Higher order numerical schemes for stochastic partial differential equations that do not possess commutative noise require the simulation of iterated stochastic integrals. In this work, we extend the algorithms derived by Kloeden, Platen, and Wright [4] and by Wiktorsson [8] for the approximation of two-times iterated stochastic integrals involved in numerical schemes for finite dimensional stochastic ordinary differential equations to an infinite dimensional setting. These methods clear the way for new types of approximation schemes for SPDEs without commutative noise. Precisely, we analyze two algorithms to approximate two-times iterated integrals with respect to an infinite dimensional Q-Wiener process in case of a trace class operator Q given the increments of the Q-Wiener process. Error estimates in the mean-square sense are derived and discussed for both methods. In contrast to the finite dimensional setting, which is contained as a special case, the optimal approximation algorithm cannot be uniquely determined but is dependent on the covariance operator Q. This difference arises as the stochastic process is of infinite dimension.
We consider a higher-order Milstein scheme for stochastic partial differential equations with trace class noise which fulfill a certain commutativity condition. A novel technique to generally improve the order of convergence of Taylor schemes for stochastic partial differential equations is introduced. The key tool is an efficient approximation of the Milstein term by particularly tailored nested derivative-free terms. For the resulting derivative-free Milstein scheme the computational cost is, in general, considerably reduced by some power. Further, a rigorous computational cost model is considered and the so called effective order of convergence is introduced which allows to directly compare various numerical schemes in terms of their efficiency. As the main result, we prove for a broad class of stochastic partial differential equations, including equations with operators that do not need to be pointwise multiplicative, that the effective order of convergence of the proposed derivative-free Milstein scheme is significantly higher than for the original Milstein scheme. In this case, the derivative-free Milstein scheme outperforms the Euler scheme as well as the original Milstein scheme due to the reduction of the computational cost. Finally, we present some numerical examples that confirm the theoretical results. * Recently, it was shown by A. Jentzen and P. E. Kloeden [21] that a higher order of convergence can be obtained when employing schemes which are developed on the basis of the mild solution of (1), that is,t 0 e A(t−s) F (X s ) ds + t 0
ObjectiveStress is associated with body mass gain in some people, but with body mass loss in others. When the stressor persists, some people adapt with their stress responses whereas others don't. Heart-rate-variability (HRV) reflects ‘autonomic variability’ and is related to stress responses to psychosocial challenges. We hypothesized that the combined effects of ‘stress exposure’ and ‘autonomic variability’ predict long-term changes in body form.MethodsData of 1369 men and 612 women from the Whitehall II cohort were analyzed. Body-mass-index, hip-to-height-ratio and waist-to-height-ratio were measured at three time points over a ten-year period. HRV and ‘psychological distress’ (General-Health-Questionnaire) were assessed.ResultsMen with high psychological distress were at risk of developing an increased waist-to-height-ratio (F=3.4,P=0.038). Men with high psychological distress and low HRV were prone to develop an increased body mass and hip-to-height-ratio (psychological distress: F=4.3,P=0.016; HRV: F=5.0,P=0.008). We found statistical trends that women displayed similar patterns of stress-related changes in body form (P=0.061;P=0.063).ConclusionAssessing ‘psychological distress’ and ‘autonomic variability’ predicts changes in body form. Psychological distress was found associated with an increased risk of developing the ‘wide-waisted phenotype’, while psychological distress combined with low autonomic variability was associated with an increased risk of developing the ‘corpulent phenotype’.
Abstract. Biogeochemical models, capturing the major feedbacks of the pelagic ecosystem of the world ocean, are today often embedded into Earth system models which are increasingly used for decision making regarding climate policies. These models contain poorly constrained parameters (e.g., maximum phytoplankton growth rate), which are typically adjusted until the model shows reasonable behavior. Systematic approaches determine these parameters by minimizing the misfit between the model and observational data. In most common model approaches, however, the underlying functions mimicking the biogeochemical processes are nonlinear and non-convex. Thus, systematic optimization algorithms are likely to get trapped in local minima and might lead to non-optimal results. To judge the quality of an obtained parameter estimate, we propose determining a preferably large lower bound for the global optimum that is relatively easy to obtain and that will help to assess the quality of an optimum, generated by an optimization algorithm. Due to the unavoidable noise component in all observations, such a lower bound is typically larger than zero. We suggest deriving such lower bounds based on typical properties of biogeochemical models (e.g., a limited number of extremes and a bounded time derivative). We illustrate the applicability of the method with two real-world examples. The first example uses realworld observations of the Baltic Sea in a box model setup. The second example considers a three-dimensional coupled ocean circulation model in combination with satellite chlorophyll a.
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