the problem of making nonparametric inference in multi-dimensional diffusion models from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of the likelihood and its gradient, and computational methods have thus far largely resorted to expensive simulation-based techniques. In this article, we propose a new computational approach which is motivated by PDE theory and is built around the characterisation of the transition densities as solutions of the associated heat (Fokker-Planck) equation. Employing optimal regularity results from the theory of parabolic PDEs, we prove a novel characterisation for the gradient of the likelihood. Using these developments, for the nonlinear inverse problem of recovering the diffusivity (in divergence form models), we then show that the numerical evaluation of the likelihood and its gradient can be reduced to standard elliptic eigenvalue problems, solvable by powerful finite element methods. This enables the efficient implementation of a large class of statistical algorithms, including (i) preconditioned Crank-Nicolson and Langevin-type methods for posterior sampling, and (ii) gradient-based descent optimisation schemes to compute maximum likelihood and maximum-a-posteriori estimates. We showcase the effectiveness of these methods via extensive simulation studies in a nonparametric Bayesian model with Gaussian process priors. Interestingly, the optimisation schemes provided satisfactory numerical recovery while exhibiting rapid convergence towards stationary points despite the problem nonlinearity; thus our approach may lead to significant computational speed-ups. The reproducible code is available online at https://github.com/MattGiord/LF-Diffusion.
- [22] arXiv:2405.01465 (cross-list from stat.CO) [pdf, other]
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Title: A Fast and Accurate Numerical Method for the Left Tail of Sums of Independent Random VariablesSubjects: Computation (stat.CO); Numerical Analysis (math.NA)
We present a flexible, deterministic numerical method for computing left-tail rare events of sums of non-negative, independent random variables. The method is based on iterative numerical integration of linear convolutions by means of Newtons-Cotes rules. The periodicity properties of convoluted densities combined with the Trapezoidal rule are exploited to produce a robust and efficient method, and the method is flexible in the sense that it can be applied to all kinds of non-negative continuous RVs. We present an error analysis and study the benefits of utilizing Newton-Cotes rules versus the fast Fourier transform (FFT) for numerical integration, showing that although there can be efficiency-benefits to using FFT, Newton-Cotes rules tend to preserve the relative error better, and indeed do so at an acceptable computational cost. Numerical studies on problems with both known and unknown rare-event probabilities showcase the method's performance and support our theoretical findings.
Replacements for Fri, 3 May 24
- [23] arXiv:1910.14093 (replaced) [pdf, other]
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Title: Superconvergence of differential structure for finite element methods on perturbed surface meshesComments: updated version, some gaps are fixedSubjects: Numerical Analysis (math.NA); Differential Geometry (math.DG)
- [24] arXiv:2103.01194 (replaced) [pdf, other]
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Title: Structure-preserving numerical schemes for Lindblad equationsSubjects: Numerical Analysis (math.NA); Quantum Physics (quant-ph)
- [25] arXiv:2403.04095 (replaced) [pdf, other]
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Title: Helmholtz preconditioning for the compressible Euler equations using mixed finite elements with Lorenz staggeringSubjects: Numerical Analysis (math.NA)
- [26] arXiv:2403.10892 (replaced) [pdf, other]
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Title: Mathematical study of a new coupled electro-thermo radiofrequency model of cardiac tissueSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
- [27] arXiv:2403.11993 (replaced) [pdf, other]
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Title: Adaptive stepsize algorithms for Langevin dynamicsSubjects: Numerical Analysis (math.NA)
- [28] arXiv:2404.05655 (replaced) [pdf, ps, other]
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Title: Convergence rates for a finite volume scheme of the stochastic heat equationComments: 27 pagesSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP); Probability (math.PR)
- [29] arXiv:2404.16794 (replaced) [pdf, other]
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Title: Structure-Preserving Oscillation-Eliminating Discontinuous Galerkin Schemes for Ideal MHD Equations: Locally Divergence-Free and Positivity-PreservingComments: 55 pagesSubjects: Numerical Analysis (math.NA); Instrumentation and Methods for Astrophysics (astro-ph.IM); Computational Physics (physics.comp-ph)
- [30] arXiv:2405.00283 (replaced) [pdf, ps, other]
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Title: An Unstructured Mesh Reaction-Drift-Diffusion Master Equation with Reversible ReactionsSubjects: Numerical Analysis (math.NA)
- [31] arXiv:2405.00569 (replaced) [pdf, other]
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Title: A novel central compact finite-difference scheme for third derivatives with high spectral resolutionComments: 28 pages, 15 figures, 10 TablesSubjects: Numerical Analysis (math.NA)
- [32] arXiv:2207.05209 (replaced) [pdf, other]
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Title: Fourier Neural Operator with Learned Deformations for PDEs on General GeometriesJournal-ref: Journal of Machine Learning Research (2023) Volume 24, Issue 1, Article No. 388, pp 18593-18618Subjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
- [33] arXiv:2310.02246 (replaced) [pdf, other]
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Title: Learning to Relax: Setting Solver Parameters Across a Sequence of Linear System InstancesComments: ICLR 2024 SpotlightSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Numerical Analysis (math.NA); Machine Learning (stat.ML)
- [34] arXiv:2312.17390 (replaced) [pdf, ps, other]
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Title: Quantum Hamiltonian Learning for the Fermi-Hubbard ModelSubjects: Quantum Physics (quant-ph); Numerical Analysis (math.NA)
- [35] arXiv:2403.14844 (replaced) [pdf, other]
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Title: Extrapolating Solution Paths of Polynomial Homotopies towards Singularities with PHCpack and phcpyComments: Accepted by the 8th International Congress on Mathematical Software 2024Subjects: Mathematical Software (cs.MS); Symbolic Computation (cs.SC); Complex Variables (math.CV); Numerical Analysis (math.NA)
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