Yanzhao Cao | Auburn University (original) (raw)

Papers by Yanzhao Cao

Research paper thumbnail of Approximating Stochastic Evolution Equations with Additive White and Rough Noises

SIAM Journal on Numerical Analysis, 2017

Research paper thumbnail of Finite element approximations for second-order stochastic differential equation driven by fractional Brownian motion

IMA Journal of Numerical Analysis, 2017

Research paper thumbnail of Adaptive Gradient Descent for Optimal Control of Parabolic Equations with Random Parameters

arXiv (Cornell University), Oct 21, 2021

Research paper thumbnail of A backward SDE method for uncertainty quantification in deep learning

Discrete and Continuous Dynamical Systems - S

We develop a backward stochastic differential equation based probabilistic machine learning metho... more We develop a backward stochastic differential equation based probabilistic machine learning method, which formulates a class of stochastic neural networks as a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced with the gradient computed through a backward stochastic differential equation. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.

Research paper thumbnail of Solving Backward Doubly Stochastic Differential Equations through Splitting Schemes

ArXiv, 2021

A splitting scheme for backward doubly stochastic differential equations is proposed. The main id... more A splitting scheme for backward doubly stochastic differential equations is proposed. The main idea is to decompose a backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation. The backward stochastic differential equation and the stochastic differential equation are then approximated by first order finite difference schemes, which results in a first order scheme for the backward doubly stochastic differential equation. Numerical experiments are conducted to illustrate the convergence rate of the proposed scheme.

Research paper thumbnail of A High Order Numerical Method for Solving Backward Doubly Stochastic Differential Equations

The numerical solution of backward doubly stochastic differential equations (BDSDES) and the rela... more The numerical solution of backward doubly stochastic differential equations (BDSDES) and the related stochastic partial differential equations (Zakai equations) are considered. First order algorithms are constructed using a generalized Itô-Taylor formula for two-sided stochastic differentials. The convergence order is proved through rigorous error analysis. Numerical experiments are carried out to demonstrate the efficiency of the proposed numerical scheme.

Research paper thumbnail of Well‐posedness and finite element approximation of time dependent generalized bioconvective flow

Numerical Methods for Partial Differential Equations, 2019

Research paper thumbnail of A First Order Scheme for Backward Doubly Stochastic Differential Equations

SIAM/ASA Journal on Uncertainty Quantification, 2016

Research paper thumbnail of An Implicit Algorithm of Solving Nonlinear Filtering Problems

Communications in Computational Physics, 2014

Nonlinear filter problems arise in many applications such as communications and signal processing... more Nonlinear filter problems arise in many applications such as communications and signal processing. Commonly used numerical simulation methods include Kalman filter method, particle filter method, etc. In this paper a novel numerical algorithm is constructed based on samples of the current state obtained by solving the state equation implicitly. Numerical experiments demonstrate that our algorithm is more accurate than the Kalman filter and more stable than the particle filter.

Research paper thumbnail of An Efficient Meshfree Implicit Filter for Nonlinear Filtering Problems

International Journal for Uncertainty Quantification, 2016

Research paper thumbnail of Efficient Particle Filtering for stochastic Korteweg-de Vries Equations

Stochastics and Dynamics, 2017

We propose an efficient algorithm to perform nonlinear data assimilation for Korteweg–de Vries so... more We propose an efficient algorithm to perform nonlinear data assimilation for Korteweg–de Vries solitons. In particular we develop a reduced particle filtering method to reduce the dimension of the problem. The method decomposes a solitonic pulse into a clean soliton and small radiative noise, and instead of inferring the complete pulse profile, we only infer the two soliton parameters with particle filter. Numerical examples are provided to demonstrate that the proposed method can provide rather accurate results, while being much more computationally affordable than a standard particle filter.

Research paper thumbnail of Numerical Solutions for Forward Backward Doubly Stochastic Differential Equations and Zakai Equations

International Journal for Uncertainty Quantification, 2011

Research paper thumbnail of A Hybrid Sparse-Grid Approach for Nonlinear Filtering Problems Based on Adaptive-Domain of the Zakai Equation Approximations

SIAM/ASA Journal on Uncertainty Quantification, 2014

Research paper thumbnail of Analysis and Finite Element Approximation of Bioconvection Flows with Concentration Dependent Viscosity

Research paper thumbnail of A Fast Algorithm for Orthogonal Polynomial Expansions on Sparse Grids

Research paper thumbnail of Orthogonal polynomial expansions on sparse grids

Journal of Complexity, 2014

Research paper thumbnail of Shape optimization for noise radiation problems

A shape design model that reduces the amount of noise radiated from aircraft turbofan engines is ... more A shape design model that reduces the amount of noise radiated from aircraft turbofan engines is studied in this paper. The model is formulated as shape control of the Helmholtz equation with radiation boundary conditions on part of the boundary and incoming waves specified as the source. Existence of optimal shape is proved to show that the model is appropriately established. A numerical experiment is conducted to demonstrate the efficiency of the model.

Research paper thumbnail of Influence of environmental parameters on the distribution of subfossil chironomids in surface sediments of Bosten lake (Xinjiang, China)

Abstract The distribution of larval subfossil chironomids in surface sediment samples obtained fr... more Abstract The distribution of larval subfossil chironomids in surface sediment samples obtained from Bosten lake was analysed, and ordination methods were used to identify the influences of physical and chemical parameters on the abundance and diversity of chironomids. A total of 18 chironomid taxa was identified across the 32 samples, 15 of which showed minimum abundance of> 1% and were present in more than one site.

Research paper thumbnail of NUMERICAL SOLUTIONS FOR FORWARD BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS AND ZAKAI EQUATIONS

The numerical solutions of decoupled forward backward doubly stochastic differential equations an... more The numerical solutions of decoupled forward backward doubly stochastic differential equations and the related stochastic partial differential equations (Zakai equations) are considered. Numerical algorithms are constructed using reference equations. Rate of convergence is obtained through rigorous error analysis. Numerical experiments are carried out to verify the rate of convergence results and to demonstrate the efficiency of the proposed numerical algorithms.

Research paper thumbnail of A MODEL FOR VECTOR-BORNE DISEASES: OPTIMAL TREATMENT AND PREVENTION

ABSTRACT In this paper we study the dynamics of a vector-transmitted disease under two assumption... more ABSTRACT In this paper we study the dynamics of a vector-transmitted disease under two assumptions. We first look at time dependent prevention and treatment efforts where optimal control theory is applied. Using analytical and numerical techniques, it is shown that there are control efforts for treatment of hosts and prevention of host-vector contacts with minimal cost and side effects. Then we considered the autonomous counter part of the first mode and here we calculated an epidemiological parameter.

Research paper thumbnail of Approximating Stochastic Evolution Equations with Additive White and Rough Noises

SIAM Journal on Numerical Analysis, 2017

Research paper thumbnail of Finite element approximations for second-order stochastic differential equation driven by fractional Brownian motion

IMA Journal of Numerical Analysis, 2017

Research paper thumbnail of Adaptive Gradient Descent for Optimal Control of Parabolic Equations with Random Parameters

arXiv (Cornell University), Oct 21, 2021

Research paper thumbnail of A backward SDE method for uncertainty quantification in deep learning

Discrete and Continuous Dynamical Systems - S

We develop a backward stochastic differential equation based probabilistic machine learning metho... more We develop a backward stochastic differential equation based probabilistic machine learning method, which formulates a class of stochastic neural networks as a stochastic optimal control problem. An efficient stochastic gradient descent algorithm is introduced with the gradient computed through a backward stochastic differential equation. Convergence analysis for stochastic gradient descent optimization and numerical experiments for applications of stochastic neural networks are carried out to validate our methodology in both theory and performance.

Research paper thumbnail of Solving Backward Doubly Stochastic Differential Equations through Splitting Schemes

ArXiv, 2021

A splitting scheme for backward doubly stochastic differential equations is proposed. The main id... more A splitting scheme for backward doubly stochastic differential equations is proposed. The main idea is to decompose a backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation. The backward stochastic differential equation and the stochastic differential equation are then approximated by first order finite difference schemes, which results in a first order scheme for the backward doubly stochastic differential equation. Numerical experiments are conducted to illustrate the convergence rate of the proposed scheme.

Research paper thumbnail of A High Order Numerical Method for Solving Backward Doubly Stochastic Differential Equations

The numerical solution of backward doubly stochastic differential equations (BDSDES) and the rela... more The numerical solution of backward doubly stochastic differential equations (BDSDES) and the related stochastic partial differential equations (Zakai equations) are considered. First order algorithms are constructed using a generalized Itô-Taylor formula for two-sided stochastic differentials. The convergence order is proved through rigorous error analysis. Numerical experiments are carried out to demonstrate the efficiency of the proposed numerical scheme.

Research paper thumbnail of Well‐posedness and finite element approximation of time dependent generalized bioconvective flow

Numerical Methods for Partial Differential Equations, 2019

Research paper thumbnail of A First Order Scheme for Backward Doubly Stochastic Differential Equations

SIAM/ASA Journal on Uncertainty Quantification, 2016

Research paper thumbnail of An Implicit Algorithm of Solving Nonlinear Filtering Problems

Communications in Computational Physics, 2014

Nonlinear filter problems arise in many applications such as communications and signal processing... more Nonlinear filter problems arise in many applications such as communications and signal processing. Commonly used numerical simulation methods include Kalman filter method, particle filter method, etc. In this paper a novel numerical algorithm is constructed based on samples of the current state obtained by solving the state equation implicitly. Numerical experiments demonstrate that our algorithm is more accurate than the Kalman filter and more stable than the particle filter.

Research paper thumbnail of An Efficient Meshfree Implicit Filter for Nonlinear Filtering Problems

International Journal for Uncertainty Quantification, 2016

Research paper thumbnail of Efficient Particle Filtering for stochastic Korteweg-de Vries Equations

Stochastics and Dynamics, 2017

We propose an efficient algorithm to perform nonlinear data assimilation for Korteweg–de Vries so... more We propose an efficient algorithm to perform nonlinear data assimilation for Korteweg–de Vries solitons. In particular we develop a reduced particle filtering method to reduce the dimension of the problem. The method decomposes a solitonic pulse into a clean soliton and small radiative noise, and instead of inferring the complete pulse profile, we only infer the two soliton parameters with particle filter. Numerical examples are provided to demonstrate that the proposed method can provide rather accurate results, while being much more computationally affordable than a standard particle filter.

Research paper thumbnail of Numerical Solutions for Forward Backward Doubly Stochastic Differential Equations and Zakai Equations

International Journal for Uncertainty Quantification, 2011

Research paper thumbnail of A Hybrid Sparse-Grid Approach for Nonlinear Filtering Problems Based on Adaptive-Domain of the Zakai Equation Approximations

SIAM/ASA Journal on Uncertainty Quantification, 2014

Research paper thumbnail of Analysis and Finite Element Approximation of Bioconvection Flows with Concentration Dependent Viscosity

Research paper thumbnail of A Fast Algorithm for Orthogonal Polynomial Expansions on Sparse Grids

Research paper thumbnail of Orthogonal polynomial expansions on sparse grids

Journal of Complexity, 2014

Research paper thumbnail of Shape optimization for noise radiation problems

A shape design model that reduces the amount of noise radiated from aircraft turbofan engines is ... more A shape design model that reduces the amount of noise radiated from aircraft turbofan engines is studied in this paper. The model is formulated as shape control of the Helmholtz equation with radiation boundary conditions on part of the boundary and incoming waves specified as the source. Existence of optimal shape is proved to show that the model is appropriately established. A numerical experiment is conducted to demonstrate the efficiency of the model.

Research paper thumbnail of Influence of environmental parameters on the distribution of subfossil chironomids in surface sediments of Bosten lake (Xinjiang, China)

Abstract The distribution of larval subfossil chironomids in surface sediment samples obtained fr... more Abstract The distribution of larval subfossil chironomids in surface sediment samples obtained from Bosten lake was analysed, and ordination methods were used to identify the influences of physical and chemical parameters on the abundance and diversity of chironomids. A total of 18 chironomid taxa was identified across the 32 samples, 15 of which showed minimum abundance of> 1% and were present in more than one site.

Research paper thumbnail of NUMERICAL SOLUTIONS FOR FORWARD BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS AND ZAKAI EQUATIONS

The numerical solutions of decoupled forward backward doubly stochastic differential equations an... more The numerical solutions of decoupled forward backward doubly stochastic differential equations and the related stochastic partial differential equations (Zakai equations) are considered. Numerical algorithms are constructed using reference equations. Rate of convergence is obtained through rigorous error analysis. Numerical experiments are carried out to verify the rate of convergence results and to demonstrate the efficiency of the proposed numerical algorithms.

Research paper thumbnail of A MODEL FOR VECTOR-BORNE DISEASES: OPTIMAL TREATMENT AND PREVENTION

ABSTRACT In this paper we study the dynamics of a vector-transmitted disease under two assumption... more ABSTRACT In this paper we study the dynamics of a vector-transmitted disease under two assumptions. We first look at time dependent prevention and treatment efforts where optimal control theory is applied. Using analytical and numerical techniques, it is shown that there are control efforts for treatment of hosts and prevention of host-vector contacts with minimal cost and side effects. Then we considered the autonomous counter part of the first mode and here we calculated an epidemiological parameter.