Journal: Journal of Nonlinear Science

Abbreviation

J. nonlinear sci.

Publisher

Springer

Journal Volumes

ISSN

1432-1467, 0938-8974

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Publications 1 - 10 of 12
  • Machine Learning Approximation Algorithms for High-Dimensional Fully Nonlinear Partial Differential Equations and Second-order Backward Stochastic Differential Equations
    Item type: Journal Article
    Beck, Christian; E, Weinan; Jentzen, Arnulf (2019)
    Journal of Nonlinear Science
    High-dimensional partial differential equations (PDEs) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work, we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (1) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (2) a merged formulation of the PDE and the 2BSDE problem, (3) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (4) a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python illustrate the efficiency and the accuracy of the method in the cases of a 100-dimensional Black–Scholes–Barenblatt equation, a 100-dimensional Hamilton–Jacobi–Bellman equation, and a nonlinear expectation of a 100-dimensional G-Brownian motion.
  • Coupled-Mode Equations and Gap Solitons in a Two-Dimensional Nonlinear Elliptic Problem with a Separable Periodic Potential
    Item type: Journal Article
    Dohnal, Tomáš; Pelinovsky, Dmitry; Schneider, Guido (2009)
    Journal of Nonlinear Science
  • Nonlinear Logarithmic Hyperelasticity with Isotropy in the Initial State
    Item type: Journal Article
    Heiduschke, Klaus (2023)
    Journal of Nonlinear Science
    Nonlinear stress–strain relations for hyperelasticity with isotropy in the initial state can be modeled by strain energies as functions of three (scalar) independent strain or deformation invariants. A model based on logarithmic strain tensors is compared to Rubin’s model based on unimodular Cauchy–Green deformation tensors. Using the three ordered eigenvalues determined by Cardano’s formula, three Mohr’s circles can be constructed for each of the stress and deformation/strain tensors involved. In Cardano’s formula and Mohr’s three circles, the spherical contributions resulting from the trace of the tensor and the deviatoric contributions resulting from the norm and determinant of its deviator are separated additively. On the one hand, when an elasticity model such as Rubin’s is based on (unimodular) Cauchy–Green deformation tensors, their traces or deviators have no simple direct interpretation, so that dilatation and distortion are coupled multiplicatively. The stress–strain relations then employ material functions of all three invariants, and Lode’s angles of the stress and deformation tensors generally differ. When, on the other hand, the elasticity model is based on logarithmic strain tensors, their traces and deviators do have the physical interpretation of finite dilatation and finite distortion. Dilatation and distortion are then decoupled additively. The question arises whether Lode’s angles of the stress and logarithmic strain should be different at all, and a simple isotropic elasticity model with matching Lode’s angles and the strain energy as a function of only two (dilatation and distortion) invariants is presented.
  • Dynamics of a rolling disk in the presence of dry friction
    Item type: Journal Article
    Le Saux C.; Leine, R.I.; Glocker, Christoph (2005)
    Journal of Nonlinear Science
  • Numerical Approaches for Investigating Quasiconvexity in the Context of Morrey's Conjecture
    Item type: Journal Article
    Voss, Jendrik; Martin, Robert J.; Sander, Oliver; et al. (2022)
    Journal of Nonlinear Science
  • The Spin-Orbit Resonances of the Solar System
    Item type: Journal Article
    Antognini, Francesco; Biasco, Luca; Chierchia, Luigi (2014)
    Journal of Nonlinear Science
  • Landscape Analysis for Shallow Neural Networks: Complete Classification of Critical Points for Affine Target Functions
    Item type: Journal Article
    Cheridito, Patrick; Jentzen, Arnulf; Rossmannek, Florian (2022)
    Journal of Nonlinear Science
    In this paper, we analyze the landscape of the true loss of neural networks with one hidden layer and ReLU, leaky ReLU, or quadratic activation. In all three cases, we provide a complete classification of the critical points in the case where the target function is affine and one-dimensional. In particular, we show that there exist no local maxima and clarify the structure of saddle points. Moreover, we prove that non-global local minima can only be caused by ‘dead’ ReLU neurons. In particular, they do not appear in the case of leaky ReLU or quadratic activation. Our approach is of a combinatorial nature and builds on a careful analysis of the different types of hidden neurons that can occur.
  • Asymptotic Dynamics of Inertial Particles with Memory
    Item type: Journal Article
    Provencher Langlois, Gabriel; Farazmand, Mohammad; Haller, George (2015)
    Journal of Nonlinear Science
  • Rigorous Model Reduction for a Damped-Forced Nonlinear Beam Model: An Infinite-Dimensional Analysis
    Item type: Journal Article
    Kogelbauer, Florian; Haller, George (2018)
    Journal of Nonlinear Science
    We use invariant manifold results on Banach spaces to conclude the existence of spectral submanifolds (SSMs) in a class of nonlinear, externally forced beam oscillations. SSMs are the smoothest nonlinear extensions of spectral subspaces of the linearized beam equation. Reduction in the governing PDE to SSMs provides an explicit low-dimensional model which captures the correct asymptotics of the full, infinite-dimensional dynamics. Our approach is general enough to admit extensions to other types of continuum vibrations. The model-reduction procedure we employ also gives guidelines for a mathematically self-consistent modeling of damping in PDEs describing structural vibrations.
  • Equilibria of Plane Convex Bodies
    Item type: Journal Article
    Allemann, Jonas; Hungerbühler, Norbert; Wasem, Micha (2021)
    Journal of Nonlinear Science
    We obtain a formula for the number of horizontal equilibria of a planar convex body K with respect to a center of mass O in terms of the winding number of the evolute of ∂K with respect to O. The formula extends to the case where O lies on the evolute of ∂K and a suitably modified version holds true for non-horizontal equilibria.
Publications 1 - 10 of 12