arXiv subjects

Leon Sallandt

Publications and source records attributed to Leon Sallandt.

6 recordsLinked to original sources

arXiv2023

From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs

The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination of Monte Carlo methods and variational formulations, using neural networks for function approximation. Extending previous work (Richter et al., 2021), we argue that tensor trains provide an appealing framework for parabolic PDEs: The combination of reformulations in terms of backward stochastic differential equations and regression-type methods holds the promise of leveraging latent low-rank structures, enabling both compression and efficient computation. Emphasizing a continuous-time viewpoint, we develop iterative schemes, which differ in terms of computational efficiency and robustness. We demonstrate both theoretically and numerically that our methods can achieve a favorable trade-off between accuracy and computational efficiency. While previous methods have been either accurate or fast, we have identified a novel numerical strategy that can often combine both of these aspects.

cs.LG↗

arXiv2021

Solving high-dimensional parabolic PDEs using the tensor train format

High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and efficient computation. Following this paradigm, we develop novel iterative schemes, involving either explicit and fast or implicit and accurate updates. We demonstrate in a number of examples that our methods achieve a favorable trade-off between accuracy and computational efficiency in comparison with state-of-the-art neural network based approaches.

stat.ML↗

arXiv2021

Approximating the Stationary Bellman Equation by Hierarchical Tensor Products

We treat infinite horizon optimal control problems by solving the associated stationary Hamilton-Jacobi-Bellman (HJB) equation numerically to compute the value function and an optimal feedback law. The dynamical systems under consideration are spatial discretizations of non linear parabolic partial differential equations (PDE), which means that the HJB is non linear and suffers from the curse of dimensionality. Its non linearity is handled by the Policy Iteration algorithm, where the problem is reduced to a sequence of linear, hyperbolic PDEs. These equations remain the computational bottleneck due to their high dimensions. By the method of characteristics these linearized HJB equations can be reformulated via the Koopman operator in the spirit of dynamic programming. The resulting operator equations are solved using a minimal residual method. To overcome numerical infeasability we use low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors), and multi-polynomials, together with high dimensional quadrature, e.g. Monte-Carlo. By controlling a destabilized version of viscous Burgers and a diffusion equation with unstable reaction term numerical evidences are given.

math.OC↗

arXiv2021

Approximating optimal feedback controllers of finite horizon control problems using hierarchical tensor formats

Controlling systems of ordinary differential equations (ODEs) is ubiquitous in science and engineering. For finding an optimal feedback controller, the value function and associated fundamental equations such as the Bellman equation and the Hamilton-Jacobi-Bellman (HJB) equation are essential. The numerical treatment of these equations poses formidable challenges due to their non-linearity and their (possibly) high-dimensionality. In this paper we consider a finite horizon control system with associated Bellman equation. After a time-discretization, we obtain a sequence of short time horizon problems which we call local optimal control problems. For solving the local optimal control problems we apply two different methods, one being the well-known policy iteration, where a fixed-point iteration is required for every time step. The other algorithm borrows ideas from Model Predictive Control (MPC), by solving the local optimal control problem via open-loop control methods on a short time horizon, allowing us to replace the fixed-point iteration by an adjoint method. For high-dimensional systems we apply low rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains (TT tensors) and multi-polynomials, together with high-dimensional quadrature, e.g. Monte-Carlo. We prove a linear error propagation with respect to the time discretization and give numerical evidence by controlling a diffusion equation with unstable reaction term and an Allen-Kahn equation.

math.OC↗

arXiv2021

Pricing high-dimensional Bermudan options with hierarchical tensor formats

An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the "curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo least-squares approach as well as the dual martingale method, both using high-dimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods.

q-fin.CP↗

arXiv2020

Approximative Policy Iteration for Exit Time Feedback Control Problems driven by Stochastic Differential Equations using Tensor Train format

We consider a stochastic optimal exit time feedback control problem. The Bellman equation is solved approximatively via the Policy Iteration algorithm on a polynomial ansatz space by a sequence of linear equations. As high degree multi-polynomials are needed, the corresponding equations suffer from the curse of dimensionality even in moderate dimensions. We employ tensor-train methods to account for this problem. The approximation process within the Policy Iteration is done via a Least-Squares ansatz and the integration is done via Monte-Carlo methods. Numerical evidences are given for the (multi dimensional) double well potential and a three-hole potential.

math.OC↗