Exponentially convergent data assimilation algorithm for Navier-Stokes equations

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in R2. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains bounded noise. The algorithm designs a time-dependent gain such that the estimation error converges to zero exponentially. The sufficient condition for the existence of the gain are formulated in the form of algebraic Riccati equations. To demonstrate the results we apply the proposed algorithm to the reconstruction a chaotic fluid flow from incomplete and noisy data

Particle based gPC methods for mean-field models of swarming with uncertainty

In this work we focus on the construction of numerical schemes for the approximation of stochastic mean--field equations which preserve the nonnegativity of the solution. The method here developed makes use of a mean-field Monte Carlo method in the physical variables combined with a generalized Polynomial Chaos (gPC) expansion in the random space. In contrast to a direct application of stochastic-Galerkin methods, which are highly accurate but lead to the loss of positivity, the proposed schemes are capable to achieve high accuracy in the random space without loosing nonnegativity of the solution. Several applications of the schemes to mean-field models of collective behavior are reported.Comment: Communications in Computational Physics, to appea

최적 투자 전략을 위한 딥러닝 알고리즘

학위논문 (석사) -- 서울대학교 대학원 : 자연과학대학 수리과학부, 2021. 2. 박형빈.This paper treats the Merton problem that how to invest in safe assets and risky assets to maximize an investor's utility, given by investment opportunities modeled by a d-dimensional state process, whose dimension is extended from Guasoni and Robertson (2015). The problem is represented by a partial differential equation with optimizing term: the Hamilton-Jacobi-Bellman equation. The main purpose of this paper is to solve partial differential equations derived from the Hamilton-Jacobi-Bellman equations with a deep learning algorithm: the Deep Galerkin method, first suggested by Sirignano and Spiliopoulos (2018). We then apply the algorithm to get the solution of the PDE based on some model settings and compare with the one from the finite difference method.본 논문은 투자자의 효용을 극대화하고자 안전 자산과 위험 자산의 투자 문제인 Merton 문제를 다룬다. 투자 기회는 Guasoni와 Robertson의 단일 상태변수의 차원을 확장한 d차원 상태변수로 주어진다. Merton 문제는 최적화 항을 포함한 편미분방정식인 Hamilton-Jacobi-Bellman(HJB) 방정식으로 표현된다. 본 논문의 주요 목적은 Hamilton-Jacobi-Bellman 방정식으로 도출한 편미분방정식의 해를 Sirignano와 Spiliopoulos가 처음 제안한 딥러닝 알고리즘인 심층 Galerkin 방법으로 구하는 것이다. 특정 조건으로 설정한 모델 하에서 알고리즘을 적용해 편미분방정식의 해를 구하고 유한차분법으로 구한 해와 비교한다.Abstract i 1 Introduction 1 2 Optimal Investment Problem 4 2.1 Market with the Merton Problem 4 2.2 The Hamilton–Jacobi–Bellman Equation 6 3 Deep Galerkin Method 10 3.1 Algorithm 10 3.2 Neural Network Approximation 13 4 Numerical Test 15 4.1 Model Settings 15 4.2 Calibration 16 4.3 Implementation 17 4.4 Comparing with the Finite Difference Method 18 5 Conclusion 26 A Proof of Theorem 3.2.1 28 A.1 Convergence of the loss functional 28 A.2 Convergence of the DNN function to the solution of PDEs 32 Abstract (in Korean) 41Maste

Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods

Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions

This paper is devoted to the construction of structure preserving stochastic Galerkin schemes for Fokker-Planck type equations with uncertainties and interacting with an external distribution, that we refer to as a background distribution. The proposed methods are capable to preserve physical properties in the approximation of statistical moments of the problem like nonnegativity, entropy dissipation and asymptotic behaviour of the expected solution. The introduced methods are second order accurate in the transient regimes and high order for large times. We present applications of the developed schemes to the case of fixed and dynamic background distribution for models of collective behaviour