20,631 research outputs found
Stochastic Optimal Prediction with Application to Averaged Euler Equations
Optimal prediction (OP) methods compensate for a lack of resolution in the
numerical solution of complex problems through the use of an invariant measure
as a prior measure in the Bayesian sense. In first-order OP, unresolved
information is approximated by its conditional expectation with respect to the
invariant measure. In higher-order OP, unresolved information is approximated
by a stochastic estimator, leading to a system of random or stochastic
differential equations.
We explain the ideas through a simple example, and then apply them to the
solution of Averaged Euler equations in two space dimensions.Comment: 13 pages, 2 figure
Mean-Field Sparse Optimal Control
We introduce the rigorous limit process connecting finite dimensional sparse
optimal control problems with ODE constraints, modeling parsimonious
interventions on the dynamics of a moving population divided into leaders and
followers, to an infinite dimensional optimal control problem with a constraint
given by a system of ODE for the leaders coupled with a PDE of Vlasov-type,
governing the dynamics of the probability distribution of the followers. In the
classical mean-field theory one studies the behavior of a large number of small
individuals freely interacting with each other, by simplifying the effect of
all the other individuals on any given individual by a single averaged effect.
In this paper we address instead the situation where the leaders are actually
influenced also by an external policy maker, and we propagate its effect for
the number of followers going to infinity. The technical derivation of the
sparse mean-field optimal control is realized by the simultaneous development
of the mean-field limit of the equations governing the followers dynamics
together with the -limit of the finite dimensional sparse optimal
control problems.Comment: arXiv admin note: text overlap with arXiv:1306.591
A Class of Mean-field LQG Games with Partial Information
The large-population system consists of considerable small agents whose
individual behavior and mass effect are interrelated via their state-average.
The mean-field game provides an efficient way to get the decentralized
strategies of large-population system when studying its dynamic optimizations.
Unlike other large-population literature, this current paper possesses the
following distinctive features. First, our setting includes the partial
information structure of large-population system which is practical from real
application standpoint. Specially, two cases of partial information structure
are considered here: the partial filtration case (see Section 2, 3) where the
available information to agents is the filtration generated by an observable
component of underlying Brownian motion; the noisy observation case (Section 4)
where the individual agent can access an additive white-noise observation on
its own state. Also, it is new in filtering modeling that our sensor function
may depend on the state-average. Second, in both cases, the limiting
state-averages become random and the filtering equations to individual state
should be formalized to get the decentralized strategies. Moreover, it is also
new that the limit average of state filters should be analyzed here. This makes
our analysis very different to the full information arguments of
large-population system. Third, the consistency conditions are equivalent to
the wellposedness of some Riccati equations, and do not involve the fixed-point
analysis as in other mean-field games. The -Nash equilibrium
properties are also presented.Comment: 19 page
Data-driven modelling of biological multi-scale processes
Biological processes involve a variety of spatial and temporal scales. A
holistic understanding of many biological processes therefore requires
multi-scale models which capture the relevant properties on all these scales.
In this manuscript we review mathematical modelling approaches used to describe
the individual spatial scales and how they are integrated into holistic models.
We discuss the relation between spatial and temporal scales and the implication
of that on multi-scale modelling. Based upon this overview over
state-of-the-art modelling approaches, we formulate key challenges in
mathematical and computational modelling of biological multi-scale and
multi-physics processes. In particular, we considered the availability of
analysis tools for multi-scale models and model-based multi-scale data
integration. We provide a compact review of methods for model-based data
integration and model-based hypothesis testing. Furthermore, novel approaches
and recent trends are discussed, including computation time reduction using
reduced order and surrogate models, which contribute to the solution of
inference problems. We conclude the manuscript by providing a few ideas for the
development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and
Multiscale Dynamics (American Scientific Publishers
Optimization and Control of Agent-Based Models in Biology: A Perspective
Agent-based models (ABMs) have become an increasingly important mode of inquiry for the life sciences. They are particularly valuable for systems that are not understood well enough to build an equation-based model. These advantages, however, are counterbalanced by the difficulty of analyzing and using ABMs, due to the lack of the type of mathematical tools available for more traditional models, which leaves simulation as the primary approach. As models become large, simulation becomes challenging. This paper proposes a novel approach to two mathematical aspects of ABMs, optimization and control, and it presents a few first steps outlining how one might carry out this approach. Rather than viewing the ABM as a model, it is to be viewed as a surrogate for the actual system. For a given optimization or control problem (which may change over time), the surrogate system is modeled instead, using data from the ABM and a modeling framework for which ready-made mathematical tools exist, such as differential equations, or for which control strategies can explored more easily. Once the optimization problem is solved for the model of the surrogate, it is then lifted to the surrogate and tested. The final step is to lift the optimization solution from the surrogate system to the actual system. This program is illustrated with published work, using two relatively simple ABMs as a demonstration, Sugarscape and a consumer-resource ABM. Specific techniques discussed include dimension reduction and approximation of an ABM by difference equations as well systems of PDEs, related to certain specific control objectives. This demonstration illustrates the very challenging mathematical problems that need to be solved before this approach can be realistically applied to complex and large ABMs, current and future. The paper outlines a research program to address them
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