Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

We live in an incredible age. Due to extraordinary advances in sciences and engineering, we better understand the world around us. At the same time, we witness profound changes in the technology, environment, societal organization, and economic well-being. We face new challenges never experienced by humans before. To efficiently address these challenges, the role of interdisciplinary interactions will continue to increase, as well as the role of mathematical, statistical, and computational models, providing a central link for such interactions.

Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here we review the theoretical and computational aspects behind these calculations, and propose an algorithm that simplifies the geometric minimum action method to minimize the action in the space of arc-length parametrized curves. We then illustrate this algorithm’s capabilities by applying it to various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable.

We discuss recent progress in finding all coherent states supported by nonlinear wave equations, their stability and the long time behavior of nearby solutions.

This paper deals with the numerical approximation of linear and non linear hyperbolic problems. We are mostly interested in the development of parameter free methods that satisfy a local maximum principle. We focus on the scalar case, but extensions to systems are relatively straightforward when these techniques are combined with the ideas contained in Abgrall (J. Comput. Phys., 214(2):773–808, 2006). In a first step, we precise the context, give conditions that guaranty that, under standard stability assumptions, the scheme will converge to weak solutions. In a second step, we provide conditions that guaranty an arbitrary order of accuracy. Then we provide several examples of such schemes and discuss in some details two versions. Numerical results support correctly our initial requirements: the schemes are accurate and satisfy a local maximum principle, even in the case of non smooth solutions.

A general method using multipliers for finding the conserved integrals admitted by any given partial differential equation (PDE) or system of partial differential equations is reviewed and further developed in several ways. Multipliers are expressions whose (summed) product with a PDE (system) yields a local divergence identity which has the physical meaning of a continuity equation involving a conserved density and a spatial flux for solutions of the PDE (system). On spatial domains, the integral form of a continuity equation yields a conserved integral. When a PDE (system) is variational, multipliers are known to correspond to infinitesimal symmetries of the variational principle, and the local divergence identity relating a multiplier to a conserved integral is the same as the variational identity used in Noether’s theorem for connecting conserved integrals to invariance of a variational principle. From this viewpoint, the general multiplier method is shown to constitute a modern form of Noether’s theorem in which the variational principle is not directly used. When a PDE (system) is non-variational, multipliers are shown to be an adjoint counterpart to infinitesimal symmetries, and the local divergence identity that relates a multiplier to a conserved integral is shown to be an adjoint generalization of the variational identity that underlies Noether’s theorem. Two main results are established for a general class of PDE systems having a solved-form for leading derivatives, which encompasses all typical PDE systems of physical interest. First, all non-trivial conserved integrals are shown to arise from non-trivial multipliers in a one-to-one manner, taking into account certain equivalence freedoms. Second, a simple scaling formula based on dimensional analysis is derived to obtain the conserved density and the spatial flux in any conserved integral, just using the corresponding multiplier and the given PDE (system). Also, a general class of multipliers that captures physically important conserved integrals such as mass, momentum, energy, angular momentum is identified. The derivations use a few basic tools from variational calculus, for which a concrete self-contained formulation is provided.

First principles calculations are computationally expensive. This information acquisition cost, combined with an exponentially high number of possible material configurations, constitutes an important roadblock towards the ultimate goal of materials by design. To overcome this barrier, one must devise schemes for the automatic and maximally informative selection of simulations. Such information acquisition decisions are task-dependent, in the sense that an optimal information acquisition policy for learning about a specific material property will not necessarily be optimal for learning about another. In this work, we develop an information acquisition policy for learning the ground state line (GSL) of binary alloys. Our approach is based on a Bayesian interpretation of the cluster expanded energy. This probabilistic surrogate of the energy enables us to quantify the epistemic uncertainty induced by the limited number of simulations which, in turn, is the key to defining a function of the configuration space that quantifies the expected improvement to the GSL resulting from a hypothetical simulation. We show that optimal information acquisition policies should balance the maximization of the expected improvement of the GSL and the minimization of the size of the simulated structure. We validate our approach by learning the GSLs of NiAl and TiAl binary alloys, where to establish the ground truth GSL we use the embedded-atom method (EAM) for the calculation of the energy of a given alloy configuration. Note that the proposed policies are directly applicable to the discovery of generic phase diagrams, if one can construct a probabilistic surrogate of the relevant thermodynamic potential.

Presented here are recent developments in spectral element methods for simulations of incompressible and low-Mach-number flows in domains with moving boundaries. Features include PDE-based mesh motion, implicit treatment of fluid–structure interaction based on a Green’s function decomposition, and an arbitrary Lagrangian-Eulerian formulation for low-Mach-number flows that includes an evolution equation for the background thermodynamic pressure. Several examples illustrate the basic principles introduced in the text.

In 2012 the UK Government identified eight great technologies which would act as a focus for future scientific research and funding. Other governments have produced similar lists. These vary from Big Data, through Agri-Science to Energy and its Storage. Mathematics lies at the heart of all of these technologies and acts to unify them all. In this paper I will review all of these technologies and look at the math behind each of them. In particular I will look in some detail at the mathematical issues involved in Big Data and energy. Overall I will aim to show that whilst it is very important that abstract mathematics is supported for its own right, the eight great technologies really do offer excellent opportunities for exciting new mathematical research and applications.

Credit Valuation Adjustment (CVA) has become an important field as its calculation is required in Basel III, issued in 2010, in the wake of the credit crisis. Exposure, which is defined as the potential future loss on a financial contract due to a default event, is one of the key elements for calculating CVA. This paper provides a backward dynamics framework for assessing exposure profiles of European, Bermudan and barrier options under the Heston and Heston Hull-White asset dynamics. We discuss the potential of the Stochastic Grid Bundling Method (SGBM), which is based on the techniques of simulation, regression and bundling (Jain and Oosterlee, Applied Mathematics and Computation, 269:412–431, 2015). By SGBM we can relatively easily compute the Potential Future Exposure (PFE) and sensitivities over the whole time horizon. Assuming independence between the default event and exposure profiles, we give here examples of calculating exposure, CVA and sensitivities for Bermudan and barrier options.

Given a family (Y ^k ,  k = 1, 2, …, N) of conditional Markov chains, we construct a conditional Markov chain X = (X ¹, …, X ^N ) such that X ^k , k = 1, 2, …, N, are conditional Markov chains, which are conditionally independent given the information contained in some filtration \(\mathbb{F}\), and such that for each k the conditional law of X ^k coincides with the conditional law of Y ^k . This is a new result that can be used to model different phenomena such as the gating behavior of multiple ion channels in a membrane patch, or credit ratings migrations.

In the new field of financial systemic risk, the network of interbank counterparty relationships can be described as a directed random graph. In cascade models of systemic risk, this skeleton acts as the medium through which financial contagion is propagated. It has been observed in real networks that such counterparty relationships exhibit negative assortativity, meaning that a bank’s counterparties are more likely to have unlike characteristics. This paper introduces and studies a general class of random graphs called the assortative configuration model, parameterized by an arbitrary node-type distribution P and edge-type distribution Q. The first main result is a law of large numbers that says the empirical edge-type distributions converge in probability to Q as the number of nodes N goes to infinity. The second main result is a formula for the large N asymptotic probability distribution of general graphical objects called configurations. This formula exhibits a key property called locally tree-like that in simpler models is known to imply strong results of percolation theory on the size of large connected clusters. Thus this paper provides the essential foundations needed to prove rigorous percolation bounds and cascade mappings in assortative networks.

Over the past century, nonlinear difference and differential equations have been used to understand conditions for coexistence of interacting populations. However, these models fail to account for random fluctuations due to demographic and environmental stochasticity which are experienced by all populations. I review some recent mathematical results about persistence and coexistence for models accounting for each of these forms of stochasticity. Demographic stochasticity stems from populations and communities consisting of a finite number of interacting individuals, and often are represented by Markovian models with a countable number of states. For closed populations in a bounded world, extinction occurs in finite time but may be preceded by long-term transients. Quasi-stationary distributions (QSDs) of these Markov models characterize this meta-stable behavior. For sufficiently large “habitat sizes”, QSDs are shown to concentrate on the positive attractors of deterministic models. Moreover, the probability extinction decreases exponentially with habitat size. Alternatively, environmental stochasticity stems from fluctuations in environmental conditions which influence survival, growth, and reproduction. Stochastic difference equations can be used to model the effects of environmental stochasticity on population and community dynamics. For these models, stochastic persistence corresponds to empirical measures placing arbitrarily little weight on arbitrarily low population densities. Sufficient and necessary conditions for stochastic persistence are reviewed. These conditions involve weighted combinations of Lyapunov exponents corresponding to “average” per-capita growth rates of rare species. The results are illustrated with how climatic variability influenced the dynamics of Bay checkerspot butterflies, the persistence of coupled sink populations, coexistence of competitors through the storage effect, and stochastic rock-paper-scissor communities. Open problems and conjectures are presented.

We discuss a new algorithm for finding all elliptic curves over \(\mathbb{Q}\) with a given conductor. Though based on (very) classical ideas, this approach appears to be computationally quite efficient. We provide details of the output from the algorithm in case of conductor p or p ², for p prime, with comparisons to existing data.

In this tutorial, we recall the main ingredients of the theory of dynamic games played over event trees and show step-by-step how to build a sustainable cooperative solution.