arXiv subjects

Honghu Liu

Publications and source records attributed to Honghu Liu.

32 records · Page 2Linked to original sources

arXiv2021

Conditional Gaussian Nonlinear System: a Fast Preconditioner and a Cheap Surrogate Model For Complex Nonlinear Systems

Developing suitable approximate models for analyzing and simulating complex nonlinear systems is practically important. This paper aims at exploring the skill of a rich class of nonlinear stochastic models, known as the conditional Gaussian nonlinear system (CGNS), as both a cheap surrogate model and a fast preconditioner for facilitating many computationally challenging tasks. The CGNS preserves the underlying physics to a large extent and can reproduce intermittency, extreme events and other non-Gaussian features in many complex systems arising from practical applications. Three interrelated topics are studied. First, the closed analytic formulae of solving the conditional statistics provide an efficient and accurate data assimilation scheme. It is shown that the data assimilation skill of a suitable CGNS approximate forecast model outweighs that by applying an ensemble method even to the perfect model with strong nonlinearity, where the latter suffers from filter divergence. Second, the CGNS allows the development of a fast algorithm for simultaneously estimating the parameters and the unobserved variables with uncertainty quantification in the presence of only partial observations. Utilizing an appropriate CGNS as a preconditioner significantly reduces the computational cost in accurately estimating the parameters in the original complex system. Finally, the CGNS advances rapid and statistically accurate algorithms for computing the probability density function and sampling the trajectories of the unobserved state variables. These fast algorithms facilitate the development of an efficient and accurate data-driven method for predicting the linear response of the original system with respect to parameter perturbations based on a suitable CGNS preconditioner.

math.NA↗

arXiv2021

Least-Squares Finite Element Method for Ordinary Differential Equations

We consider the least-squares finite element method (lsfem) for systems of nonlinear ordinary differential equations and establish an optimal error estimate for this method when piecewise linear elements are used. The main assumptions are that the vector field is sufficiently smooth and that the local Lipschitz constant, as well as the operator norm of the Jacobian matrix associated with the nonlinearity, are sufficiently small when restricted to a suitable neighborhood of the true solution for the considered initial value problem. This theoretic optimality is further illustrated numerically, along with evidence of possible extension to higher-order basis elements. Examples are also presented to show the advantages of lsfem compared with finite difference methods in various scenarios. Suitable modifications for adaptive time-stepping are discussed as well.

math.NA↗

arXiv2020

Optimal management of harvested population at the edge of extinction

Optimal control of harvested population at the edge of extinction in an unprotected area, is considered. The underlying population dynamics is governed by a Kolmogorov-Petrovsky-Piskunov equation with a harvesting term and space-dependent coefficients while the control consists of transporting individuals from a natural reserve. The nonlinear optimal control problem is approximated by means of a Galerkin scheme. Convergence result about the optimal controlled solutions and error estimates between the corresponding optimal controls, are derived. For certain parameter regimes, nearly optimal solutions are calculated from a simple logistic ordinary differential equation (ODE) with a harvesting term, obtained as a Galerkin approximation of the original partial differential equation (PDE) model. A critical allowable fraction $\underlineα$ of the reserve's population is inferred from the reduced logistic ODE with a harvesting term. This estimate obtained from the reduced model allows us to distinguish sharply between survival and extinction for the full PDE itself, and thus to declare whether a control strategy leads to success or failure for the corresponding rescue operation while ensuring survival in the reserve's population. In dynamical terms, this result illustrates that although continuous dependence on the forcing may hold on finite-time intervals, a high sensitivity in the system's response may occur in the asymptotic time. We believe that this work, by its generality, establishes bridges interesting to explore between optimal control problems of ODEs with a harvesting term and their PDE counterpart.

math.OC↗

arXiv2019

Variational approach to closure of nonlinear dynamical systems: Autonomous case

A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the more classical notion of slow manifold or invariant manifold when breakdown of slaving occurs. An optimal PM provides the manifold that describes the average motion of the neglected scales as a function of the resolved scales and allows, in principle, for determining the best vector field of the reduced state space that describes e.g. the dynamics' slow motion. The underlying optimal parameterizations are approximated by dynamically-based formulas derived analytically from the original equations. These formulas are contingent upon the determination of only a few (scalar) parameters obtained from minimization of cost functionals, depending on training dataset collected from direct numerical simulation. In practice, a training period of length comparable to a characteristic recurrence or decorrelation time of the dynamics, is sufficient for the efficient derivation of optimized parameterizations. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-Bénard convection are then discussed. The approach is finally illustrated --- in the context of the Kuramoto-Sivashinsky turbulence --- as providing efficient closures without slaving for a cutoff scale $k_c$ placed within the inertial range and the reduced state space is just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by classical invariant manifold approximations in their standard forms when the latter are used at this cutoff wavelength.

math.DS↗

arXiv2019

Data-Driven Correction Reduced Order Models for the Quasi-Geostrophic Equations: A Numerical Investigation

This paper investigates the recently introduced data-driven correction reduced order model (DDC-ROM) in the numerical simulation of the quasi-geostrophic equations. The DDC-ROM uses available data to model the correction term that is generally used to represent the missing information in low-dimensional ROMs. Physical constraints are added to the DDC-ROM to create the constrained data-driven correction reduced order model (CDDC-ROM) in order to further improve its accuracy and stability. Finally, the DDC-ROM is tested on time intervals that are longer than the time interval over which it was trained. The numerical investigation shows that, for low-dimensional ROMs, both the DDC-ROM and CDDC-ROM perform better than the standard Galerkin ROM (G-ROM) and the CDDC-ROM provides the best results.

physics.flu-dyn↗

arXiv2017

Galerkin approximations of nonlinear optimal control problems in Hilbert spaces

Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated for the first time in terms of optimal control of energy balance climate models posed on the sphere $\mathbb{S}^2$.

math.OC↗

arXiv2017

Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.

math.OC↗

arXiv2017

Regularized Reduced Order Models for a Stochastic Burgers Equation

In this paper, we study the numerical stability of reduced order models for convection-dominated stochastic systems in a relatively simple setting: a stochastic Burgers equation with linear multiplicative noise. Our preliminary results suggest that, in a convection-dominated regime, standard reduced order models yield inaccurate results in the form of spurious numerical oscillations. To alleviate these oscillations, we use the Leray reduced order model, which increases the numerical stability of the standard model by smoothing (regularizing) the convective term with an explicit spatial filter. The Leray reduced order model yields significantly better results than the standard reduced order model and is more robust with respect to changes in the strength of the noise.

physics.flu-dyn↗

arXiv2015

Low-Dimensional Galerkin Approximations of Nonlinear Delay Differential Equations

This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point mass, and introduce polynomials orthogonal with respect to such an inner product that live in the domain of the linear operator associated with the underlying DDE. These polynomials are then used to design a general Galerkin scheme for which we derive rigorous convergence results and show that it can be numerically implemented via simple analytic formulas. The scheme so obtained is applied to three nonlinear DDEs, two autonomous and one forced: (i) a simple DDE with distributed delays whose solutions recall Brownian motion; (ii) a DDE with a discrete delay that exhibits bimodal and chaotic dynamics; and (iii) a periodically forced DDE with two discrete delays arising in climate dynamics. In all three cases, the Galerkin scheme introduced in this article provides a good approximation by low-dimensional ODE systems of the DDE's strange attractor, as well as of the statistical features that characterize its nonlinear dynamics.

nlin.CD↗

arXiv2014

Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

This article proposes a new approach based on finite-horizon parameterizing manifolds (PMs) for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. Given a finite horizon $[0,T]$ and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the uncontrolled high modes by the controlled low ones so that the unexplained high-mode energy is reduced, in an $L^2$-sense, when this parameterization is applied. Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes (Chekroun, Liu, and Wang, 2013, http://arxiv.org/pdf/1310.3896v1.pdf). These formulas allow for an effective derivation of reduced ODE systems, aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. A priori error estimates between the resulting PM-based low-dimensional suboptimal controller $u_R^\ast$ and the optimal controller $u^*$ are derived. These estimates demonstrate that the closeness of $u_R^\ast$ to $u^*$ is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with $u_R^\ast$ and $u^*$; and (ii) the energy kept in the high modes of the PDE solution either driven by $u_R^\ast$ or $u^*$ itself. The practical performances of such PM-based suboptimal controllers are numerically assessed for various optimal control problems associated with a Burgers-type equation. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.

math.OC↗

arXiv2013

Non-Markovian Reduced Systems for Stochastic Partial Differential Equations: The Additive Noise Case

This article proposes for stochastic partial differential equations (SPDEs) driven by additive noise, a novel approach for the approximate parameterizations of the ``small'' scales by the ``large'' ones, along with the derivaton of the corresponding reduced systems. This is accomplished by seeking for stochastic parameterizing manifolds (PMs) introduced in a previous work by the authors(*), which are random manifolds aiming to provide --- in a mean square sense --- such approximate parameterizations. Backward-forward systems are designed to give access to such PMs as pullback limits depending through the nonlinear terms on the time-history of the dynamics of the low modes when the latter is simply approximated by its stochastic linear component. It is shown that the corresponding pullback limits can be efficiently determined, leading in turn to an operational procedure for the derivation of non-Markovian reduced systems able to achieve good modeling performances in practice. This is illustrated on a stochastic Burgers-type equation, where it is shown that the corresponding non-Markovian features of these reduced systems play a key role to reach such performances.

math.AP↗

arXiv2013

On stochastic parameterizing manifolds: Pullback characterization and Non-Markovian reduced equations

A general approach to provide approximate parameterizations of the "small" scales by the "large" ones, is developed for stochastic partial differential equations driven by linear multiplicative noise. This is accomplished via the concept of parameterizing manifolds (PMs) that are stochastic manifolds which improve in mean square error the partial knowledge of the full SPDE solution $u$ when compared to the projection of $u$ onto the resolved modes, for a given realization of the noise. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers (as parameterized by the sought PM) as a pullback limit depending on the time-history of the modes with low wave numbers. The resulting manifolds obtained by such a procedure are not subject to a spectral gap condition such as encountered in the classical theory. Instead, certain PMs can be determined under weaker non-resonance conditions. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. Such reduced systems take the form of SDEs involving random coefficients that convey memory effects via the history of the Wiener process, and arise from the nonlinear interactions between the low modes, embedded in the "noise bath." These random coefficients follow typically non-Gaussian statistics and exhibit an exponential decay of correlations whose rate depends explicitly on gaps arising in the non-resonances conditions. It is shown on a stochastic Burgers-type equation, that such PM-based reduced systems can achieve very good performance in reproducing statistical features of the SPDE dynamics projected onto the resolved modes, such as the autocorrelations and probability functions of the corresponding modes amplitude.

math.AP↗

arXiv2011

Dynamic Transitions for Quasilinear Systems and Cahn-Hilliard equation with Onsager mobility

The main objectives of this article are two-fold. First, we study the effect of the nonlinear Onsager mobility on the phase transition and on the well-posedness of the Cahn-Hilliard equation modeling a binary system. It is shown in particular that the dynamic transition is essentially independent of the nonlinearity of the Onsager mobility. However, the nonlinearity of the mobility does cause substantial technical difficulty for the well-posedness and for carrying out the dynamic transition analysis. For this reason, as a second objective, we introduce a systematic approach to deal with phase transition problems modeled by quasilinear partial differential equation, following the ideas of the dynamic transition theory developed recently by Ma and Wang.

math-ph↗

arXiv2011

Dynamic Transitions and Pattern Formations for Cahn-Hilliard Model with Long-Range Repulsive Interactions

The main objective of this article is to study the order-disorder phase transition and pattern formation for systems with long-range repulsive interactions. The main focus is on the Cahn-Hilliard model with a nonlocal term in the corresponding energy functional, representing the long-range repulsive interaction. First, we show that as soon as the linear problem loses stability, the system always undergoes a dynamic transition to one of the three types, forming different patterns/structures. The types of transition are then dictated by a nondimensional parameter, measuring the interactions between the long-range repulsive term and the quadratic and cubic nonlinearities in the model. The derived explicit form of this parameter offers precise information for the phase diagrams. Second, we obtain a novel and explicit pattern selection mechanism associated with the competition between the long-range repulsive interaction and the short-range attractive interactions. In particular, the hexagonal pattern is unique to the long-range interaction, and is associated with a novel two-dimensional reduced transition equations on the center manifold generated by the unstable modes, consisting of (degenerate) quadratic terms and non-degenerate cubic terms. Finally, explicit information on the metastability and basin of attraction of different disordered/ordered states and patterns are derived as well.

cond-mat.stat-mech↗