ALBERT

Description: 〈span〉〈div〉Abstract〈/div〉Habitat patterns of subtropical and tropical planktic foraminifers in the Caribbean Sea were obtained from plankton samples collected in spring 2009 and 2013. The spatial distribution in surface waters (3.5 m water depth) and depth habitat patterns (surface to 400 m) of 33 species were compared with prevailing water-mass conditions (temperature, salinity, and chlorophyll-〈span〉a〈/span〉 concentration) and planktic foraminiferal test assemblages in surface sediments. Distribution patterns indicate a significant relationship with seawater temperature and trophic conditions. A reduction in standing stocks was observed close to the Orinoco River plume and in the Gulf of Paria, associated with high turbidity and concomitant low surface-water salinity. In contrast, a transient mesoscale patch of high chlorophyll concentration in the eastern Caribbean Sea was associated with higher standing stocks in near surface waters, including high abundances of 〈span〉Globigerinita glutinata〈/span〉 and 〈span〉Neogloboquadrina dutertrei〈/span〉. 〈span〉Globorotalia truncatulinoides〈/span〉 mainly lives close to the seasonal pycnocline and can be linked to winter conditions indicated by lower sea-surface temperatures (SSTs) of ∼20°C. 〈span〉Globigerinoides sacculifer〈/span〉 and 〈span〉Globoturborotalita rubescens〈/span〉 were associated with oligotrophic conditions in the pelagic Caribbean Sea during early spring and showed a synodic lunar reproduction cycle. The live assemblages in the water column from 2009 and 2013 were similar to those reported in earlier studies from the 1960s and 1990s and to assemblages of tests in the surface sediments. Minor differences in faunal proportions were attributed to seasonal variability and environmental differences at the local scale. An exception was the low relative abundance of 〈span〉Globigerinoides ruber〈/span〉 in the Caribbean Sea in 2009 compared to surface sediment samples and plankton net samples collected in the 1960s and 1990s. Decreasing abundance of 〈span〉Gs. ruber〈/span〉 white in the Caribbean Sea may be associated with increasing SSTs over past decades and changes in nutrient flux and primary production.〈/span〉

Description: In this article we propose a new, explicit and easily implementable numerical method for approximating a class of semilinear stochastic evolution equations with non-globally Lipschitz continuous nonlinearities. We establish strong convergence rates for this approximation method in the case of semilinear stochastic evolution equations with globally monotone coefficients. Our strong convergence result, in particular, applies to a class of stochastic reaction–diffusion partial differential equations.

Description: Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform a rigorous strong error analysis for SGD optimization algorithms. In particular, we prove for every arbitrarily small $varepsilon in (0,infty )$ and every arbitrarily large $p{,in,} (0,infty )$ that the considered SGD optimization algorithm converges in the strong $L^p$-sense with order $1/2-varepsilon $ to the global minimum of the objective function of the considered stochastic optimization problem under standard convexity-type assumptions on the objective function and relaxed assumptions on the moments of the stochastic errors appearing in the employed SGD optimization algorithm. The key ideas in our convergence proof are, first, to employ techniques from the theory of Lyapunov-type functions for dynamical systems to develop a general convergence machinery for SGD optimization algorithms based on such functions, then, to apply this general machinery to concrete Lyapunov-type functions with polynomial structures and, thereafter, to perform an induction argument along the powers appearing in the Lyapunov-type functions in order to achieve for every arbitrarily large $ p in (0,infty ) $ strong $ L^p $-convergence rates.

Description: We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process that is strongly Hölder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Hölder norms and the convergence rate is essentially reduced by the Hölder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach-space-valued stochastic processes.

Description: Recently, artificial neural networks (ANNs) in conjunction with stochastic gradient descent optimization methods have been employed to approximately compute solutions of possibly rather high-dimensional partial differential equations (PDEs). Very recently, there have also been a number of rigorous mathematical results in the scientific literature, which examine the approximation capabilities of such deep learning-based approximation algorithms for PDEs. These mathematical results from the scientific literature prove in part that algorithms based on ANNs are capable of overcoming the curse of dimensionality in the numerical approximation of high-dimensional PDEs. In these mathematical results from the scientific literature, usually the error between the solution of the PDE and the approximating ANN is measured in the $L^p$-sense, with respect to some $p in [1,infty )$ and some probability measure. In many applications it is, however, also important to control the error in a uniform $L^infty $-sense. The key contribution of the main result of this article is to develop the techniques to obtain error estimates between solutions of PDEs and approximating ANNs in the uniform $L^infty $-sense. In particular, we prove that the number of parameters of an ANN to uniformly approximate the classical solution of the heat equation in a region $ [a,b]^d $ for a fixed time point $ T in (0,infty ) $ grows at most polynomially in the dimension $ d in {mathbb {N}} $ and the reciprocal of the approximation precision $ varepsilon〉 0 $. This verifies that ANNs can overcome the curse of dimensionality in the numerical approximation of the heat equation when the error is measured in the uniform $L^infty $-norm.

Description: Habitat patterns of subtropical and tropical planktic foraminifers in the Caribbean Sea were obtained from plankton samples collected in spring 2009 and 2013. The spatial distribution in surface waters (3.5 m water depth) and depth habitat patterns (surface to 400 m) of 33 species were compared with prevailing water-mass conditions (temperature, salinity, and chlorophyll-a concentration) and planktic foraminiferal test assemblages in surface sediments. Distribution patterns indicate a significant relationship with seawater temperature and trophic conditions. A reduction in standing stocks was observed close to the Orinoco River plume and in the Gulf of Paria, associated with high turbidity and concomitant low surface-water salinity. In contrast, a transient mesoscale patch of high chlorophyll concentration in the eastern Caribbean Sea was associated with higher standing stocks in near surface waters, including high abundances of Globigerinita glutinata and Neogloboquadrina dutertrei. Globorotalia truncatulinoides mainly lives close to the seasonal pycnocline and can be linked to winter conditions indicated by lower sea-surface temperatures (SSTs) of ∼20°C. Globigerinoides sacculifer and Globoturborotalita rubescens were associated with oligotrophic conditions in the pelagic Caribbean Sea during early spring and showed a synodic lunar reproduction cycle. The live assemblages in the water column from 2009 and 2013 were similar to those reported in earlier studies from the 1960s and 1990s and to assemblages of tests in the surface sediments. Minor differences in faunal proportions were attributed to seasonal variability and environmental differences at the local scale. An exception was the low relative abundance of Globigerinoides ruber in the Caribbean Sea in 2009 compared to surface sediment samples and plankton net samples collected in the 1960s and 1990s. Decreasing abundance of Gs. ruber white in the Caribbean Sea may be associated with increasing SSTs over past decades and changes in nutrient flux and primary production.