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Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations
(2018)SAM Research ReportReport -
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A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
(2018)SAM Research ReportArtificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate ...Report -
DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing
(2018)Research ReportReport -
Solving stochastic differential equations and Kolmogorov equations by means of deep learning
(2018)Research ReportReport -
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Strong error analysis for stochastic gradient descent optimization algorithms
(2018)SAM Research ReportStochastic 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 ε ∈ (0,∞) and every arbitrarily large p ∈ (0,∞) that the considered SGD optimization algorithm converges in the strong Lp-sense with order 1/2 − ε to the global minimum of ...Report