arXiv subjects

Yufei Zhang

Publications and source records attributed to Yufei Zhang.

154 records · Page 9Linked to original sources

arXiv2020

Adaptive sampling of pressure distribution features for the airfoil statistical analysis

In the area of supercritical wing design, a variety of principles, laws and rules have been summarized by scholars who perform theoretical and experimental analyses. The applicability of these rules is usually restricted by the airfoil samples investigated. With the advance of computational fluid dynamics and computational intelligence, such work can be better conducted on computers. The present paper proposes an output space sampling method to generate airfoil samples that have specified pressure distributions or meet certain special requirements. The well-selected and distributed samples are then utilized for statistical studies to obtain more reliable or more universal aerodynamics rules that can be used as guidance in the process of supercritical airfoil design. The capabilities of the output space sampling method in regard to filling the space and exploring the boundaries are also improved. The output space sampling method is employed to generate supercritical airfoil samples with different requirements, allowing the relationships between the shock wave location and the drag divergence Mach number as well as the drag creep characteristic to be revealed.

physics.flu-dyn↗

arXiv2020

Feature selection and processing of turbulence modeling based on an artificial neural network

Data-driven turbulence modeling has been considered an effective method for improving the prediction accuracy of Reynolds-averaged Navier-Stokes equations. Related studies aimed to solve the discrepancy of traditional turbulence modeling by acquiring specific patterns from high-fidelity data through machine learning methods, such as artificial neural networks. The present study focuses on the unsmoothness and prediction error problems from the aspect of feature selection and processing. The selection criteria for the input features are summarized, and an effective input set is constructed. The effect of the computation grid on the smoothness is studied. A modified feature decomposition method for the spatial orientation feature of the Reynolds stress is proposed. The improved machine learning framework is then applied to the periodic hill database with notably varying geometries. The results of the modified method show significant enhancement in the prediction accuracy and smoothness, including the shape and size of separation areas and the friction and pressure distributions on the wall, which confirms the validity of the approach.

physics.flu-dyn↗

arXiv2020

Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general Lévy noise.

math.NA↗

arXiv2020

Total Difference Chromatic Numbers of Graphs

Inspired by graceful labelings and total labelings of graphs, we introduce the idea of total difference labelings. A $k$-total labeling of a graph $G$ is an assignment of $k$ distinct labels to the edges and vertices of a graph so that adjacent vertices, incident edges, and an edge and its incident vertices receive different labels. A $k$-total difference labeling of a graph $G$ is a function $f$ from the set of edges and vertices of $G$ to the set $\{1,2,\ldots,k\}$, that is a $k$-total labeling of $G$ and for which $f(\{u,v\})=|f(u)-f(v)|$ for any two adjacent vertices $u$ and $v$ of $G$ with incident edge $\{u,v\}$. The least positive integer $k$ for which $G$ has a $k$-total difference labeling is its total difference chromatic number, $χ_{td}(G)$. We determine the total difference chromatic number of paths, cycles, stars, wheels, gears and helms. We also provide bounds for total difference chromatic numbers of caterpillars, lobsters, and general trees.

math.GM↗

arXiv2020

A neural network based policy iteration algorithm with global $H^2$-superlinear convergence for stochastic games on domains

In this work, we propose a class of numerical schemes for solving semilinear Hamilton-Jacobi-Bellman-Isaacs (HJBI) boundary value problems which arise naturally from exit time problems of diffusion processes with controlled drift. We exploit policy iteration to reduce the semilinear problem into a sequence of linear Dirichlet problems, which are subsequently approximated by a multilayer feedforward neural network ansatz. We establish that the numerical solutions converge globally in the $H^2$-norm, and further demonstrate that this convergence is superlinear, by interpreting the algorithm as an inexact Newton iteration for the HJBI equation. Moreover, we construct the optimal feedback controls from the numerical value functions and deduce convergence. The numerical schemes and convergence results are then extended to HJBI boundary value problems corresponding to controlled diffusion processes with oblique boundary reflection. Numerical experiments on the stochastic Zermelo navigation problem are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

math.NA↗

arXiv2020

Error estimates of penalty schemes for quasi-variational inequalities arising from impulse control problems

This paper proposes penalty schemes for a class of weakly coupled systems of Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) arising from stochastic hybrid control problems of regime-switching models with both continuous and impulse controls. We show that the solutions of the penalized equations converge monotonically to those of the HJBQVIs. We further establish that the schemes are half-order accurate for HJBQVIs with Lipschitz coefficients, and first-order accurate for equations with more regular coefficients. Moreover, we construct the action regions and optimal impulse controls based on the error estimates and the penalized solutions. The penalty schemes and convergence results are then extended to HJBQVIs with possibly negative impulse costs. We also demonstrate the convergence of monotone discretizations of the penalized equations, and establish that policy iteration applied to the discrete equation is monotonically convergent with an arbitrary initial guess in an infinite dimensional setting. Numerical examples for infinite-horizon optimal switching problems are presented to illustrate the effectiveness of the penalty schemes over the conventional direct control scheme.

math.OC↗

arXiv2019

A penalty scheme for monotone systems with interconnected obstacles: convergence and error estimates

We present a novel penalty approach for a class of quasi-variational inequalities (QVIs) involving monotone systems and interconnected obstacles. We show that for any given positive switching cost, the solutions of the penalized equations converge monotonically to those of the QVIs. We estimate the penalization errors and are able to deduce that the optimal switching regions are constructed exactly. We further demonstrate that as the switching cost tends to zero, the QVI degenerates into an equation of HJB type, which is approximated by the penalized equation at the same order (up to a log factor) as that for positive switching cost. Numerical experiments on optimal switching problems are presented to illustrate the theoretical results and to demonstrate the effectiveness of the method.

math.NA↗

arXiv2018

A penalty scheme and policy iteration for nonlocal HJB variational inequalities with monotone drivers

We propose a class of numerical schemes for nonlocal HJB variational inequalities (HJBVIs) with monotone drivers. The solution and free boundary of the HJBVI are constructed from a sequence of penalized equations, for which a continuous dependence result is derived and the penalization error is estimated. The penalized equation is then discretized by a class of semi-implicit monotone approximations. We present a novel analysis technique for the well-posedness of the discrete equation, and demonstrate the convergence of the scheme, which subsequently gives a constructive proof for the existence of a solution to the penalized equation and variational inequality. We further propose an efficient iterative algorithm with local superlinear convergence for solving the discrete equation. Numerical experiments are presented for an optimal investment problem under ambiguity and a recursive consumption-portfolio allocation problem.

math.NA↗

arXiv2018

Fully Discrete Schemes and Their Analyses for Forward-Backward Stochastic Differential Equations

We propose some numerical schemes for forward-backward stochastic differential equations (FBSDEs) based on a new fundamental concept of transposition solutions. These schemes exploit time-splitting methods for the variation of constants formula of the associated partial differential equations and a discrete representation of the transition semigroups. The convergence of the schemes is established for FBSDEs with uniformly Lipschitz drivers, locally Lipschitz and maximal monotone drivers. Numerical experiments are presented for several nonlinear financial derivative pricing problems to demonstrate the adaptivity and effectiveness of the new schemes. The ideas here can be applied to construct high-order schemes for FBSDEs with general Markov forward processes.

math.NA↗

arXiv2018

Approximation schemes for mixed optimal stopping and control problems with nonlinear expectations and jumps

We propose a class of numerical schemes for mixed optimal stopping and control of processes with infinite activity jumps and where the objective is evaluated by a nonlinear expectation. Exploiting an approximation by switching systems, piecewise constant policy timestepping reduces the problem to nonlocal semi-linear equations with different control parameters, uncoupled over individual time steps, which we solve by fully implicit monotone approximations to the controlled diffusion and the nonlocal term, and specifically the Lax-Friedrichs scheme for the nonlinearity in the gradient. We establish a comparison principle for the switching system and demonstrate the convergence of the schemes, which subsequently gives a constructive proof for the existence of a solution to the switching system. Numerical experiments are presented for a recursive utility maximization problem to demonstrate the effectiveness of the new schemes.

math.NA↗