High-dimensional and higher-order multifidelity Monte Carlo estimators

Abstract

Multifidelity Monte Carlo methods rely on a hierarchy of possibly less accurate but statistically correlated simplified or reduced models, in order to accelerate the estimation of statistics of high-fidelity models without compromising the accuracy of the estimates. This approach has recently gained widespread attention in uncertainty quantification [1]. This is partly due to the availability of optimal strategies for the estimation of the expectation of scalar quantities-of-interest [2]. In practice, the optimal strategy for the expectation is also used for the estimation of variance and sensitivity indices [3]. However, a general strategy is still lacking for vector-valued problems, nonlinearly statistically-dependent models, and estimators for which a closed-form expression of the error is unavailable. The focus of the present work is to generalize the standard multifidelity estimators to the above cases. The proposed generalized estimators lead to an optimization problem that can be solved analytically and whose coefficients can be estimated numerically with few runs of the highand low-fidelity models. We analyze the performance of the proposed approach on a selected number of experiments, with a particular focus on cardiac electrophysiology, where a hierarchy of physics-based low-fidelity models is readily available.

Fluid dynamics in syringomyelia cavities: Effects of heart rate, CSF velocity, CSF velocity waveform and craniovertebral decompression

“…The intramural periarterial drainage pathway In contrast to the glymphatic system, clearance along the IPAD pathway is in the reverse direction, out along the basement membranes of capillaries and arteries [20]. This pathway has been identified by the injection of fluorscent tracers into the brain [20,57]. Morris et al [57] also injected nanoparticles into mouse CSF in the SAS.…”

“…This pathway has been identified by the injection of fluorscent tracers into the brain [20,57]. Morris et al [57] also injected nanoparticles into mouse CSF in the SAS. Within 5 minutes after injection, nanoparticles were present along basement membranes on the outer aspects of cortical arteries, between the pia mater and the glia limitans, what has been referred to by others as paravascular spaces [34].…”

“…Replacing partial differential equation (PDE) parameters subject to uncertainty with spatially correlated random fields is a common modelling choice in the uncertainty quantification (UQ) literature [21,19,70] and Monte Carlo methods have been successfully used in biology to quantify how uncertainty in model input propagates to uncertainty in model output. However, these methods have mainly been applied to simulations of the cardiovascular system [57,17] and, to our knowledge, there has only been one study in which Monte Carlo methods have been used for UQ in brain modelling [33]. To the authors' knowledge, there has been no previous work on stochastic forward uncertainty quantification for simulations of tracer transport with the brain parenchyma.…”

Fast uncertainty quantification of tracer distribution in the brain interstitial fluid with multilevel and quasi Monte Carlo

“…The computational resources available for this study were limited and with more computational power we could have obtained more accurate results by adding a finer level to the hierarchy and/or increasing the number of samples. However, we did not encounter any significant issues with setting up a hierarchy on which good MLMC convergence could be observed, a problem mentioned in recent work by Quaglino et al, 24 in which they apply MFMC for a cardiac electrophysiology application.…”

Uncertainty quantification of parenchymal tracer distribution using random diffusion and convective velocity fields

“…Replacing partial differential equation (PDE) parameters subject to uncertainty with spatially correlated random fields is a common modelling choice in the uncertainty quantification (UQ) literature [21,19,70] and Monte Carlo methods have been successfully used in biology to quantify how uncertainty in model input propagates to uncertainty in model output. However, these methods have mainly been applied to simulations of the cardiovascular system [57,17] and, to our knowledge, there has only been one study in which Monte Carlo methods have been used for UQ in brain modelling [33]. To the authors' knowledge, there has been no previous work on stochastic forward uncertainty quantification for simulations of tracer transport with the brain parenchyma.…”