# Web page for Angxiu Ni

I am a graduate student at the mathematics department, UC Berkeley.
My person can sometimes be found in 854 Evans.
E-mail: niangxiu at berkeley dot edu

## Teaching

### Fall 2017:

• GSI for Math 54: Linear Algebra and Differential Equations.
Discussion session: MWF 12:00-1:00 in 75 Evans, 1:00-2:00 in 71 Evans.
Office hour: Tue 9:00-11:00 in 854 Evans.

## Research

My research interest is in numerical analysis.
Currently, I am working on computing sensitivity of long-time averaged objectives in chaotic dynamical systems.
My main contribution to this problem is developing the Non-intrusive Least Squares Shadowing (NILSS) algorithm.

### Non-Intrusive Least Squares Shadowing (NILSS)

NILSS is an algorithm for computing the sensitivity of long-time averaged objective in chaotic dynamical systems.
Its theory makes the assumption that the dynamical system is uniformly hyperbolic, and ergodic.
There are three major ideas which leads to NILSS:
• The sensitivity can be reflected by comparing a base trajectory and its shadowing trajectory.
The shadowing trajectory is one with a perturbed parameter, but the first order approximation of its difference with the base trajectory does not diverge.
This is a classical result.
• Such shadowing trajectory can be found through a minimization on the L-2 norm of the distance between the base trajectory and the shadowing trajectory.
This idea is due to Qiqi Wang.
The corresponding numerical algorithm is called the Least Squares Shadowing (LSS).
• With a particular presentation, the minimization problem in LSS can be carried out only in the unstable directions.
This greatly reduces the computational cost.
NILSS is one of the first algorithms efficient enough to be applied on complicated problems like 3-D chaotic flows.
I developed NILSS with help from Qiqi.

## Publications

The arxiv versions may be more correct than the journal versions.

### NILSS related

1. "Sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Shadowing (NILSS)", Angxiu Ni, Qiqi Wang, Journal of Computational Physics, Volume 347, Page 56-77, 2017.
• This paper contains an intuitive explanation of NILSS; it also briefly mentions the finite difference NILSS.
2. "Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NI-LSS)", Angxiu Ni, Patrick J. Blonigan, Mario Chater, Qiqi Wang, and Zhichao Zhang, 46th AIAA Fluid Dynamics Conference, AIAA AVIATION Forum, (AIAA 2016-4399).
• This paper is the first public appearance of the NILSS algorithm; it also contains an attempt of the adjoint version of NILSS.
3. "Characteristic Lyapunov Vectors and the Shadowing Direction of a Chaotic Flow past a 3-D Cylinder at Reynolds Number 525", Angxiu Ni, to be submitted to Physical Review Fluids.
• This paper shows this 3-D flow problem is uniformly hyperbolic as a dynamical system.
• Flow feature of Characteristic Lyapunov Vectors (CLV) move towards the flow region with higher dissipation, as the CLV growth rate becomes more nagative.
• The shadowing direction exists for this flow problem, and it can be computed by NILSS.
• The shadowing direction has physical meaning, and can be used to compute sensitivity of long-time averaged objectives in chaotic systems.
• Here are some YouTube videos of the CLVs and shadowing directions computed in this paper:
4. "Sensitivity Analysis for a Chaotic Flow past a 3-D Cylinder by the Finite Difference Non-Intrusive Least Squares Shadowing (NILSS) Algorithm", Angxiu Ni, Chaitanya Talnikar, Pablo Fernandez, Qiqi Wang, submitted to 2018 AIAA AVIATION forum.
• This paper contains a coherent presentation of the finite difference NILSS.

### Other topics

1. "Simplified Least Squares Shadowing sensitivity analysis for chaotic ODEs and PDEs", Mario Chater, Angxiu Ni, Qiqi Wang, Journal of Computational Physics 329, 126-140, 2017.
2. "Least Squares Shadowing method for sensitivity analysis of differential equations", Mario Chater, Angxiu Ni, Patrick Blonigan, Qiqi Wang, arXiv:1509.02882, 2015.
3. "An improvement to NSGA-II algorithm and its application in optimization design of multi-element airfoil", Angxiu Ni, Yufei Zhang, Haixin Chen, Acta Aerodynamica Sinica 32 (2), 252-257, 2014.

## Conference Presentations

I try to go to two or three conferences a year. Please talk to me if you see me :)
1. (might go) "Sensitivity Analysis for a Chaotic Flow past a 3-D Cylinder by the Finite Difference Non-Intrusive Least Squares Shadowing (NILSS) Algorith", Angxiu Ni, Chaitanya Talnikar, Pablo Fernandez, Qiqi Wang, 2018 AIAA AVIATION forum, Atlanta, Georgia, USA.
2. "Characteristic Lyapunov Vectors and the Shadowing direction of a 3-D cylinder flow at Reynolds number 525", Angxiu Ni, Chaitanya Talnikar, Pablo Fernandez, Qiqi Wang, 70th Annual Meeting of the APS Division of Fluid Dynamics, November 19–21, 2017, Denver, Colorado, USA.
3. "Computation of Sensitivity and Stability in Chaotic Flow by Non-Intrusive Least Squares Shadowing (NILSS)", Angxiu Ni, Qiqi Wang, 2017 SIAM Annual Meeting, July 10-14, 2017, Pittsburgh, Pennsylvania, USA.
4. "Sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Shadowing (NILSS), and challenges", Angxiu Ni, Qiqi Wang, invited talk at the National Institute of Aerospace, June 14, 2017.
• There is a good video recording of this talk, thank you Boris and Eric!
5. "Sensitivity analysis on chaotic dynamical systems by Non-Intrusive Least Squares Shadowing (NILSS)", Angxiu Ni, Qiqi Wang, 2017 SIAM Conference on Computational Science and Engineering, Feb 27 - March 3, 2017, Atlanta, Georgia, USA.
6. "Sensitivity analysis for chaotic dynamical systems using Non-Intrusive Least Square Shadowing", Angxiu Ni, Patrick J. Blonigan, Mario Chater, Qiqi Wang, and Zhichao Zhang, 2016 SIAM Annual Meeting, July 11-15, 2016, Boston, Massachusetts, USA.
7. "Sensitivity analysis for chaotic aeroelastic oscillations using Non-Intrusive Least Square Shadowing", Angxiu Ni, Patrick J. Blonigan, Mario Chater, Qiqi Wang, and Zhichao Zhang, 16th AIAA Aviation, June 13-17, 2016, Washington, D.C..
8. "Dynamical behaviors of a thin plate under bypassing flow", Angxiu Ni, Qiqi Wang, 68th Annual Meeting of the APS Division of Fluid Dynamics, November 22-24, 2015, Boston, USA.
9. "A GA optimizer enhanced by DOE and RS for aerodynamic design", Angxiu Ni, Haixin Chen, 13th Russian-Chinese Conference on Aviation Science and Technology, September 2-3, 2013, TsAGI, Zhukovsky, Russia.
10. "Optimization of slot parameters for multi-element airfoil using NSGA-II", Angxiu Ni, Yufei Zhang, Haixin Chen, National Fluid Mechanics Youth Workshop, May 18-19, 2013, Shanghai, China.