Related papers: The Bayesian Finite Element Method in Inverse Prob…
We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows…
Bayesian inverse problems are an important application for probabilistic solvers of partial differential equations: when fully resolving numerical error is computationally infeasible, probabilistic solvers can be used to consistently model…
The increased availability of observation data from engineering systems in operation poses the question of how to incorporate this data into finite element models. To this end, we propose a novel statistical construction of the finite…
In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated…
Statistical learning additions to physically derived mathematical models are gaining traction in the literature. A recent approach has been to augment the underlying physics of the governing equations with data driven Bayesian statistical…
The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite…
The abundance of observed data in recent years has increased the number of statistical augmentations to complex models across science and engineering. By augmentation we mean coherent statistical methods that incorporate measurements upon…
The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the unknown true system response. We provide a probabilistic error analysis…
Motivated by problems where the response is needed at select localized regions in a large computational domain, we devise a novel finite element discretization that results in exponential convergence at pre-selected points. The two key…
In this work, we have developed a variational Bayesian inference theory of elasticity, which is accomplished by using a mixed Variational Bayesian inference Finite Element Method (VBI-FEM) that can be used to solve the inverse deformation…
Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods (MFEM) in space for simulating…
The Finite Element Method (FEM) is a widely used technique for simulating crash scenarios with high accuracy and reliability. To reduce the significant computational costs associated with FEM, the Finite Element Method Integrated Networks…
A well-established approach for inferring full displacement and stress fields from possibly sparse data is to calibrate the parameter of a given constitutive model using a Bayesian update. After calibration, a (stochastic) forward…
We consider a randomised implementation of the finite element method (FEM) for elliptic partial differential equations on high-dimensional models. This is motivated by applications where model predictions are essential for real-time process…
The Finite Element Method (FEM) is a well-established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with…
The Finite Element Method (FEM) is the gold standard for spatial discretization in numerical simulations for a wide spectrum of real-world engineering problems. Prototypical areas of interest include linear heat transfer and linear…
The statistical finite element method (StatFEM) is an emerging probabilistic method that allows observations of a physical system to be synthesised with the numerical solution of a PDE intended to describe it in a coherent statistical…
Using the standard finite element method (FEM) to solve general partial differential equations, the round-off error is found to be proportional to $N^{\beta_{\rm R}}$, with $N$ the number of degrees of freedom (DoFs) and $\beta_{\rm R}$ a…
The recently proposed soft finite element method (SoftFEM) reduces the stiffness (condition numbers), consequently improving the overall approximation accuracy. The method subtracts a least-square term that penalizes the gradient jumps…
We introduce a novel hybrid methodology combining classical finite element methods (FEM) with neural networks to create a well-performing and generalizable surrogate model for forward and inverse problems. The residual from finite element…