# Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing

## Abstract

This paper presents a discrete adjoint version of the recently developed non-intrusive least squares shadowing (NILSS) algorithm, which circumvents the instability that conventional adjoint methods encounter for chaotic systems. The NILSS approach involves solving a smaller minimization problem than other shadowing approaches and can be implemented with only minor modifications to preexisting tangent and adjoint solvers. Adjoint NILSS is demonstrated on a small chaotic ODE, a one-dimensional scalar PDE, and a direct numerical simulation (DNS) of the minimal flow unit, a turbulent channel flow on a small spatial domain. This is the first application of an adjoint shadowing-based algorithm to a three-dimensional turbulent flow. - Highlights: • A discrete adjoint non-intrusive least squares shadowing (NILSS) is presented. • The NILSS approach is closely related to multiple shooting shadowing (MSS). • Adjoint NILSS prevents exponential growth in time of the adjoint field. • Adjoint NILSS is demonstrated on a simulation of wall-bounded turbulent flow.

- Authors:

- Publication Date:

- OSTI Identifier:
- 22701627

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 348; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; COMPUTERIZED SIMULATION; ONE-DIMENSIONAL CALCULATIONS; PARTIAL DIFFERENTIAL EQUATIONS; SENSITIVITY ANALYSIS; THREE-DIMENSIONAL CALCULATIONS; TURBULENT FLOW; DYNAMICAL SYSTEMS

### Citation Formats

```
Blonigan, Patrick J., E-mail: patrick.j.blonigan@nasa.gov.
```*Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.08.002.

```
Blonigan, Patrick J., E-mail: patrick.j.blonigan@nasa.gov.
```*Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing*. United States. doi:10.1016/J.JCP.2017.08.002.

```
Blonigan, Patrick J., E-mail: patrick.j.blonigan@nasa.gov. Wed .
"Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing". United States. doi:10.1016/J.JCP.2017.08.002.
```

```
@article{osti_22701627,
```

title = {Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing},

author = {Blonigan, Patrick J., E-mail: patrick.j.blonigan@nasa.gov},

abstractNote = {This paper presents a discrete adjoint version of the recently developed non-intrusive least squares shadowing (NILSS) algorithm, which circumvents the instability that conventional adjoint methods encounter for chaotic systems. The NILSS approach involves solving a smaller minimization problem than other shadowing approaches and can be implemented with only minor modifications to preexisting tangent and adjoint solvers. Adjoint NILSS is demonstrated on a small chaotic ODE, a one-dimensional scalar PDE, and a direct numerical simulation (DNS) of the minimal flow unit, a turbulent channel flow on a small spatial domain. This is the first application of an adjoint shadowing-based algorithm to a three-dimensional turbulent flow. - Highlights: • A discrete adjoint non-intrusive least squares shadowing (NILSS) is presented. • The NILSS approach is closely related to multiple shooting shadowing (MSS). • Adjoint NILSS prevents exponential growth in time of the adjoint field. • Adjoint NILSS is demonstrated on a simulation of wall-bounded turbulent flow.},

doi = {10.1016/J.JCP.2017.08.002},

journal = {Journal of Computational Physics},

number = ,

volume = 348,

place = {United States},

year = {Wed Nov 01 00:00:00 EDT 2017},

month = {Wed Nov 01 00:00:00 EDT 2017}

}