# On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations

## Abstract

The Karhunen–Lòeve (KL) decomposition provides a low-dimensional representation for random fields as it is optimal in the mean square sense. Although for many stochastic systems of practical interest, described by stochastic partial differential equations (SPDEs), solutions possess this low-dimensional character, they also have a strongly time-dependent form and to this end a fixed-in-time basis may not describe the solution in an efficient way. Motivated by this limitation of standard KL expansion, Sapsis and Lermusiaux (2009) [26] developed the dynamically orthogonal (DO) field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty ‘lives’ but also the stochastic characteristics of uncertainty. Recently, Cheng et al. (2013) [28] introduced an alternative approach, the bi-orthogonal (BO) method, which performs the exact same tasks, i.e. it evolves the spatial basis and the stochastic characteristics of uncertainty. In the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence, in the sense that one method is an exact reformulation of the other. We show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the BO and the DO solutions. We also examine a pathologymore »

- Authors:

- Division of Applied Mathematics, Brown University, Providence, RI 02912 (United States)
- Massachusetts Institute of Technology, Cambridge, MA 02139 (United States)

- Publication Date:

- OSTI Identifier:
- 22314878

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 270; Other Information: Copyright (c) 2014 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; COMPUTERIZED SIMULATION; EIGENVALUES; FIELD EQUATIONS; MATHEMATICAL SOLUTIONS; MATRICES; NONLINEAR PROBLEMS; PARTIAL DIFFERENTIAL EQUATIONS; PERFORMANCE; RANDOMNESS; ROTATION; SINGULARITY; STOCHASTIC PROCESSES; TIME DEPENDENCE; TRANSFORMATIONS

### Citation Formats

```
Choi, Minseok, Sapsis, Themistoklis P., and Karniadakis, George Em, E-mail: george_karniadakis@brown.edu.
```*On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations*. United States: N. p., 2014.
Web. doi:10.1016/J.JCP.2014.03.050.

```
Choi, Minseok, Sapsis, Themistoklis P., & Karniadakis, George Em, E-mail: george_karniadakis@brown.edu.
```*On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations*. United States. doi:10.1016/J.JCP.2014.03.050.

```
Choi, Minseok, Sapsis, Themistoklis P., and Karniadakis, George Em, E-mail: george_karniadakis@brown.edu. Fri .
"On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations". United States.
doi:10.1016/J.JCP.2014.03.050.
```

```
@article{osti_22314878,
```

title = {On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations},

author = {Choi, Minseok and Sapsis, Themistoklis P. and Karniadakis, George Em, E-mail: george_karniadakis@brown.edu},

abstractNote = {The Karhunen–Lòeve (KL) decomposition provides a low-dimensional representation for random fields as it is optimal in the mean square sense. Although for many stochastic systems of practical interest, described by stochastic partial differential equations (SPDEs), solutions possess this low-dimensional character, they also have a strongly time-dependent form and to this end a fixed-in-time basis may not describe the solution in an efficient way. Motivated by this limitation of standard KL expansion, Sapsis and Lermusiaux (2009) [26] developed the dynamically orthogonal (DO) field equations which allow for the simultaneous evolution of both the spatial basis where uncertainty ‘lives’ but also the stochastic characteristics of uncertainty. Recently, Cheng et al. (2013) [28] introduced an alternative approach, the bi-orthogonal (BO) method, which performs the exact same tasks, i.e. it evolves the spatial basis and the stochastic characteristics of uncertainty. In the current work we examine the relation of the two approaches and we prove theoretically and illustrate numerically their equivalence, in the sense that one method is an exact reformulation of the other. We show this by deriving a linear and invertible transformation matrix described by a matrix differential equation that connects the BO and the DO solutions. We also examine a pathology of the BO equations that occurs when two eigenvalues of the solution cross, resulting in an instantaneous, infinite-speed, internal rotation of the computed spatial basis. We demonstrate that despite the instantaneous duration of the singularity this has important implications on the numerical performance of the BO approach. On the other hand, it is observed that the BO is more stable in nonlinear problems involving a relatively large number of modes. Several examples, linear and nonlinear, are presented to illustrate the DO and BO methods as well as their equivalence.},

doi = {10.1016/J.JCP.2014.03.050},

journal = {Journal of Computational Physics},

number = ,

volume = 270,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}