@article{19027,
  abstract     = {Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean–Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean–Kawasaki equation being highly singular.},
  author       = {Cornalba, Federico and Fischer, Julian L},
  issn         = {1095-7170},
  journal      = {SIAM Journal on Numerical Analysis},
  number       = {1},
  pages        = {262--287},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics}},
  doi          = {10.1137/23M1617345},
  volume       = {63},
  year         = {2025},
}

@article{9335,
  abstract     = {Various degenerate diffusion equations exhibit a waiting time phenomenon: depending on the “flatness” of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin film equations, and we obtain sufficient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. For the spatially discrete solution, the waiting time phenomenon refers to a deviation of the edge of support from its original position by a quantity comparable to the mesh width, over a mesh-independent time interval. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique à la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.},
  author       = {Fischer, Julian L and Matthes, Daniel},
  issn         = {0036-1429},
  journal      = {SIAM Journal on Numerical Analysis},
  number       = {1},
  pages        = {60--87},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{The waiting time phenomenon in spatially discretized porous medium and thin film equations}},
  doi          = {10.1137/19M1300017},
  volume       = {59},
  year         = {2021},
}

@article{9352,
  abstract     = {This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H+(\varepsilon/H)^{d/2}$, $\varepsilon$ being the small correlation length of the random coefficient and $H$ the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization.},
  author       = {Fischer, Julian L and Gallistl, Dietmar and Peterseim, Dietmar},
  issn         = {0036-1429},
  journal      = {SIAM Journal on Numerical Analysis},
  number       = {2},
  pages        = {660--674},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{A priori error analysis of a numerical stochastic homogenization method}},
  doi          = {10.1137/19M1308992},
  volume       = {59},
  year         = {2021},
}