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Register a new userOptimising the relaxation route with optimal control
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A. Prados
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We look into the minimisation of the connection time between non-equilibrium steady states. As a prototypical example of an intrinsically non-equilibrium system, a driven granular gas is considered. For time-independent driving, its natural time scale for relaxation is characterised from an empirical -- the relaxation function -- and a theoretical -- the recently derived classical speed limits -- point of view. Using control theory, we find that bang-bang protocols -- comprising two steps, heating with the largest possible value of the driving and cooling with zero driving -- minimise the connecting time. The bang-bang time is shorter than both the empirical relaxation time and the classical speed limit: in this sense, the natural time scale for relaxation is beaten. Information theory quantities stemming from the Fisher information are also analysed over these optimal protocols. The implementation of the bang-bang processes in numerical simulations of the dynamics of the granular gas show an excellent agreement with the theoretical predictions. Moreover, general implications of our results are discussed for a wide class of driven non-equilibrium systems. Specifically, we show that analogous bang-bang protocols, with a number of bangs equal to the number of relevant physical variables, give the minimum connecting time under quite general conditions.
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We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high--order accuracy. Subsequently we extend these results to semi--lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings.
The mathematical modeling of tumor growth has a long history, and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using mathematical tools from statistical physics. To this extent, we introduce a novel kinetic model of growth which highlights the role of microscopic transitions in determining a variety of equilibrium distributions. At variance with other approaches, the mesoscopic description in terms of elementary interactions allows to design precise microscopic feedback control therapies, able to influence the natural tumor growth and to mitigate the risk factors involved in big sized tumors. We further show that under a suitable scaling both the free and controlled growth models correspond to Fokker--Planck type equations for the growth distribution with variable coefficients of diffusion and drift, whose steady solutions in the free case are given by a class of generalized Gamma densities which can be characterized by fat tails. In this scaling the feedback control produces an explicit modification of the drift operator, which is shown to strongly modify the emerging distribution for the tumor size. In particular, the size distributions in presence of therapies manifest slim tails in all growth models, which corresponds to a marked mitigation of the risk factors. Numerical results confirming the theoretical analysis are also presented.
The computation of free energies is a common issue in statistical physics. A natural technique to compute such high dimensional integrals is to resort to Monte Carlo simulations. However these techniques generally suffer from a high variance in the low temperature regime, because the expectation is often dominated by high values corresponding to rare system trajectories. A standard way to reduce the variance of the estimator is to modify the drift of the dynamics with a control enhancing the probability of rare event, leading to so-called importance sampling estimators. In theory, the optimal control leads to a zero-variance estimator; it is however defined implicitly and computing it is of the same difficulty as the original problem. We propose here a general strategy to build approximate optimal controls in the small temperature limit for diffusion processes, with the first goal to reduce the variance of free energy Monte Carlo estimators. Our construction builds upon low noise asymptotics by expanding the optimal control around the instanton, which is the path describing most likely fluctuations at low temperature. This technique not only helps reducing variance, but it is also interesting as a theoretical tool since it differs from usual small temperature expansions (WKB ansatz). As a complementary consequence of our expansion, we provide a perturbative formula for computing the free energy in the small temperature regime, which refines the now standard Freidlin--Wentzell asymptotics. We compute this expansion explicitly for lower orders, and explain how our strategy can be extended to an arbitrary order of accuracy. We support our findings with illustrative numerical examples.
Difference imaging is a technique for obtaining precise relative photometry of variable sources in crowded stellar fields and, as such, constitutes a crucial part of the data reduction pipeline in surveys for microlensing events or transiting extrasolar planets. The Optimal Image Subtraction (OIS) algorithm permits the accurate differencing of images by determining convolution kernels which, when applied to reference images of particularly good quality, provide excellent matches to the point-spread functions (PSF) in other images of the time series to be analysed. The convolution kernels are built as linear combinations of a set of basis functions, conventionally bivariate Gaussians modulated by polynomials. The kernel parameters must be supplied by the user and should ideally be matched to the PSF, pixel-sampling, and S/N of the data to be analysed. We have studied the outcome of the reduction as a function of the kernel parameters using our implementation of OIS within the TRIPP package. From the analysis of noise-free PSF simulations as well as test images from the ISIS OIS package, we derive qualitative and quantitative relations between the kernel parameters and the success of the subtraction as a function of the PSF sizes and sampling in reference and data images and compare the results to those of implementations in the literature. On this basis, we provide recommended parameters for data sets with different S/N and sampling.
The coupling of human movement dynamics with the function and design of wearable assistive devices is vital to better understand the interaction between the two. Advanced neuromuscular models and optimal control formulations provide the possibility to study and improve this interaction. In addition, optimal control can also be used to generate predictive simulations that generate novel movements for the human model under varying optimization criterion.
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