INDAM-GNCS 2017: Advanced numerical methods combined with computational reduction techniques for parameterised PDEs and applications

Introduction: The objective of this project is to design and analyse innovative numerical methods for the approximation of partial differential equations (PDEs) in computational science and engineering. The increasing complexity of realistic models and the evolution of computational platforms and architectures are challenging the numerical analysis community to develop more efficient, effective, and innovative methods. We share a consolidated expertise on advanced discretisation schemes based on variational approaches, such as h-type finite elements (FEM), isogeometric analysis (IGA), spectral, and boundary el- ements (BEM). In several applied fields and scenarios a successive resolution of PDEs with different values of control or design variables or different physical/geometric quantities is required, thus demanding high computational efficiency. We plan to study suitable Reduced Order Methods (ROM), such as reduced basis methods (RBM), proper orthogonal decomposition (POD) and hierarchical model (HiMod) reduction, representing effective strategies to contain the overall computational costs. We expect that our results will open new scenarios, making possible the solution of more complex problems with significantly reduced computational effort. Applications will play a leading role in the project to highlight demonstrative proofs of concept related with methodological advances, as well as software developments, in order to enhance efficient scientific computing on modern platforms/technological devices.
Description: Several phenomena of interest in applied sciences are modelled as coupled systems of parametrized PDEs. Industrial and clinical researches show nowadays a growing demand of efficient computational tools for multi-query and real-time computations for parametrized systems, such as sensitivity analysis of the haemodynamics in patient specific biomedical devices or shape optimization of structural parts. Relying on classical high-fidelity discretisation methods for each new query is usually unaffordable due to high computational costs, especially in iterative procedures used for optimization, control, uncertainty quantification, general inverse problems. For these reasons, an increasing amount of research has been devoted in the last decade to reduced order methods. These allow us to obtain accurate and reliable results at greatly reduced computational costs thanks to the selection of few basis functions, representing the most relevant features of the phenomena involved, and to an efficient separation of expensive computations (offline) from the inexpensive (online) reduced-order queries. Reduced order methods (ROMs) proposed in this context, are for instance reduced basis (RB), proper orthogonal decomposition (POD) (and their combination: POD in time and RB in parameters, for instance), hierarchical model reduction (HiMOD), and proper generalized decomposition (PGD) methods.

ROM methodology: We aim at consolidating the development of computational reduction techniques for problems described by parametrized mathematical models, governed by partial differential equations (PDEs). Parameters might be both physical (material properties, nondimensional coefficients such as Reynolds number, boundary conditions, forcing terms) and geometric (i.e., quantities which characterize the shape of the domain). The focus of this project concerns scientific computing and modelling, with a special interest in computational mechanics, electromagnetism, optimization, and control. In all these cases, iterative minimization procedures entailing several numerical resolutions of PDEs (each time with different values of control or design variables or different physical/geometric scenarios) are involved, thus requiring high computational efficiency. For this reason, suitable reduced order model (ROM) techniques, such as reduced basis methods (RBM) and hierarchical model reduction (HiMod), provide an effective strategy to contain the overall computational cost. HiMod, for instance, provides surrogate models for problems characterized by an intrinsic directionality, with transverse components relevant locally. The idea of HiMod is to discretize the full model via a combination of a 1D finite element approximation along the leading direction with a modal expansion for the transverse dynamics. This leads to solving a system of 1D models. Research activities have led to a significant development of ROM for many problems, and to real-life applications in several scenarios. However, to make these techniques more efficient and viable in a more applied and technological context, several methodological issues are still to be investigated and developed. For instance, in order to perform efficient numerical simulations in complex and variable geometric configurations, as required for instance in engineering or medical applications, ROMs need to be coupled with efficient adaptive and/or parameterization techniques. In order to treat complex geometric properties, the recent developments of the Isogeometric Analysis for curves and surfaces, based on B-Spline or NURBS (Non-Uniform Rational B-Spline), can help ROM to become more performing. This guarantees an efficient and highly integrated treatment and processing of geometries developed within CAD systems, as well as the generation of high-quality computational grids, without resorting to any geometric approximation. To make ROM even more efficient, the combination with BEM (Boundary Element Method) is advisable in linear problems instead of the classical finite elements. Another possible tool to save on the computational cost is represented by the employment of anisotropic adaptive meshes, whose elements are automatically tuned in terms of size, shape, and orientation.

ROM implementation: We plan to develop a versatile and automatic software library for the optimal management of the problem parameters, especially the geometric parametrization, and the coupling of this library with the main packages implementing ROMs. The resulting software will comprise an offline optimized module for the generation of the basis functions and of high-quality adapted meshes, possibly anisotropic, and an online module that will be designed to be used also remotely on a variety of platforms (from laptops to tablets, delocalized), including a graphical interface. The new supercomputer installed at SISSA/ICTP, Ulysses, will be also used for the offline part.
ROM applications: We will focus on: optimal control and inverse problems in electromagnetism; integration of high order methods (IGA), lower dimensionality methods (BEM), and reduced order methods for the efficient management of geometries of industrial and medical interest; computational reduction techniques (reduced basis methods, proper orthogonal decomposition and hierarchical model reduction) and possible interplay. We aim at reducing the online (and possibly the offline) computational time in computational mechanics, fluid dynamics and electromagnetism, in order to address more and more complex problems, taking into account also uncertainty scenarios, and a certain versatility. Other possible applications of interest include naval and mechanical engineering (flows around ships), medicine (fluid-structure interaction problems in the cardiovascular system, as well as the electroencephalography inverse problem) and electric engineering (source identification). All these fields reflect a deep know-how of the senior participants of the project.
ROM Objectives: The research project intends to further improve the capabilities of the reduced order methods in different fields, in order to face more demanding and complex applications arising in industrial, medical and applied sciences contexts. This research will involve human resources from SISSA, University of Trento, Politecnico di Milano-MOX, University of Pavia. We expect this collaboration to be fruitful in the framework of computational electromagnetism and anisotropic mesh adaptation. More in general, we believe that anisotropic mesh adaptation will improve the computational efficiency in any applicative field exhibiting strong directional features. Developments of advanced reduced order methods will be carefully crafted and adapted in several fields where there is a strong need of computational reduction strategies as well as of parametric studies. In the project the developments of ROM for complex problems in the framework of computational fluid-dynamics and fluid-structure interactions will take advantage of the combined and diversified expertise of the participants of the projects: isogeometric analysis, immersed boundary method, fluid-dynamics and fluid-structure interactions.

Our research project is largely of fundamental and methodological nature, hence we expect to contribute to the advance of knowledge in the framework of numerical analysis for the approximation of PDEs. Nevertheless, we expect that our results will provide new insights for the design of numerical schemes in several application areas. The novelties of our research will concern new analysis tools for the adopted methodologies and new software for scientific computing involving problems ranging from academic examples to real life applications in the fields of fluid-dynamics, electromagnetism, structural mechanics, cardiology and in general multiphysics problems in computational science and engineering.

Models and methods developed within the research program will contribute to open new scenarios in the numerical approximation of PDEs, making it possible the solution of more complex problems with significantly reduced computational times (order of seconds). In addition to industry, the techniques developed within this project have a great potential impact in biomedical and biological fields, ensuring the possibility to increase the complexity of the problems to be addressed, for example by including also uncertainty quantification of some parameters of interest. A remarkable strength of the proposed methodology is the possibility to offer a clear separation between computational resources, together with computational stages. Clusters and supercomputers are useful in the offline stage to generate a basis of solutions, by means of repetitive and expensive computations. Less powerful devices, such as laptops but even tablets or smartphones, enable during the online stage to obtain real-time calculations. The development of the proposed methods is complementary to important investigations dealing with high-performance, large scale, parallel computing environments. By easily accessing a database of pre-computed solutions, reduced-order methods make it promptly available both large computing platforms and increasingly complex mathematical models, such as for example in healthcare or shipyard facilities.

Responsabile: Gianluigi Rozza (P.I), SISSA mathLab
Participants: Luca Heltai (SISSA mathLab), Stefano Micheletti (MOX, Dipartimento di Matematica, Politecnico di Milano), Simona Perotto (MOX, Dipartimento di Matematica, Politecnico di Milano), Alessandro Reali (Universit`a degli Studi di Pavia), Simone Morganti (Universit`a degli Studi di Pavia), Ana Alonso Rodrıguez (Universit`a degli Studi di Trento), Alberto Valli (Universit`a degli Studi di Trento), Francesco Ballarin (SISSA mathLab), Giovanni Stabile (SISSA mathLab), Marco Tezzele (SISSA mathLab), Federico Pichi (SISSA mathLab), Saddam Hijazi (SISSA mathLab), Zakia Zainib (SISSA mathLab), Shafqat Ali (SISSA mathLab), Massimo Carraturo (Universit`a degli Studi di Pavia), Nicola Ferro (Politecnico di Milano), Yves Antonio Brandes Costa Barbosa (Politecnico di Milano), Juan Luis Valerdi Cabrera (Universit`a degli Studi di Trento)