Simulation of stochastic Volterra equations driven by space--time L\'evy noise

In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space--time L\'evy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods $L^p$- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes

Importance sampling of heavy-tailed iterated random functions

We consider a stochastic recurrence equation of the form $Z_{n+1} = A_{n+1} Z_n+B_{n+1}$, where $\mathbb{E}[\log A_1]<0$, $\mathbb{E}[\log^+ B_1]<\infty$ and $\{(A_n,B_n)\}_{n\in\mathbb{N}}$ is an i.i.d. sequence of positive random vectors. The stationary distribution of this Markov chain can be represented as the distribution of the random variable $Z \triangleq \sum_{n=0}^\infty B_{n+1}\prod_{k=1}^nA_k$. Such random variables can be found in the analysis of probabilistic algorithms or financial mathematics, where $Z$ would be called a stochastic perpetuity. If one interprets $-\log A_n$ as the interest rate at time $n$, then $Z$ is the present value of a bond that generates $B_n$ unit of money at each time point $n$. We are interested in estimating the probability of the rare event $\{Z>x\}$, when $x$ is large; we provide a consistent simulation estimator using state-dependent importance sampling for the case, where $\log A_1$ is heavy-tailed and the so-called Cram\'{e}r condition is not satisfied. Our algorithm leads to an estimator for $P(Z>x)$. We show that under natural conditions, our estimator is strongly efficient. Furthermore, we extend our method to the case, where $\{Z_n\}_{n\in\mathbb{N}}$ is defined via the recursive formula $Z_{n+1}=\Psi_{n+1}(Z_n)$ and $\{\Psi_n\}_{n\in\mathbb{N}}$ is a sequence of i.i.d. random Lipschitz functions

Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes

We propose a class of strongly efficient rare event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on the heavy-tailed sample path large deviations result recently established in Rhee, Blanchet, and Zwart (2016). The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are "universal" in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with several applications that arise in the context of mathematical finance, actuarial science, and queueing theory

Preparation of Monodomain Liquid Crystal Elastomers and Liquid Crystal Elastomer Nanocomposites

LCEs are shape-responsive materials with fully reversible shape change and potential applications in medicine, tissue engineering, artificial muscles, and as soft robots. Here, we demonstrate the preparation of shape-responsive liquid crystal elastomers (LCEs) and LCE nanocomposites along with characterization of their shape-responsiveness, mechanical properties, and microstructure. Two types of LCEs — polysiloxane-based and epoxy-based — are synthesized, aligned, and characterized. Polysiloxane-based LCEs are prepared through two crosslinking steps, the second under an applied load, resulting in monodomain LCEs. Polysiloxane LCE nanocomposites are prepared through the addition of conductive carbon black nanoparticles, both throughout the bulk of the LCE and to the LCE surface. Epoxy-based LCEs are prepared through a reversible esterification reaction. Epoxy-based LCEs are aligned through the application of a uniaxial load at elevated (160 °C) temperatures. Aligned LCEs and LCE nanocomposites are characterized with respect to reversible strain, mechanical stiffness, and liquid crystal ordering using a combination of imaging, two-dimensional X-ray diffraction measurements, differential scanning calorimetry, and dynamic mechanical analysis. LCEs and LCE nanocomposites can be stimulated with heat and/or electrical potential to controllably generate strains in cell culture media, and we demonstrate the application of LCEs as shape-responsive substrates for cell culture using a custom-made apparatus

Progressive buckling of large-diameter monopiles due to boulder collision during installation

This research aims for an improved description of the progressive buckling phenomena of large-diameter monopiles due to boulder collision during the installation progress. This description contributes to a more reliable predictions of the possibility and the severity of progressive buckling in situ during the design process. This improved description benefits from establishing a calculation approach that leads to reliable 3D simulations of progressive buckling, given 2D benchmark models. Adopting the numerical technique, Coupled Eulerian-Lagrangian (CEL) method, the deliverables are generated from three stages of research. The first stage contains parametric studies based on 2D plane-strain plate installation models exploring variation of material properties, installation conditions, boulder shapes, and tip geometries. It is found that the boulder size was most influential to the tip deflections, followed by the boulder embedment and the soil strength. Laboratory-scale physical modelling is then introduced during the second stage to ensure validation of the 2D models. Soil movements during plate installation are revealed using Particle Image Velocimetry (PIV) analysis. Together, these two stages provide confidence in the use of numerical methods for 3D geometries. The third stage explored truncated 3D simulations, reducing the problem to a quarter section of the pile, and examining the impact of the ratio between the pile diameter and the pile wall thickness. These 3D models build links between the 2D benchmarks and 3D simulations, to complete the calculation approach for reliable simulation of the progressive buckling phenomena of monopiles during installation. The numerical results allow safety envelopes to be established with respect to the boulder diameter, the contact eccentricity, the boulder depth, and the relative strength between the pile and the soil. These envelopes provided a quick evaluation of whether the tip deflection would happen with corresponding parameters

Attenuated Responses To Inflammatory Cytokines In Mouse Embryonic Stem Cells: Biological Implications And The Molecular Basis

Embryonic stem cells (ESCs) have attracted intense interest due to their great potential for regenerative medicine. However, their immune property is an overlooked but a significant issue that needs to be thoroughly investigated not only to resolve the concern for therapeutic applications but also for further understanding the early stage of organismal development. Recent studies demonstrated that ESCs are deficient in innate immune responses to viral/bacterial infections and inflammatory cytokines. Inflammatory conditions generally inhibit cell proliferation, which could be detrimental to ESCs, since cell proliferation is their dedicated task during early embryogenesis. Thus, I hypothesize that the attenuated innate immunity in ESCs could allow them to evade the cytotoxicity caused by immune reactions and is, therefore, a self-protective mechanism during early embryogenesis. We have differentiated mouse ESCs (mESCs) to fibroblast-like cells (mESC-FBs) which were proved to have partially developed innate immunity. Using these cells as a model for comparison with mESCs, the insensitivity of mESCs to the cytotoxic effects from IFNg, which is an inflammatory cytokine highly presented during early embryogenesis, and other inflammatory conditions were demonstrated, including attenuated expressions of inflammatory and signaling molecules, inactivated transcription factor and unaffected cell viability. Furthermore, basal expressions of protein phosphatases that inhibit IFNg pathway were higher in mESCs than mESC-FBs. Treating mESCs with protein phosphatases inhibitor upregulated the expression of IFNg induced signaling molecule. In all, the attenuated inflammatory responses are beneficial for mESCs, and the inhibition effects from protein phosphatases could, at least, partially explain their attenuated responses to IFNg

Potential landscape-scale pollinator networks across Great Britain: structure, stability and influence of agricultural land cover

Understanding spatial variation in the structure and stability of plant-pollinator networks, and their relationship with anthropogenic drivers, is key to maintaining pollination services and mitigating declines. Constructing sufficient networks to examine patterns over large spatial scales remains challenging. Using biological records (citizen science), we constructed potential plant-pollinator networks at 10km resolution across Great Britain, comprising all potential interactions inferred from recorded floral visitation and species co-occurrence. We calculated network metrics (species richness, connectance, pollinator and plant generality) and adapted existing methods to assess robustness to sequences of simulated plant extinctions across multiple networks. We found positive relationships between agricultural land cover and both pollinator generality and robustness to extinctions under several extinction scenarios. Increased robustness was attributable to changes in plant community composition (fewer extinction-prone species) and network structure (increased pollinator generality). Thus, traits enabling persistence in highly agricultural landscapes can confer robustness to potential future perturbations on plant-pollinator networks