Multilevel Monte Carlo Approaches for Numerical Homogenization

International audienceIn this article, we study the application of Multi-Level Monte Carlo (MLMC) approaches to numerical random homogenization. Our objective is to compute the expectation of some functionals of the homogenized coefficients, or of the homogenized solutions. This is accomplished within MLMC by considering different levels of representative volumes (RVE), and, when it comes to homogenized solutions, different levels of coarse-grid meshes. Many inexpensive computations with the smallest RVE size and the largest coarse mesh are combined with fewer expensive computations performed on larger RVEs and smaller coarse meshes. We show that, by carefully selecting the number of realizations at each level, we can achieve a speed-up in the computations in comparison to a standard Monte Carlo method. Numerical results are presented both for one-dimensional and two-dimensional test-cases

On selected efficient numerical methods for multiscale problems with stochastic coefficients

Many real life problems have multiple spatial scales. In addition to the multiscale nature one has to take uncertainty into account. In this work we consider multiscale problems with stochastic coefficients. We combine multiscale methods, e.g., mixed multiscale finite elements or homogenization, which are used for deterministic problems with stochastic methods, such as multi-level Monte Carlo or polynomial chaos methods. The work is divided into three parts. In the first two parts we study homogenization with different stochastic methods. Therefore we consider elliptic stationary diffusion equations with stochastic coefficients. The last part is devoted to the study of mixed multiscale finite elements in combination with multi-level Monte Carlo methods. In the third part we consider multi-phase flow and transport equations

On selected efficient numerical methods for multiscale problems with stochastic coefficients

Many real life problems have multiple spatial scales. In addition to the multiscale nature one has to take uncertainty into account. In this work we consider multiscale problems with stochastic coefficients. We combine multiscale methods, e.g., mixed multiscale finite elements or homogenization, which are used for deterministic problems with stochastic methods, such as multi-level Monte Carlo or polynomial chaos methods. The work is divided into three parts. In the first two parts we study homogenization with different stochastic methods. Therefore we consider elliptic stationary diffusion equations with stochastic coefficients. The last part is devoted to the study of mixed multiscale finite elements in combination with multi-level Monte Carlo methods. In the third part we consider multi-phase flow and transport equations

Reduced decline of lung diffusing capacity in COPD patients with diabetes and metformin treatment

We studied whether in patients with COPD the use of metformin for diabetes treatment was linked to a pattern of lung function decline consistent with the hypothesis of anti-aging effects of metformin. Patients of GOLD grades 1-4 of the COSYCONET cohort with follow-up data of up to 4.5 y were included. The annual decline in lung function (FEV1, FVC) and CO diffusing capacity (KCO, TLCO) in %predicted at baseline was evaluated for associations with age, sex, BMI, pack-years, smoking status, baseline lung function, exacerbation risk, respiratory symptoms, cardiac disease, as well as metformin-containing therapy compared to patients without diabetes and metformin. Among 2741 patients, 1541 (mean age 64.4 y, 601 female) fulfilled the inclusion criteria. In the group with metformin treatment vs. non-diabetes the mean annual decline in KCO and TLCO was significantly lower (0.2 vs 2.3, 0.8 vs. 2.8%predicted, respectively;p < 0.05 each), but not the decline of FEV1 and FVC. These results were confirmed using multiple regression and propensity score analyses. Our findings demonstrate an association between the annual decline of lung diffusing capacity and the intake of metformin in patients with COPD consistent with the hypothesis of anti-aging effects of metformin as reflected in a surrogate marker of emphysema