35 research outputs found
Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations
We introduce a \textit{non-modal} analysis technique that characterizes the
diffusion properties of spectral element methods for linear
convection-diffusion systems. While strictly speaking only valid for linear
problems, the analysis is devised so that it can give critical insights on two
questions: (i) Why do spectral element methods suffer from stability issues in
under-resolved computations of nonlinear problems? And, (ii) why do they
successfully predict under-resolved turbulent flows even without a
subgrid-scale model? The answer to these two questions can in turn provide
crucial guidelines to construct more robust and accurate schemes for complex
under-resolved flows, commonly found in industrial applications. For
illustration purposes, this analysis technique is applied to the hybridized
discontinuous Galerkin methods as representatives of spectral element methods.
The effect of the polynomial order, the upwinding parameter and the P\'eclet
number on the so-called \textit{short-term diffusion} of the scheme are
investigated. From a purely non-modal analysis point of view, polynomial orders
between and with standard upwinding are well suited for under-resolved
turbulence simulations. For lower polynomial orders, diffusion is introduced in
scales that are much larger than the grid resolution. For higher polynomial
orders, as well as for strong under/over-upwinding, robustness issues can be
expected. The non-modal analysis results are then tested against under-resolved
turbulence simulations of the Burgers, Euler and Navier-Stokes equations. While
devised in the linear setting, our non-modal analysis succeeds to predict the
behavior of the scheme in the nonlinear problems considered
A fast multi-resolution lattice Green's function method for elliptic difference equations
We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast multipole method in conjunction with fast Fourier transforms to yield linear complexity and decrease time-to-solution. We extend this method to a multi-resolution scheme and allow for locally refined Cartesian blocks embedded in the computational domain. Appropriately chosen interpolation and regularization operators retain consistency between the discrete Laplace operator and its inverse on the unbounded domain. Second-order accuracy and linear complexity are maintained, while significantly reducing the number of degrees of freedom and hence the computational cost
FinXABSA: Explainable Finance through Aspect-Based Sentiment Analysis
This paper presents a novel approach for explainability in financial analysis
by deriving financially-explainable statistical relationships through
aspect-based sentiment analysis, Pearson correlation, Granger causality &
uncertainty coefficient. The proposed methodology involves constructing an
aspect list from financial literature and applying aspect-based sentiment
analysis on social media text to compute sentiment scores for each aspect.
Pearson correlation is then applied to uncover financially explainable
relationships between aspect sentiment scores and stock prices. Findings for
derived relationships are made robust by applying Granger causality to
determine the forecasting ability of each aspect sentiment score for stock
prices. Finally, an added layer of interpretability is added by evaluating
uncertainty coefficient scores between aspect sentiment scores and stock
prices. This allows us to determine the aspects whose sentiment scores are most
statistically significant for stock prices. Relative to other methods, our
approach provides a more informative and accurate understanding of the
relationship between sentiment analysis and stock prices. Specifically, this
methodology enables an interpretation of the statistical relationship between
aspect-based sentiment scores and stock prices, which offers explainability to
AI-driven financial decision-making
A fast multi-resolution lattice Green's function method for elliptic difference equations
We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast multipole method in conjunction with fast Fourier transforms to yield linear complexity and decrease time-to-solution. We extend this method to a multi-resolution scheme and allow for locally refined Cartesian blocks embedded in the computational domain. Appropriately chosen interpolation and regularization operators retain consistency between the discrete Laplace operator and its inverse on the unbounded domain. Second-order accuracy and linear complexity are maintained, while significantly reducing the number of degrees of freedom and hence the computational cost
Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations
High-order spectral element methods (SEM) for large-eddy simulation (LES) are still very limited in industry. One of the main reasons behind this is the lack of robustness of SEM for under-resolved simulations, which can lead to the failure of the computation or to inaccurate results, aspects that are critical in an industrial setting. To help address this issue, we introduce a non-modal analysis technique that characterizes the numerical diffusion properties of spectral element methods for linear convectionâdiffusion problems, including the scales affected by numerical diffusion and the relationship between the amount of numerical diffusion and the level of under-resolution in the simulation. This framework differs from traditional eigenanalysis techniques in that all eigenmodes are taken into account with no need to differentiate them as physical or unphysical. While strictly speaking only valid for linear problems, the non-modal analysis is devised so that it can give critical insights for under-resolved nonlinear problems. For example, why do SEM sometimes suffer from numerical stability issues in LES? And, why do they at other times be robust and successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these questions in turn provides crucial guidelines to construct more robust and accurate schemes for LES.
For illustration purposes, the non-modal analysis is applied to the hybridized discontinuous Galerkin methods as representatives of SEM. The effects of the polynomial order, the upwinding parameter and the PĂ©clet number on the so-called short-term diffusion of the scheme are investigated. From a non-modal analysis point of view, and for the particular case of hybridized discontinuous Galerkin methods, polynomial orders between 2 and 4 with standard upwinding are found to be well suited for under-resolved turbulence simulations. For lower polynomial orders, diffusion is introduced in scales that are much larger than the grid resolution. For higher polynomial orders, as well as for strong under/over-upwinding, robustness issues can be expected due to low and non-monotonic numerical diffusion. The non-modal analysis results are tested against under-resolved turbulence simulations of the Burgers, Euler and NavierâStokes equations. While devised in the linear setting, non-modal analysis successfully predicts the behavior of the scheme in the nonlinear problems considered. Although the focus of this paper is on LES, the non-modal analysis can be applied to other simulation fields characterized by under-resolved scales
Immersed Boundary Lattice Green Function methods for External Aerodynamics
In this paper, we document the capabilities of a novel numerical approach - the immersed boundary lattice Green's function (IBLGF) method - to simulate external incompressible flows over complex geometries. This new approach is built upon the immersed boundary method and lattice Green's functions to solve the incompressible Navier-Stokes equations. We show that the combination of these two concepts allows the construction of an efficient and robust numerical framework for the direct numerical and large-eddy simulation of external aerodynamic problems at moderate to high-Reynolds numbers
Spatial eigensolution analysis of energy-stable flux reconstruction schemes and influence of the numerical flux on accuracy and robustness
This study focuses on the dispersion and diffusion characteristics of high-order energy-stable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter âcâ dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of âcâ, two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from âunder-â to âover-upwindingâ. In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving non-periodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics