Goal-oriented error control of stochastic system approximations using metric-based anisotropic adaptations

International audienceThe simulation of complex nonlinear engineering systems such as compressible fluid flows may be targeted to make more efficient and accurate the approximation of a specific (scalar) quantity of interest of the system. Putting aside modeling error and parametric uncertainty, this may be achieved by combining goal-oriented error estimates and adaptive anisotropic spatial mesh refinements. To this end, an elegant and efficient framework is the one of (Riemannian) metric-based adaptation where a goal-based a priori error estimation is used as indicator for adaptivity. This work proposes a novel extension of this approach to the case of aforementioned system approximations bearing a stochastic component. In this case, an optimisation problem leading to the best control of the distinct sources of errors is formulated in the continuous framework of the Riemannian metric space. Algorithmic developments are also presented in order to quantify and adaptively adjust the error components in the deterministic and stochastic approximation spaces. The capability of the proposed method is tested on various problems including a supersonic scramjet inlet subject to geometrical and operational parametric uncertainties. It is demonstrated to accurately capture discontinuous features of stochastic compressible flows impacting pressure-related quantities of interest, while balancing computational budget and refinements in both spaces

Anisotropic Adaptivity and Subgrid Scale Modelling for the Solution of the Neutron Transport Equation with an Emphasis on Shielding Applications

This thesis demonstrates advanced new discretisation and adaptive meshing technologies that improve the accuracy and stability of using finite element discretisations applied to the Boltzmann transport equation (BTE). This equation describes the advective transport of neutral particles such as neutrons and photons within a domain. The BTE is difficult to solve, due to its large phase space (three dimensions of space, two of angle and one each of energy and time) and the presence of non-physical oscillations in many situations. This work explores the use of a finite element method that combines the advantages of the two schemes: the discontinuous and continuous Galerkin methods. The new discretisation uses multiscale (subgrid) finite elements that work locally within each element in the finite element mesh in addition to a global, continuous, formulation. The use of higher order functions that describe the variation of the angular flux over each element is also explored using these subgrid finite element schemes. In addition to the spatial discretisation, methods have also been developed to optimise the finite element mesh in order to reduce resulting errors in the solution over the domain, or locally in situations where there is a goal of specific interest (such as a dose in a detector region). The chapters of this thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 introduces the reader to motivation behind the research contained within this thesis. Chapter 2 introduces the forms of the BTE that are used within this thesis. Chapter 3 provides the methods that are used, together with examples, of the validation and verification of the software that was developed as a result of this work, the transport code RADIANT. Chapter 4 introduces the inner element subgrid scale finite element discretisation of the BTE that forms the basis of the discretisations within RADIANT and explores its convergence and computational times on a set of benchmark problems. Chapter 5 develops the error metrics that are used to optimise the mesh in order to reduce the discretisation error within a finite element mesh using anisotropic adaptivity that can use elongated elements that accurately resolves computational demanding regions, such as in the presence of shocks. The work of this chapter is then extended in Chapter 6 that forms error metrics for goal based adaptivity to minimise the error in a detector response. Finally, conclusions from this thesis and suggestions for future work that may be explored are discussed in Chapter 7.Open Acces

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

We introduce a data-driven version of the plus Cartan connection on the homogeneous space $\mathbb{M}_2$ of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on $\mathbb{M}_{2}$. The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on $\mathbb{M}_2$ that we compute by a new modified anisotropic fast-marching method. Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented Mathematica notebooks available at GitHub (https://github.com/NickyvdBerg/DataDrivenTracking)

Connecting mathematical models for image processing and neural networks

This thesis deals with the connections between mathematical models for image processing and deep learning. While data-driven deep learning models such as neural networks are flexible and well performing, they are often used as a black box. This makes it hard to provide theoretical model guarantees and scientific insights. On the other hand, more traditional, model-driven approaches such as diffusion, wavelet shrinkage, and variational models offer a rich set of mathematical foundations. Our goal is to transfer these foundations to neural networks. To this end, we pursue three strategies. First, we design trainable variants of traditional models and reduce their parameter set after training to obtain transparent and adaptive models. Moreover, we investigate the architectural design of numerical solvers for partial differential equations and translate them into building blocks of popular neural network architectures. This yields criteria for stable networks and inspires novel design concepts. Lastly, we present novel hybrid models for inpainting that rely on our theoretical findings. These strategies provide three ways for combining the best of the two worlds of model- and data-driven approaches. Our work contributes to the overarching goal of closing the gap between these worlds that still exists in performance and understanding.Gegenstand dieser Arbeit sind die Zusammenhänge zwischen mathematischen Modellen zur Bildverarbeitung und Deep Learning. Während datengetriebene Modelle des Deep Learning wie z.B. neuronale Netze flexibel sind und gute Ergebnisse liefern, werden sie oft als Black Box eingesetzt. Das macht es schwierig, theoretische Modellgarantien zu liefern und wissenschaftliche Erkenntnisse zu gewinnen. Im Gegensatz dazu bieten traditionellere, modellgetriebene Ansätze wie Diffusion, Wavelet Shrinkage und Variationsansätze eine Fülle von mathematischen Grundlagen. Unser Ziel ist es, diese auf neuronale Netze zu übertragen. Zu diesem Zweck verfolgen wir drei Strategien. Zunächst entwerfen wir trainierbare Varianten von traditionellen Modellen und reduzieren ihren Parametersatz, um transparente und adaptive Modelle zu erhalten. Außerdem untersuchen wir die Architekturen von numerischen Lösern für partielle Differentialgleichungen und übersetzen sie in Bausteine von populären neuronalen Netzwerken. Daraus ergeben sich Kriterien für stabile Netzwerke und neue Designkonzepte. Schließlich präsentieren wir neuartige hybride Modelle für Inpainting, die auf unseren theoretischen Erkenntnissen beruhen. Diese Strategien bieten drei Möglichkeiten, das Beste aus den beiden Welten der modell- und datengetriebenen Ansätzen zu vereinen. Diese Arbeit liefert einen Beitrag zum übergeordneten Ziel, die Lücke zwischen den zwei Welten zu schließen, die noch in Bezug auf Leistung und Modellverständnis besteht.ERC Advanced Grant INCOVI

Complexity reduction in parametric flow problems via Nonintrusive Proper Generalised Decomposition in OpenFOAM

Tesi en modalitat cotutela: Universitat Politècnica de Catalunya i Swansea University. Programa Erasmus Mundus en Simulació en Enginyeria i Desenvolupament de l'Emprenedoria (SEED)The present thesis explores the viability of the proper generalised decomposition (PGD) as a tool for parametric studies in a daily industrial environment. Starting from the equations modelling incompressible flows, the separated formulation of the equations, the development of a parametric solver, the implementation in a commercial computational fluid dynamics (CFD) software, OpenFOAM, and a numerical validation are presented. The parametrised Stokes and Oseen flows are used as an initial step to test the applicability of the PGD to flow problems. The rationale for the construction of a separable approximation is described and implemented in OpenFOAM. For the numerical validation of the developed strategy analytical test cases are solved. Then, the parametrised steady laminar incompressible Navier-Stokes equations are considered. The nonintrusive implementation of PGD in OpenFOAM is formulated, focusing on the seamless integration of a reduced order model (ROM) in the framework of an industrially validated CFD software. The proposed strategy exploits classical solution strategies in OpenFOAM to solve the PGD spatial iteration, while the parametric one is solved via a collocation approach. Such nonintrusiveness represents an important step towards the industrialisation of PGD-based approaches. The capabilities of the methodology are tested by applying it to benchmark tests in the literature and solving a parametrised flow control problem in a realistic geometry of interest for the automotive industry. Finally, the PGD framework is extended to turbulent Navier-Stokes problems. The separable form of an industrially popular turbulence model, namely Spalart-Allmaras model, is formulated and a PGD strategy for the construction of a parametric turbulent eddy viscosity is devised. Different implementation possibilities in the nonintrusive PGD for parametrised Navier-Stokes equations are explored and the proposed strategy is applied to well-documented turbulent flow control benchmark cases in both two and three dimensions.La tesis explora la viabilidad del método de reducción de modelos Proper Generalised Decomposition (PGD) como herramienta habitual en un entorno industrial para obtener soluciones de problemas de flujo viscoso incompresible que dependan de parámetros. En este documento, partiendo de las ecuaciones que modelan el flujo viscoso e incompresible, se describe en detalle la formulación en forma separada, espacio-parámetros, de las ecuaciones para el método PGD, se desarrolla el algoritmo de resolución teniendo en cuenta los parámetros, se detalla como realizar la implementación en OpenFOAM, que es un software comercial de dinámica de fluidos computacional (CFD por sus siglas en inglés) y se discuten las validaciones numéricas correspondientes. Como paso previo para probar la viabilidad de la PGD a problemas de interés, se estudian flujos de Stokes y Oseen con datos parametrizados. De esta forma, se desarrollan las bases para la construcción de una aproximación separada, espacio-parámetros, de la solución numérica velocidad-presión, todo ello implementado en OpenFOAM. Para estas formulaciones se valida la aproximación numérica de la estrategia desarrollada con ejemplos cuya solución analítica es conocida, lo que permite analizar los errores cometidos, y se presentan ejemplos numéricos de referencia ampliamente estudiados en la literatura para mostrar su viabilidad. Seguidamente se consideran las ecuaciones de Navier-Stokes para flujo incompresible, estacionario y laminar de nuevo dependiendo de parámetros de diseño. La implementación no intrusiva de la PGD en OpenFOAM está formulada para obtener integración perfecta de un modelo de orden reducido (ROM por sus siglas en inglés) con un software CFD validado industrialmente. La metodología propuesta explota las estrategias de solución clásicas ya existentes en OpenFOAM para resolver la iteración espacial de la PGD, mientras que la iteración de las funciones que dependen de los parámetros se realiza de forma externa a OpenFOAM (empleando formulaciones basadas en la colocación puntual). La no-intrusividad es crítica para una cualquier estrategia que pretenda emplear la formulación PGD en la práctica diaria de la producción y diseño industrial. Para justificar la metodología propuesta así como su viabilidad, se muestra la solución de problemas de referencia clásicos y habituales en la literatura así como la resolución de un problema de control de flujo parametrizado en una geometría realista de interés para la industria de la automoción. Finalmente, es importante resaltar que se extiende a flujos turbulentos la metodología propuesta para trabajar con la PGD de manera no-intrusiva. Más concretamente, las ecuaciones de Navier-Stokes se complementan con un modelo de turbulencia habitual en aplicaciones industriales: el modelo de Spalart-Allmaras. En este caso, se propone una extensión de la estructura separada de las aproximaciones (velocidad y presión), y se diseña una estrategia PGD para la construcción de una viscosidad turbulenta paramétrica. Se exploran diferentes posibilidades de implementación de la PGD no intrusiva para las ecuaciones de Navier-Stokes para flujo turbulento y dependiendo de parámetros. La estrategia propuesta se aplica a casos de referencia de control de flujo turbulento bien documentados en dos y tres dimensiones.Postprint (published version

Complexity reduction in parametric flow problems via Nonintrusive Proper Generalised Decomposition in OpenFOAM

The present thesis explores the viability of the proper generalised decomposition (PGD) as a tool for parametric studies in a daily industrial environment. Starting from the equations modelling incompressible flows, the separated formulation of the equations, the development of a parametric solver, the implementation in a commercial computational fluid dynamics (CFD) software, OpenFOAM, and a numerical validation are presented. The parametrised Stokes and Oseen flows are used as an initial step to test the applicability of the PGD to flow problems. The rationale for the construc- tion of a separable approximation is described and implemented in OpenFOAM. For the numerical validation of the developed strategy analytical test cases are solved. Then, the parametrised steady laminar incompressible Navier-Stokes equations are considered. The nonintrusive implementation of PGD in OpenFOAM is formulated, focusing on the seamless integration of a reduced order model (ROM) in the framework of an industrially validated CFD software. The proposed strategy exploits classical solution strategies in OpenFOAM to solve the PGD spatial iteration, while the parametric one is solved via a collocation approach. Such nonintrusiveness represents an important step towards the industrialisation of PGD-based approaches. The capabilities of the methodology are tested by applying it to benchmark tests in the literature and solving a parametrised flow control problem in a realistic geometry of interest for the automotive industry. Finally, the PGD framework is extended to turbulent Navier-Stokes problems. The separable form of an industrially popular turbulence model, namely Spalart-Allmaras model, is formulated and a PGD strategy for the construction of a parametric turbulent eddy viscosity is devised. Different im- plementation possibilities in the nonintrusive PGD for parametrised Navier- Stokes equations are explored and the proposed strategy is applied to well-documented turbulent flow control benchmark cases in both two and three dimensions.La tesis explora la viabilidad del método de reducción de modelos Proper Generalised Decomposition (PGD) como herramienta habitual en un entorno industrial para obtener soluciones de problemas de flujo viscoso incompresible que dependan de parámetros. En este documento, partiendo de las ecuaciones que modelan el flujo viscoso e incompresible, se describe en detalle la formulación en forma separada, espacio-parámetros, de las ecuaciones para el método PGD, se desarrolla el algoritmo de resolución teniendo en cuenta los parámetros, se detalla como realizar la implementación en OpenFOAM, que es un software comercial de dinámica de fluidos computacional (CFD por sus siglas en inglés) y se discuten las validaciones numéricas correspondientes. Como paso previo para probar la viabilidad de la PGD a problemas de interés, se estudian flujos de Stokes y Oseen con datos parametrizados. De esta forma, se desarrollan las bases para la construcción de una aproximación separada, espacio-parámetros, de la solución numérica velocidad-presión, todo ello implementado en OpenFOAM. Para estas formulaciones se valida la aproximación numérica de la estrategia desarrollada con ejemplos cuya solución analítica es conocida, lo que permite analizar los errores cometidos, y se presentan ejemplos numéricos de referencia ampliamente estudiados en la literatura para mostrar su viabilidad. Seguidamente se consideran las ecuaciones de Navier-Stokes para flujo incompresible, estacionario y laminar de nuevo dependiendo de parámetros de diseño. La implementación no intrusiva de la PGD en OpenFOAM está formulada para obtener integración perfecta de un modelo de orden reducido (ROM por sus siglas en inglés) con un software CFD validado industrialmente. La metodología propuesta explota las estrategias de solución clásicas ya existentes en OpenFOAM para resolver la iteración espacial de la PGD, mientras que la iteración de las funciones que dependen de los parámetros se realiza de forma externa a OpenFOAM (empleando formulaciones basadas en la colocación puntual). La no-intrusividad es crítica para una cualquier estrategia que pretenda emplear la formulación PGD en la práctica diaria de la producción y diseño industrial. Para justificar la metodología propuesta así como su viabilidad, se muestra la solución de problemas de referencia clásicos y habituales en la literatura así como la resolución de un problema de control de flujo parametrizado en una geometría realista de interés para la industria de la automoción. Finalmente, es importante resaltar que se extiende a flujos turbulentos la metodología propuesta para trabajar con la PGD de manera no-intrusiva. Más concretamente, las ecuaciones de Navier-Stokes se complementan con un modelo de turbulencia habitual en aplicaciones industriales: el modelo de Spalart-Allmaras. En este caso, se propone una extensión de la estructura separada de las aproximaciones (velocidad y presión), y se diseña una estrategia PGD para la construcción de una viscosidad turbulenta paramétrica. Se exploran diferentes posibilidades de implementación de la PGD no intrusiva para las ecuaciones de Navier-Stokes para flujo turbulento y dependiendo de parámetros. La estrategia propuesta se aplica a casos de referencia de control de flujo turbulento bien documentados en dos y tres dimensiones

Adaptive Algorithms

Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations