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Register a new userAccurate macroscale modelling of spatial dynamics in multiple dimensions
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Developments in dynamical systems theory provides new support for the macroscale modelling of pdes and other microscale systems such as Lattice Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically resolving subgrid microscale dynamics the dynamical systems approach constructs accurate closures of macroscale discretisations of the microscale system. Here we specifically explore reaction-diffusion problems in two spatial dimensions as a prototype of generic systems in multiple dimensions. Our approach unifies into one the modelling of systems by a type of finite elements, and the `equation free macroscale modelling of microscale simulators efficiently executing only on small patches of the spatial domain. Centre manifold theory ensures that a closed model exist on the macroscale grid, is emergent, and is systematically approximated. Dividing space either into overlapping finite elements or into spatially separated small patches, the specially crafted inter-element/patch coupling also ensures that the constructed discretisations are consistent with the microscale system/PDE to as high an order as desired. Computer algebra handles the considerable algebraic details as seen in the specific application to the Ginzburg--Landau PDE. However, higher order models in multiple dimensions require a mixed numerical and algebraic approach that is also developed. The modelling here may be straightforwardly adapted to a wide class of reaction-diffusion PDEs and lattice equations in multiple space dimensions. When applied to patches of microscopic simulations our coupling conditions promise efficient macroscale simulation.
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Developments in dynamical systems theory provides new support for the discretisation of pde{}s and other microscale systems. By systematically resolving subgrid microscale dynamics the new approach constructs asymptotically accurate, macroscale closures of discrete models of the pde. Here we explore reaction-diffusion problems in two spatial dimensions. Centre manifold theory ensures that slow manifold, holistic, discretisations exists, are quickly attractive, and are systematically approximated. Special coupling of the finite elements ensures that the resultant discretisations are consistent with the pde to as high an order as desired. Computer algebra handles the enormous algebraic details as seen in the specific application to the Ginzburg--Landau equation. However, higher order models in 2D appear to require a mixed numerical and algebraic approach that is also developed. Being driven by the residuals of the equations, the modelling here may be straightforwardly adapted to a wide class of reaction-diffusion differential and lattice equations in multiple space dimensions.
Developments in dynamical systems theory provides new support for the discretisation of pde{}s and other microscale systems. Here we explore the methodology applied to the gap-tooth scheme in the equation-free approach of Kevrekidis in two spatial dimensions. The algebraic detail is enormous so we detail computer algebra procedures to handle the enormity. However, modelling the dynamics on 2D spatial patches appears to require a mixed numerical and algebraic approach that is detailed in this report. Being based upon the computation of residuals, the procedures here may be simply adapted to a wide class of reaction-diffusion equations.
We present a new computational scheme that enables efficient and reliable Quantitative Trait Loci (QTL) scans for experimental populations. Using a standard brute-force exhaustive search effectively prohibits accurate QTL scans involving more than two loci to be performed in practice, at least if permutation testing is used to determine significance. Some more elaborate global optimization approaches, e.g. DIRECT, have earlier been adopted to QTL search problems. Dramatic speedups have been reported for high-dimensional scans. However, since a heuristic termination criterion must be used in these types of algorithms the accuracy of the optimization process cannot be guaranteed. Indeed, earlier results show that a small bias in the significance thresholds is sometimes introduced. Our new optimization scheme, PruneDIRECT, is based on an analysis leading to a computable (Lipschitz) bound on the slope of a transformed objective function. The bound is derived for both infinite size and finite size populations. Introducing a Lipschitz bound in DIRECT leads to an algorithm related to classical Lipschitz optimization. Regions in the search space can be permanently excluded (pruned) during the optimization process. Heuristic termination criteria can thus be avoided. Hence, PruneDIRECT has a well-defined error bound and can in practice be guaranteed to be equivalent to a corresponding exhaustive search. We present simulation results that show that for simultaneous mapping of three QTL using permutation testing, PruneDIRECT is typically more than 50 times faster than exhaustive search. The speedup is higher for stronger QTL. This could be used to quickly detect strong candidate eQTL networks.
Finite difference/element/volume methods of discretising PDEs impose a subgrid scale interpolation on the dynamics. In contrast, the holistic discretisation approach developed herein constructs a natural subgrid scale field adapted to the whole system out-of-equilibrium dynamics. Consequently, the macroscale discretisation is fully informed by the underlying microscale dynamics. We establish a new proof that in principle there exists an exact closure of the dynamics of a general class of reaction-advection-diffusion PDEs, and show how our approach constructs new systematic approximations to the in-principle closure starting from a simple, piecewise-linear, continuous approximation. Under inter-element coupling conditions that guarantee continuity of several field properties, the holistic discretisation possesses desirable properties such as a natural cubic spline first-order approximation to the field, and the self-adjointness of the diffusion operator under periodic, Dirichlet and Neumann macroscale boundary conditions. As a concrete example, we demonstrate the holistic discretisation procedure on the well-known Burgers PDE, and compare the theoretical and numerical stability of the resulting discretisation to other approximations. The approach developed here promises to be able to systematically construct automatically good, macroscale discretisations to a wide range of PDEs, including wave PDEs.
The estimation of the frequencies of multiple superimposed exponentials in noise is an important research problem due to its various applications from engineering to chemistry. In this paper, we propose an efficient and accurate algorithm that estimates the frequency of each component iteratively and consecutively by combining an estimator with a leakage subtraction scheme. During the iterative process, the proposed method gradually reduces estimation error and improves the frequency estimation accuracy. We give theoretical analysis where we derive the theoretical bias and variance of the frequency estimates and discuss the convergence behaviour of the estimator. We show that the algorithm converges to the asymptotic fixed point where the estimation is asymptotically unbiased and the variance is just slightly above the Cramer-Rao lower bound. We then verify the theoretical results and estimation performance using extensive simulation. The simulation results show that the proposed algorithm is capable of obtaining more accurate estimates than state-of-art methods with only a few iterations.
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