Lucas Lestandi

Bio: Lucas Lestandi is an academic researcher from University of Bordeaux. The author has contributed to research in topics: Low-rank approximation & Curse of dimensionality. The author has an hindex of 4, co-authored 7 publications receiving 30 citations. Previous affiliations of Lucas Lestandi include Agency for Science, Technology and Research & Nanyang Technological University.

Low rank approximation of multidimensional data

Abstract: In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics community and the computational mechanics one. The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in the literature but either on separate papers or into a pure applied mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.

Reduced order model of flows by time-scaling interpolation of DNS data

Abstract: A new reduced order model (ROM) is proposed here for reconstructing super-critical flow past circular cylinder and lid driven cavity using time-scaling of vorticity data directly. The present approach is a significant improvement over instability-mode (developed from POD modes) based approach implemented in Sengupta et al. [Phys Rev E 91(4):043303, 2015], where governing Stuart–Landau–Eckhaus equations are solved. In the present method, we propose a novel ROM that uses relation between Strouhal number (St) and Reynolds number (Re). We provide a step by step approach for this new ROM for any Re and is a general procedure with vorticity data requiring very limited storage as well as being extremely fast. We emphasize on the scientific aspects of developing ROM by taking data from close proximity of the target Re to produce DNS-quality reconstruction, while the applied aspect is also shown. All the donor points need not be immediate neighbors and the reconstructed solution has equivalent relaxed accuracy. However, one would restrain the range where the flow behavior is coherent between donors. The reported work is a proof of concept utilizing the external and internal flow examples, and this can be extended for other flows characterized by appropriate Re–St data.

POD Applied to Numerical Study of Unsteady Flow Inside Lid-driven Cavity

Abstract: Flow inside a lid-driven cavity (LDC) is studied here to elucidate bifurcation sequences of the flow at super-critical Reynolds numbers (Recr1) with the help of analyzing the time series at most energetic points in the flow domain. The implication of Recr1 in the context of direct simulation of Navier-Stokes equation is presented here for LDC, with or without explicit excitation inside the LDC. This is aided further by performing detailed enstrophy-based proper orthogonal decomposition (POD) of the flow field. The flow has been computed by an accurate numerical method for two different uniform grids. POD of results of these two grids help us understand the receptivity aspects of the flow field, which give rise to the computed bifurcation sequences by understanding the similarity and differences of these two sets of computations. We show that POD modes help one understand the primary and secondary instabilities noted during the bifurcation sequences. AMS subject classifications: 65M12, 65M15, 65M60, 76D05, 76F20, 76F65

Low rank approximation techniques and reduced order modeling applied to some fluid dynamics problems

Abstract: Numerical simulation has experienced tremendous improvements in the last decadesdriven by massive growth of computing power. Exascale computing has beenachieved this year and will allow solving ever more complex problems. But suchlarge systems produce colossal amounts of data which leads to its own difficulties.Moreover, many engineering problems such as multiphysics or optimisation andcontrol, require far more power that any computer architecture could achievewithin the current scientific computing paradigm. In this thesis, we proposeto shift the paradigm in order to break the curse of dimensionality byintroducing decomposition and building reduced order models (ROM) for complexfluid flows.This manuscript is organized into two parts. The first one proposes an extendedreview of data reduction techniques and intends to bridge between appliedmathematics community and the computational mechanics one. Thus, foundingbivariate separation is studied, including discussions on the equivalence ofproper orthogonal decomposition (POD, continuous framework) and singular valuedecomposition (SVD, discrete matrices). Then a wide review of tensor formats andtheir approximation is proposed. Such work has already been provided in theliterature but either on separate papers or into a purely applied mathematicsframework. Here, we offer to the data enthusiast scientist a comparison ofCanonical, Tucker, Hierarchical and Tensor train formats including theirapproximation algorithms. Their relative benefits are studied both theoreticallyand numerically thanks to the python library texttt{pydecomp} that wasdeveloped during this thesis. A careful analysis of the link between continuousand discrete methods is performed. Finally, we conclude that for mostapplications ST-HOSVD is best when the number of dimensions $d$ lower than fourand TT-SVD (or their POD equivalent) when $d$ grows larger.The second part is centered on a complex fluid dynamics flow, in particular thesingular lid driven cavity at high Reynolds number. This flow exhibits a seriesof Hopf bifurcation which are known to be hard to capture accurately which iswhy a detailed analysis was performed both with classical tools and POD. Oncethis flow has been characterized, emph{time-scaling}, a new ``physics based''interpolation ROM is presented on internal and external flows. This methodsgives encouraging results while excluding recent advanced developments in thearea such as EIM or Grassmann manifold interpolation.

Reduced Basis Methods for Partial Differential Equations: Evaluation of multiple non-compliant flux-type output functionals for a non-affine electrostatics problem

Abstract: A method for rapid evaluation of flux-type outputs of interest from solutions to partial differential equations (PDEs) is presented within the reduced basis framework for linear, elliptic PDEs. The central point is a Neumann-Dirichlet equivalence that allows for evaluation of the output through the bilinear form of the weak formulation of the PDE. Through a comprehensive example related to electrostatics, we consider multiple outputs, a posteriori error estimators and empirical interpolation treatment of the non-affine terms in the bilinear form. Together with the considered Neumann-Dirichlet equivalence, these methods allow for efficient and accurate numerical evaluation of a relationship mu->s(mu), where mu is a parameter vector that determines the geometry of the physical domain and s(mu) is the corresponding flux-type output matrix of interest. As a practical application, we lastly employ the rapid evaluation of s-> s(mu) in solving an inverse (parameter-estimation) problem.

Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier-Stokes equations

Abstract: In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier–Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.

A High Accuracy Preserving Parallel Algorithm for Compact Schemes for DNS

Abstract: A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorithm with two benefits. First, the number of overlap points is reduced to a single common boundary point between any two neighboring sub-domains, thereby saving the number of points used, with resultant speed-up. Second, with a proper design, errors arising due to sub-domain boundary closure schemes are reduced to a user designed error tolerance, bringing the new parallel scheme on par with sequential computing. Error reduction is achieved by using global spectral analysis, introduced in “Analysis of central and upwind compact schemes,” J. Comput. Phys. 192, 2, (2003) 677--694, which analyzes any discrete computing method in the full domain integrally. The design of the parallel compact scheme is explained, followed by a demonstration of the accuracy of the method by solving benchmark flows: (1) periodic two-dimensional Taylor-Green vortex problem; (2) flow inside two-dimensional square lid-driven cavity (LDC) at high Reynolds number; and (3) flow inside a non-periodic three-dimensional cubic LDC with the staggered grid arrangement.

Grid sensitivity and role of error in computing a lid-driven cavity problem.

Abstract: The investigation on grid sensitivity for the bifurcation problem of the canonical lid-driven cavity (LDC) flow results is reported here with very fine grids. This is motivated by different researchers presenting different first bifurcation critical Reynolds number (${\text{Re}}_{\text{cr}1}$), which appears to depend on the formulation, numerical method, and choice of grid. By using a very-high-accuracy parallel algorithm, and the same method with which sequential results were presented by Lestandi et al. [Comput. Fluids 166, 86 (2018)] [for (257 $\ifmmode\times\else\texttimes\fi{}$ 257) and (513 $\ifmmode\times\else\texttimes\fi{}$ 513) uniformly spaced grid], we present results using ($1025\ifmmode\times\else\texttimes\fi{}1025$) and ($2049\ifmmode\times\else\texttimes\fi{}2049$) grid points. Detailed results presented using these grids help us understand the computational physics of the numerical receptivity of the LDC flow, with and without explicit excitation. The mathematical physics of the investigated problem will become apparent when we identify the roles of numerical errors with the ambient omnipresent disturbances in real physical flows as interchangeable. In physical or in numerical setups, presence of disturbances cannot be ignored. In this context, the need for explicit excitation for the used compact scheme arises for a definitive threshold amplitude, below which the flow relaxes back to quiescent state after the excitation is removed in computations. We also implement the present parallel method to show the physical aspects of primary and secondary instabilities to be maintained for other numerical schemes, and we show the results to reflect the complex physics during multiple subcritical Hopf bifurcation. Also, we relate the various sources of errors during computations that is typical of such shear-driven flow. These results, with near spectral accuracy, constitute universal benchmark results for the solution of Navier-Stokes equation for LDC.

Generalization of the Neville-Aitken Interpolation Algorithm on Grassmann Manifolds : Applications to Reduced Order Model

Abstract: The interpolation on Grassmann manifolds in the framework of parametric evolution partial differential equations is presented. Interpolation points on the Grassmann manifold are the subspaces spanned by the POD bases of the available solutions corresponding to the chosen parameter values. The well-known Neville-Aitken's algorithm is extended to Grassmann manifold, where interpolation is performed in a recursive way via the geodesic barycenter of two points. The performances of the proposed method are illustrated through three independent CFD applications, namely: the Von Karman vortex shedding street, the lid-driven cavity with inflow and the flow induced by a rotating solid. The obtained numerical simulations are pertinent both in terms of the accuracy of results and the time computation.