Roadmap
This page is still under construction.
Work Packages
Work Package #1 – Baseline models
The parameterization strategy proposed in this research project applies to a wide range of linear models. Our main focus is however on control-oriented data-driven models. To assess the performances and robustness of these models, reference data must obtained. Running over the first year of the project, this WP deals with the implementation of state-of-the-art linear methods for flow control. These developments will mostly happen in nekStab, an open-source toolbox for large-scale stability and bifurcation analysis of incompressible flows. Relying on the spectral element solver Nek5000, this library is being developed by the principal investigator. Of interest for this work package are Nek5000’s capabilities in solving the direct and adjoint linearized Navier-Stokes equations and the coupling with nekStab’s large-scale eigenvalue and singular value solvers.
Consider a multiple-input multiple output linear time-invariant (MIMO LTI) system
\[ \Sigma = \begin{pmatrix} \begin{array}{c|c} \mathbf{A} & \mathbf{B} \\ \hline \mathbf{C} \end{array} \end{pmatrix} \]
with \(\mathbf{x} \in \mathbb{R}^n\) being the state vector of the system, \(\mathbf{u} \in \mathbb{R}^p\) its inputs, and \(\mathbf{y} \in \mathbb{R}^q\) its outputs. A feedback control law of the form \(\mathbf{u} = - \mathbf{Kx}\) can stabilize the otherwise unstable dynamics. For linear quadratic regulators (resp. linear quadratic estimators), synthesizing \(\mathbf{K} \in \mathbb{R}^{n \times p}\) (resp. \(\mathbf{L} \in \mathbb{R}^{n \times q}\)) requires the solution of an algebraic Ricatti equation. Due to the sheer size of the discretizations in fluid dynamics, this is however computationally intractable even for moderately complex flow configurations. Two main strategies have been proposed in the literature to overcome these limitations, including Balanced Proper Orthgonal Decomposition (Rowley 2005), the Adjoint of the Direct-Adjoint and Adjoint of the Adjoint-Direct (Bewley, Luchini, and Pralits 2016; Semeraro and Pralits 2018). The implementation of these strategies, both in sandboxes and in nekStab, forms the main tasks of this work package.
Task 1.1 – Adjoint of the direct-adjoint (ADA)
- ADA in Julia/Python for prototyping
- Implement ADA for the Ginzburg-Landau equation.
- Implement classic LQR for the sake of reference.
- Validate/benchmark the two approaches against one another.
- ADA in
nekStab- Implement ADA for the hydrodynamic equations only.
- Validate the implementation based on
- Plane Poiseuille flow
- Shear-driven cavity flow
- Cylinder flow
Task 1.2 – Adjoint of the adjoint-direct (AAD)
- AAD in Julia/Python for prototyping
- Implement AAD for the Ginzburg-Landau equation.
- Implement classic LQE for the sake of reference.
- Validate/benchmark the two approaches against one another.
- AAD in
nekStab- Implement AAD for the hydrodynamic equations only.
- Validate the implementation based on
- Plane Poiseuille flow
- Shear-driven cavity flow
- Cylinder flow
- Combine AAD and ADA for LQG control
- Validate the combined approach for closed-look feedback control.
- State feedback:
- Plane Poiseuille flow
- Output feedback:
- Shear-driven cavity flow
- Cylinder flow
- State feedback:
- Validate the combined approach for closed-look feedback control.
Task 1.3 – Balanced Proper Orthogonal Decomposition (BPOD)
- Implementation of BPOD in
nekStab- Impulse response of the direct system.
- Impulse response of the adjoint system.
- Singular Value Decomposition of the cross Gramian.
- Petrov-Galerkin projection of the linearized operator.
- Output of the reduced-order model in a
Juliacompatible format.
- Validation of the implementation
- Plane Poiseuille flow
- Shear-driven cavity flow
- Cylinder flow
Task 1.4 – Optimal sensor and actuator placement
- Sandbox implementation of the \(\mathcal{H}_2\) optimal sensor and actuator placement
- Ginzburg-Landau sandbox with standard solvers.
- Linearized operator in matrix form.
- Adjoint operator in matrix form.
- Ricatti and Lyapunov direct solvers.
- Generic optimization loop.
- Ginzburg-Laudau sandbox with time-stepper formulation.
- LQR implementation based on ADA.
- LQE implementation based on AAD.
- Generic optimization loop.
- Benchmark the two approaches with a variety of optimizers.
- Ginzburg-Landau sandbox with standard solvers.
- \(\mathcal{H}_2\) optimal sensor/actuator placement in
nekStab- LQR implementation based on ADA.
- LQE implementation based on AAD.
- Optimizer.
- Submodular sensor/actuator placement
- Implement standard algorithms in
SubmodularOpt.jl. - Benchmark submodular sensor placement vs. \(\mathcal{H}_2\) optimal vs. Manohar et al. on the Ginzburg-Landau equation.
- Couple
SubmodularOpt.jlandnekStabfor sensor/actuator placement of high-dimensional systems.
- Implement standard algorithms in
Work Package #2 – Data-driven linear modeling
This second work package forms one of the main contributions of this project. It covers most of the first year of the Ph. D. candidate. Given two sets of data \(\mathbf{X} \in \mathbb{R}^{n \times k}\) and \(\mathbf{Y} \in \mathbb{R}^{m \times k}\), many data-driven linear models can be formulated as a reduced-rank regression problem. Of particular interest here is that many linear system identification techniques can be cast into this framework, e.g. DMD (Schmid 2010), DMDc (Proctor, Brunton, and Kutz 2016), ioDMD(Benner, Himpe, and Mitchell 2018), ERA (Juang and Pappa 1985), or N4SID (Van Overschee and Moor 1993). The core of this work package is thus unification of all of these approaches under the umbrella of reduced-rank regression and benchmarking the resulting performances against classical algorithms or heuristics used in the literature for the related tasks.
Task 2.1 – Unifying framework for data-driven linear modeling
- Literature review on data-driven linear models.
- Unification of the most important modeling techniques
- POD
- DMD, DMDc, ioDMD
- ERA, N4SID, MOESP, CVA
- BPOD
- Linear Stochastic Estimation
- Extended DMD, Kernel DMD
Task 2.2 – Implementation in nekStab + development of an external OSS toolbox
Task 2.3 – Benchmarking data-driven models
- Identify the most relevent benchmarks from the literature review.
- Sandbox implementation of the identified benchmarks.
- Exhaustive comparison of the performances of vanilla vs. RRR reformulations
- POD
- DMD, KMD
- DMDc, ioDMD
- ERA, N4SID, MOESP, CVA
- Extended DMD, Kernel DMD
- LSE, CCA
Task 2.4 – Data-driven optimal sensors and actuators placement
Work Package #3 – Parameterization and interpolation on the Grassman and Stiefel manifolds
This third work package constitutes the major contribution of this project: the parameterization of data-driven models with respect to the operating conditions. For simplicity, parameterization with respect to a single parameter will be considered, albeit the proposed methodology is more general. Given training data ate different operating conditions, a data-driven linear model can be identified for each sampled point in the parameter space using the methodology developed in WP2. Yet, interpolating the models matrices in an entry wise fashion for a new operating condition cannot be used as it would not preserve the underlying low-rank structure of the problem. This structured interpolation needs to take place on a particular matrix manifold, either the Grassman or Stiefel manifold. Because they are Riemannian manifolds, both are equipped with an intrinsic notion of distance.
This work package aims at exploring the possibility to use the tools from differential geometry to obtain parameterized data-driven linear models. Our attention will be focused on two particular points:
- How to sample properly the parameter space for the particular problem of control-oriented models?
- Defining a priori and a posteriori error estimates of the interpolated models.
As for WP2, the performances of the parameterized data-driven models will be benchmarked and compared against those of the high-fidelity models. The differnt algorithms for the parametrization will be implemented in an open-source library. This work package mostly covers the third year of the project, corresponding to the second year of the Ph. D. thesis.
Task 3.1 – Parameterization of BPOD on the Grassman manifold
- Literature review on the different sampling strategies for matrix-valued manifolds.
- Parameterization of the BPOD subspaces vs. direct parameterization of the cross gramian matrix.
- a priori and a posteriori error estimates for the \(\mathcal{H}_2\) norm of the interpolated model.
- Benchmark the performances of the interpolated models against their ground truth counterparts
- Sandbox implementation on the Ginzburg-Landau equation.
- Using
nekStab:- Plane Poiseuille flow
- Cylinder flow
- Shear-driven cavity flow
Task 3.2 – Parameterization of ADA and AAD on the Stiefel manifold
Task 3.3 – Parameterization of data-driven models on the Stiefel manifold
Work Package #4 – Sensitivity, uncertainty quantification, and applications
Tasks included in this work package are quite prospective. An aspect often overlooked in data-driven models applied to fluid dynamics is their robustness and statistical significance. Assessing the prediction uncertainty is also critical, particularly for feedback control using an imperfect model. While techniques falling under the umbrella of robust control or model predictive control exist, a different route will be explored, combining ideas from sensitivity analysis and uncertainty quantification with statistics on the Stiefel manifold.
Task 4.1 – Sensitivity of data-driven models to varying parameters
Task 4.2 – Uncertainty quantification on the Stiefel manifold
Task 4.3 – Application to a fully three-dimensional flow configuration
Deliverables
- D0.1 Progress report (01/2023 - 06/2023)
- D0.2 Progress report (07/2023 - 12/2023)
- D0.3 Progress report (01/2024 - 06/2024)
- D0.4 Progress report (07/2024 - 12/2024)
- D0.5 Progress report (01/2025 - 06/2025)
- D0.6 Progress report (07/2025 - 12/2025)
- D0.7 Progress report (01/2026 - 06/2026)
- D0.8 Final report