Towards understanding tree root profiles: simulating hydrologically optimal strategies for root distribution

In this modelling study differences in vertical root distributions measured in four contrasting forest locations in the Netherlands were investigated. Root distributions are seen as a reflection of the plant’s optimisation strategy, based on hydrological grounds. The 'optimal' root distribution is defined as the one that maximises the water uptake from the root zone over a period of ten years. The optimal root distributions of four forest locations with completely different soil physical characteristics are calculated using the soil hydrological model SWIF. Two different model configurations for root interactions were tested: the standard model configuration in which one single root profile was used (SWIF-NC), and a model configuration in which two root profiles compete for the same available water (SWIF-C). The root profiles were parameterised with genetic algorithms. The fitness of a certain root profile was defined as the amount of water uptake over a simulation period of ten years. The root profiles of SWIF-C were optimised using an evolutionary game. The results showed clear differences in optimal root distributions between the various sites and also between the two model configurations. Optimisation with SWIF-C resulted in root profiles that were easier to interpret in terms of feasible biological strategies. Preferential water uptake in wetter soil regions was an important factor for interpretation of the simulated root distributions. As the optimised root profiles still showed differences with measured profiles, this analysis is presented, not as the final solution for explaining differences in root profiles of vegetation but as a first step using an optimisation theory to increase understanding of the root profiles of trees.</p> <p style='line-height: 20px;'><b>Keywords:</b> forest hydrology, optimisation, root

Towards understanding tree root profiles: simulating hydrologically optimal strategies for root distribution

International audienceIn this modelling study differences in vertical root distributions measured in four contrasting forest locations in the Netherlands were investigated. Root distributions are seen as a reflection of the plant's optimisation strategy, based on hydrological grounds. The "optimal" root distribution is defined as the one that maximises the water uptake from the root zone over a period of ten years. The optimal root distributions of four forest locations with completely different soil physical characteristics are calculated using the soil hydrological model SWIF. Two different model configurations for root interactions were tested: the standard model configuration in which one single root profile was used (SWIF-NC), and a model configuration in which two root profiles compete for the same available water (SWIF-C). The root profiles were parameterised with genetic algorithms. The fitness of a certain root profile was defined as the amount of water uptake over a simulation period of ten years. The root profiles of SWIF-C were optimised using an evolutionary game. The results showed clear differences in optimal root distributions between the various sites and also between the two model configurations. Optimisation with SWIF-C resulted in root profiles that were easier to interpret in terms of feasible biological strategies. Preferential water uptake in wetter soil regions was an important factor for interpretation of the simulated root distributions. As the optimised root profiles still showed differences with measured profiles, this analysis is presented, not as the final solution for explaining differences in root profiles of vegetation but as a first step using an optimisation theory to increase understanding of the root profiles of trees. Keywords: forest hydrology, optimisation, root

Storage capacity of correlated perceptrons

We consider an ensemble of $K$ single-layer perceptrons exposed to random inputs and investigate the conditions under which the couplings of these perceptrons can be chosen such that prescribed correlations between the outputs occur. A general formalism is introduced using a multi-perceptron costfunction that allows to determine the maximal number of random inputs as a function of the desired values of the correlations. Replica-symmetric results for $K=2$ and $K=3$ are compared with properties of two-layer networks of tree-structure and fixed Boolean function between hidden units and output. The results show which correlations in the hidden layer of multi-layer neural networks are crucial for the value of the storage capacity.Comment: 16 pages, Latex2

Weak measurement and rapid state reduction in bipartite quantum systems

In this paper we consider feedback control algorithms for the rapid purification of a bipartite state consisting of two qubits, when the observer has access to only one of the qubits. We show 1) that the algorithm that maximizes the average purification rate is not the same as that that for a single qubit, and 2) that it is always possible to construct an algorithm that generates a deterministic rate of purification for {\em both} qubits. We also reveal a key difference between projective and continuous measurements with regard to state-purification.Comment: 4 pages, 3 figure

A discrete invitation to quantum filtering and feedback control

The engineering and control of devices at the quantum-mechanical level--such as those consisting of small numbers of atoms and photons--is a delicate business. The fundamental uncertainty that is inherently present at this scale manifests itself in the unavoidable presence of noise, making this a novel field of application for stochastic estimation and control theory. In this expository paper we demonstrate estimation and feedback control of quantum mechanical systems in what is essentially a noncommutative version of the binomial model that is popular in mathematical finance. The model is extremely rich and allows a full development of the theory, while remaining completely within the setting of finite-dimensional Hilbert spaces (thus avoiding the technical complications of the continuous theory). We introduce discretized models of an atom in interaction with the electromagnetic field, obtain filtering equations for photon counting and homodyne detection, and solve a stochastic control problem using dynamic programming and Lyapunov function methods.Comment: 76 pages, 12 figures. A PDF file with high resolution figures can be found at http://minty.caltech.edu/papers.ph

Training a perceptron in a discrete weight space

On-line and batch learning of a perceptron in a discrete weight space, where each weight can take $2 L+1$ different values, are examined analytically and numerically. The learning algorithm is based on the training of the continuous perceptron and prediction following the clipped weights. The learning is described by a new set of order parameters, composed of the overlaps between the teacher and the continuous/clipped students. Different scenarios are examined among them on-line learning with discrete/continuous transfer functions and off-line Hebb learning. The generalization error of the clipped weights decays asymptotically as $exp(-K \alpha^2)$/$exp(-e^{|\lambda| \alpha})$ in the case of on-line learning with binary/continuous activation functions, respectively, where $\alpha$ is the number of examples divided by N, the size of the input vector and $K$ is a positive constant that decays linearly with 1/L. For finite $N$ and $L$, a perfect agreement between the discrete student and the teacher is obtained for $\alpha \propto \sqrt{L \ln(NL)}$. A crossover to the generalization error $\propto 1/\alpha$, characterized continuous weights with binary output, is obtained for synaptic depth $L > O(\sqrt{N})$.Comment: 10 pages, 5 figs., submitted to PR