Stochastic two-scale convergence and Young measures

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence

Homogenization of elliptic systems with non-periodic, state dependent coefficients

In this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided

Plasticity in Cancer Cell Populations: Biology, Mathematics and Philosophy of Cancer

International audienc

Homogenization of elliptic systems with non-periodic, state dependent coefficients

In this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided

Stochastic unfolding and homogenization

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogenization result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method introduced in this paper extends extitdiscrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition

Stochastic unfolding and homogenization

The notion of periodic two-scale convergence and the method of periodic un- folding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coe cients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coe cients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in di erent ways to the stochastic case. In this work we introduce a stochastic unfolding method that fea- tures many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogeniza- tion result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method descibed in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition

Stochastic unfolding and homogenization

The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere weak convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a new result on stochastic homogenization for a non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method described in the present paper extends to the continuum setting the notion of discrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.Comment: 46 page

Derivation of an effective damage model with evolving micro-structure

In this paper rate-independent damage models for elastic materials are considered. The aim is the derivation of an effective damage model by investigating the limit process of damage models with evolving micro-defects. In all presented models the damage is modeled via a unidirectional change of the material tensor. With progressing time this tensor is only allowed to decrease in the sense of quadratic forms. The magnitude of the damage is given by comparing the actual material tensor with two reference configurations, denoting completely undamaged material and maximally damaged material (no complete damage). The starting point is a microscopic model, where the underlying micro-defects, describing the distribution of either undamaged material or maximally damaged material (but nothing in between), are of a given time-dependent shape but of different sizes. Scaling the microstructure of this microscopic model by a parameter ε>0 the limit passage ε→0 is preformed via two-scale convergence techniques. Therefore, a regularization approach for piecewise constant functions is introduced to guaranty enough regularity for identifying the limit model. In the limit model the material tensor depends on a damage variable z:[0,T]→ W1,p(Ω) taking values between 0 and 1 such that, in contrast to the microscopic model, some kind of intermediate damage for a material point x∈Ω is possible. Moreover, this damage variable is connected to the material tensor via an explicit formula, namely, a unit cell formula known from classical homogenization results

Crohn's disease: an in vitro analysis of T lymphocyte function and response to commensal microbes

Inflammatory bowel disease is a chronic, relapsing inflammation of the intestine. Approximately 0.5% of the western world is estimated to suffer from the disease. Crohn’s disease, a type of inflammatory bowel disease, consists of a patchy inflammation, that can occur throughout the entire gastrointestinal tract. Due to the heterogeneous nature of Crohn’s disease, little is known about the mechanisms of disease pathogenesis. Compared to healthy individuals, patients with Crohn’s disease often have elevated levels of inflammatory T lymphocyte subsets, their associated effector cytokines, and higher intestinal permeability. Crohn’s disease has been associated with genetic risk factors, sedentary lifestyles, and a loss of tolerance to the commensal members of the microbiome. In this research, T lymphocytes from non-IBD and Crohn’s disease patient peripheral blood mononuclear cells (PBMCs) were analysed for comparison. Flow cytometry was used to fluorescently label cells and analytes of interest. Visual analytic tools, such as CytoAnalytics’ EarlyBird and viSNE were used to visualise patient variation within, and between groups. It was hypothesised that Crohn’s disease T lymphocytes would have a bias for inflammation-inducing T lymphocyte subsets, and that these cells would have a higher capacity for inflammation, than non-IBD controls. Indeed, inflammatory T lymphocytes were present at higher frequencies in Crohn’s disease patient blood samples. Furthermore, Crohn’s disease patient PBMCs had a higher capacity for proliferation in response to T lymphocyte stimulation. In responses to the commensal bacterium, Faecalibacterium prausnitzii, Crohn’s disease patient PBMC Tregs increased in frequency, whereas CD4+ inflammatory T lymphocyte subsets maintained their frequency. In contrast, non-IBD PBMC Tregs were not influenced by co-culture with F. prausnitzii, and CD4+ inflammatory T lymphocyte subsets decreased in the presence of F. prausnitzii. Together, these data suggest that Crohn’s disease patient PBMCs respond abnormally to commensal bacteria, and this could play a role in disease pathogenesis. Intestinal organoids are derived from patient colonic biopsy stem cells. Organoids provide a unique in vitro model for the observation of intestinal interactions. In this thesis, a 2D intestinal monolayer was developed from the culture of 3D intestinal organoids. 2D monolayers emulate epithelial intestinal barrier cellular organisation. A difficult question to answer in Crohn’s disease research is whether intestinal permeability and inflammation is a cause or consequence of disease. As such, patient organoids and bacteria were integrated into the model stepwise, to analyse the influence each component has on the system, in the absence, or presence, on an active immune presence. Using this model, it was found that monolayers derived from Crohn’s disease patients were more permeable than non-IBD controls. Furthermore, Crohn’s disease patient monolayers had deleterious interactions in co-culture with F. prausnitzii, resulting in reduced epithelial integrity; in contrast to non-IBD patient monolayers which were unaffected. The addition of matched PBMCs into Crohn’s disease patient monolayers exacerbated epithelial degradation in the presence of F. prausnitzii. Taken together, it was found that Crohn’s disease patient PBMCs had greater inflammatory capacity than non-IBD control PBMCs. These data suggest that intestinal cells in patients with Crohn’s disease may have deleterious interactions with commensal bacteria, which could result in increased intestinal permeability. Furthermore, Crohn’s disease patient Treg responses to commensal bacteria suggest that intestinal areas of immune suppression may promote bacteria success, and potential dysbiosis. Translocation of the luminal microbiota into the lamina propria, combined with excessive T lymphocyte inflammatory capacity, could be an initial driving factor for Crohn’s disease pathogenesis. Data in this thesis provide valuable insight into the initiation of epithelial degradation in patients with Crohn’s disease. These data suggest that epithelial degradation may occur in Crohn’s disease patients in response to commensal bacteria, even in the absence of an immune cell influence. With further development, the intestinal organoid monolayer model may provide insight into the mechanisms of epithelial degradation in individual patients. Further, the monolayer model may also provide a tool to optimise individual patient treatments, in vitro