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425 records · Page 24Linked to original sources

arXiv2025

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification

The article introduces a method to learn dynamical systems that are governed by Euler--Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

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arXiv2025

Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise

We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations (SPDEs) driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen--Cahn equation indicates that these schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in (J. Cui, J. Hong, and L. Sun, Stochastic Process. Appl. (2021): 55--93), provided that the interface thickness is not too small.

math.NA↗

arXiv2024

Density estimation for elliptic PDE with random input by preintegration and quasi-Monte Carlo methods

In this paper, we apply quasi-Monte Carlo (QMC) methods with an initial preintegration step to estimate cumulative distribution functions and probability density functions in uncertainty quantification (UQ). The distribution and density functions correspond to a quantity of interest involving the solution to an elliptic partial differential equation (PDE) with a lognormally distributed coefficient and a normally distributed source term. There is extensive previous work on using QMC to compute expected values in UQ, which have proven very successful in tackling a range of different PDE problems. However, the use of QMC for density estimation applied to UQ problems will be explored here for the first time. Density estimation presents a more difficult challenge compared to computing the expected value due to discontinuities present in the integral formulations of both the distribution and density. Our strategy is to use preintegration to eliminate the discontinuity by integrating out a carefully selected random parameter, so that QMC can be used to approximate the remaining integral. First, we establish regularity results for the PDE quantity of interest that are required for smoothing by preintegration to be effective. We then show that an $N$-point lattice rule can be constructed for the integrands corresponding to the distribution and density, such that after preintegration the QMC error is of order $\mathcal{O}(N^{-1+ε})$ for arbitrarily small $ε>0$. This is the same rate achieved for computing the expected value of the quantity of interest. Numerical results are presented to reaffirm our theory.

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arXiv2024

Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems

We propose a machine learning method for computing eigenvalues and eigenfunctions of the Schrödinger operator on a $d$-dimensional hypercube with Dirichlet boundary conditions. The cut-off function technique is employed to construct trial functions that precisely satisfy the homogeneous boundary conditions. This approach eliminates the error caused by the standard boundary penalty method, improves the overall accuracy of the method, as demonstrated by the typical numerical examples. Under the assumption that the eigenfunctions belong to a spectral Barron space, we derive an explicit convergence rate of the generalization error of the proposed method, which does not suffer from the curse of dimensionality. We verify the assumption by proving a new regularity shift result for the eigenfunctions when the potential function belongs to an appropriate spectral Barron space. Moreover, we extend the generalization error bound to the normalized penalty method, which is widely used in practice.

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arXiv2024

A Neural-preconditioned Poisson Solver for Mixed Dirichlet and Neumann Boundary Conditions

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner's success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.

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arXiv2024

The Ramshaw-Mesina Hybrid Algorithm applied to the Navier Stokes Equations

In 1991, Ramshaw and Mesina proposed a novel synthesis of penalty methods and artificial compression methods. When the two were balanced they found the combination was 3-4 orders more accurate than either alone. This report begins the study of their interesting method applied to the Navier-Stokes equations. We perform stability analysis, semi-discrete error analysis, and tests of the algorithm. Although most of the results for implicit time discretizations of our numerical tests comply with theirs for explicit time discretizations, the behavior in damping pressure oscillations and violations of incompressibility are different from their findings and our heuristic analysis.

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arXiv2024

Convergence of a Ramshaw-Mesina Iteration

In 1991 Ramshaw and Mesina introduced a clever synthesis of penalty methods and artificial compression methods. Its form makes it an interesting option to replace the pressure update in the Uzawa iteration. The result, for the Stokes problem, is \begin{equation} \left\{ \begin{array} [c]{cc} Step\ 1: & -\triangle u^{n+1}+\nabla p^{n}=f(x),\ {\rm in}\ Ω,\ u^{n+1}|_{\partialΩ}=0,\\ Step\ 2: & p^{n+1}-p^{n}+β\nabla\cdot(u^{n+1}-u^{n})+α^{2}\nabla\cdot u^{n+1}=0. \end{array} \right. \end{equation} For saddle point problems, including Stokes, this iteration converges under a condition similar to the one required for Uzawa iteration.

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arXiv2024

Time transient Simulations via Finite Element Network Analysis: Theoretical Formulation and Numerical Validation

This paper extends the finite element network analysis (FENA) to include a dynamic time-transient formulation. FENA was initially formulated in the context of the linear static analysis of 1D and 2D elastic structures. By introducing the concept of super finite network element, this paper provides the necessary foundation to extend FENA to linear time-transient simulations for both homogeneous and inhomogeneous domains. The concept of neural network concatenation, originally formulated to combine networks representative of different structural components in space, is extended to the time domain. Network concatenation in time enables training neural network models based on data available in a limited time frame and then using the trained networks to simulate the system evolution beyond the initial time window characteristic of the training data set. The proposed methodology is validated by applying FENA to the transient simulation of one-dimensional structural elements (such as rods and beams) and by comparing the results with either analytical or finite element solutions. Results confirm that FENA accurately predicts the dynamic response of the physical system and, while introducing an error on the order of 1% (compared to analytical or computational solutions of the governing differential equations), it is capable of delivering extreme computational efficiency.

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arXiv2022

An empirical study of various candidate selection and partitioning techniques in the DIRECT framework

Over the last three decades, many attempts have been made to improve the DIRECT (DIviding RECTangles) algorithm's efficiency. Various novel ideas and extensions have been suggested. The main two steps of DIRECT-type algorithms are selecting and partitioning potentially optimal rectangles. However, the most efficient combination of these two steps is an area that has not been investigated so far. This paper presents a study covering an extensive examination of various candidate selection and partitioning techniques within the same DIRECT algorithmic framework. Twelve DIRECT-type algorithmic variations are compared on 800 randomly generated GKLS-type test problems and 96 box-constrained global optimization problems from DIRECTGOLib v1.1 with varying complexity. Based on these studies, we have identified the most efficient selection and partitioning combinations leading to new, more efficient, DIRECT-type algorithms. All these algorithms are included in the latest version of DIRECTGO v1.1.0 and are publicly available.

math.OC↗

arXiv2021

Least-squares spectral methods for ODE eigenvalue problems

We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices and objects combining quasimatrices and matrices. The strength of the approach is its flexibility that lies in the quasimatrix formulation allowing the basis functions to be chosen arbitrarily (e.g. those obtained by solving nearby problems), and often giving high accuracy. We also show how our algorithm can easily be modified to solve problems with eigenvalue-dependent boundary conditions, and discuss reformulations as an integral equation, which often improves the accuracy.

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arXiv2021

$L^p$-Convergence Rate of Backward Euler Schemes for Monotone SDEs

We give a unified method to derive the strong convergence rate of the backward Euler scheme for monotone SDEs in $L^p(Ω)$-norm, with general $p \ge 4$. The results are applied to the backward Euler scheme of SODEs with polynomial growth coefficients. We also generalize the argument to the Galerkin-based backward Euler scheme of SPDEs with polynomial growth coefficients driven by multiplicative trace-class noise.

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