23,576 research outputs found
A machine learning framework for data driven acceleration of computations of differential equations
We propose a machine learning framework to accelerate numerical computations
of time-dependent ODEs and PDEs. Our method is based on recasting
(generalizations of) existing numerical methods as artificial neural networks,
with a set of trainable parameters. These parameters are determined in an
offline training process by (approximately) minimizing suitable (possibly
non-convex) loss functions by (stochastic) gradient descent methods. The
proposed algorithm is designed to be always consistent with the underlying
differential equation. Numerical experiments involving both linear and
non-linear ODE and PDE model problems demonstrate a significant gain in
computational efficiency over standard numerical methods
Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem
In this paper we propose and analyze a method based on the Riccati
transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation
arising from the stochastic dynamic optimal allocation problem. We show how the
fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a
quasi-linear parabolic equation whose diffusion function is obtained as the
value function of certain parametric convex optimization problem. Although the
diffusion function need not be sufficiently smooth, we are able to prove
existence, uniqueness and derive useful bounds of classical H\"older smooth
solutions. We furthermore construct a fully implicit iterative numerical scheme
based on finite volume approximation of the governing equation. A numerical
solution is compared to a semi-explicit traveling wave solution by means of the
convergence ratio of the method. We compute optimal strategies for a portfolio
investment problem motivated by the German DAX 30 Index as an example of
application of the method
Thermo-Mechanical Wave Propagation In Shape Memory Alloy Rod With Phase Transformations
Many new applications of ferroelastic materials require a better
understanding of their dynamics that often involve phase transformations. In
such cases, an important prerequisite is the understanding of wave propagation
caused by pulse-like loadings. In the present study, a mathematical model is
developed to analyze the wave propagation process in shape memory alloy rods.
The first order martensite transformations and associated thermo-mechanical
coupling effects are accounted for by employing the modified
Ginzburg-Landau-Devonshire theory. The Landau-type free energy function is
employed to characterize different phases, while a Ginzburg term is introduced
to account for energy contributions from phase boundaries. The effect of
internal friction is represented by a Rayleigh dissipation term. The resulted
nonlinear system of PDEs is reduced to a differential-algebraic system, and
Chebyshev's collocation method is employed together with the backward
differentiation method. A series of numerical experiments are performed. Wave
propagations caused by impact loadings are analyzed for different initial
temperatures. It is demonstrated that coupled waves will be induced in the
material. Such waves will be dissipated and dispersed during the propagation
process, and phase transformations in the material will complicate their
propagation patterns. Finally, the influence of internal friction and capillary
effects on the process of wave propagation is analyzed numerically.Comment: Keywords: nonlinear waves, thermo-mechanical coupling, martensite
transformations, Ginzburg-Landau theory, Chebyshev collocation metho
Second-order Stable Finite Difference Schemes for the Time-fractional Diffusion-wave Equation
We propose two stable and one conditionally stable finite difference schemes
of second-order in both time and space for the time-fractional diffusion-wave
equation. In the first scheme, we apply the fractional trapezoidal rule in time
and the central difference in space. We use the generalized Newton-Gregory
formula in time for the second scheme and its modification for the third
scheme. While the second scheme is conditionally stable, the first and the
third schemes are stable. We apply the methodology to the considered equation
with also linear advection-reaction terms and also obtain second-order schemes
both in time and space. Numerical examples with comparisons among the proposed
schemes and the existing ones verify the theoretical analysis and show that the
present schemes exhibit better performances than the known ones
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