Partial Differential Equation-based numerical simulators are ubiquitous in engineering. However, the scaling of PDE solvers with increasing spatial, temporal and parametric resolutions is rather poor, leading to computational costs that are exponentially increasing with the complexity of the physical system of interest. As a consequence, discretisation schemes are often coarser than desired, in a pragmatic push towards applications such as physics-based modelling in interaction with reality, aka digital twins.
A way forward is a consistent treatment of all sources of uncertainty and a subsequently approach model refinement as a unified, uncertainty-driven task. To take modelling error into account, classical Bayesian model calibration and state estimation methodologies treat model parameters and model outputs as random variables, which are then conditioned to data in order to yield posterior distributions with appropriate credible intervals. However, the traditional way to quantify discretisation errors is through deterministic numerical analysis, yielding point estimates or bounds, without distribution, making these approaches incompatible with a Bayesian treatment of model uncertainty.
Recently, significant developments have been made in the area of probabilistic solvers for PDEs. The idea is to formulate discretisation schemes as Bayesian estimation problems, yielding not a single parametrised/spatio/temporal field but a distribution of such fields. Most methods use Gaussian Processes as fundamental building block. The basic idea is to condition a Gaussian random field to satisfy the PDE at particular points of the computational domain. This gives rise to probabilistic variants of meshless methods traditionally used to solve PDEs. To date however, such approaches are not available for finite element solvers, which are typically based on integral formulations over arbitrary simplexes, leading to analytically intractable integrals.
We propose what we believe is the first probabilistic finite element methodology and apply it to steady heat diffusion. It is based on the definition of a discrete Gaussian prior over a p-refined finite element space. This prior is conditioned to satisfy the PDE weakly, using the non-refined finite element space to generate a linear observation operator. The Hyperparameters of the Gaussian process are optimised using maximum likelihood. We also provide an efficient solver based on Monte- Carlo sampling of the analytical posterior, coupled with an approximate multigrid sampler for the p- refined gaussian prior. We show that this sampler ensures that the overall cost of the methodology is of the order the p-refined deterministic FE technology, whilst delivering valuable probability distributions for the continuous solution to the PDE system.
A way forward is a consistent treatment of all sources of uncertainty and a subsequently approach model refinement as a unified, uncertainty-driven task. To take modelling error into account, classical Bayesian model calibration and state estimation methodologies treat model parameters and model outputs as random variables, which are then conditioned to data in order to yield posterior distributions with appropriate credible intervals. However, the traditional way to quantify discretisation errors is through deterministic numerical analysis, yielding point estimates or bounds, without distribution, making these approaches incompatible with a Bayesian treatment of model uncertainty.
Recently, significant developments have been made in the area of probabilistic solvers for PDEs. The idea is to formulate discretisation schemes as Bayesian estimation problems, yielding not a single parametrised/spatio/temporal field but a distribution of such fields. Most methods use Gaussian Processes as fundamental building block. The basic idea is to condition a Gaussian random field to satisfy the PDE at particular points of the computational domain. This gives rise to probabilistic variants of meshless methods traditionally used to solve PDEs. To date however, such approaches are not available for finite element solvers, which are typically based on integral formulations over arbitrary simplexes, leading to analytically intractable integrals.
We propose what we believe is the first probabilistic finite element methodology and apply it to steady heat diffusion. It is based on the definition of a discrete Gaussian prior over a p-refined finite element space. This prior is conditioned to satisfy the PDE weakly, using the non-refined finite element space to generate a linear observation operator. The Hyperparameters of the Gaussian process are optimised using maximum likelihood. We also provide an efficient solver based on Monte- Carlo sampling of the analytical posterior, coupled with an approximate multigrid sampler for the p- refined gaussian prior. We show that this sampler ensures that the overall cost of the methodology is of the order the p-refined deterministic FE technology, whilst delivering valuable probability distributions for the continuous solution to the PDE system.
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