Courses
Numerical methods for flow in porous media
Isabelle Faille (IFP Énergies Nouvelles)
This short course deals with modeling subsurface fl ow, mainly in the context of reservoir and sedimentary basin simulation. We will start by deriving the basic equations that govern single and two-phase flow in porous media. After a short description of the modeling work flows, we will present the basic numerical methods that are used in the oil industry simulators and which are based upon finite volume techniques. We will then discuss and illustrate some recent extensions designed to overcome the limitations of the standard approaches.
Convolution quadrature for wave simulations
Francisco Javier Sayas (University of Delaware)
Several problems in propagation and scattering of linear waves can be expressed as convolution equations combined with forward convolutions. These problems include interesting realistic models in acoustics, seismology, and electromagnetism.
The course will start with two formal examples from acoustics, motivating the introduction of the concept of causal convolutions in the sense of distributions. The mathematics involved are very rich: Laplace transforms, operator-valued distributions, resolvent estimates, boundary integral operators. We will boldly approach them, giving precise definitions and relying on the wisdom of our ancestors for the theory behind.
Continuous time convolutions can then be approximated by Lubich’s Convolution Quadrature techniques. We thus end up with discrete time convolution equations and operators. Two families of CQ methods will be explored: those based on multistep methods, and those based in multistage (Runge-Kutta) schemes.
We will discuss implementation of discrete convolutions in a black-box environment and play a little bit with them using the deltaBEM Matlab collection that gives really easy access to the operators for wave propagation around multiple obstacles in the plane.
An introduction to the Reduced Basis Method
Benjamin Stamm (Université Pierre et Marie Curie – CNRS)
This course gives an introduction to the Reduced Basis Method (RBM). This method gains popularity in all kind of applications and targets parametrized problems stated as a Partial Differential Equation (PDE) in a context of many-query systems, optimization or uncertainty quantification where a fast input (parameter)-output (a functional of the solution of the PDE) procedure needs to be efficiently solved many times.
The RBM is embedded in an offline-online setting. During the offline stage, a suitable linear low-dimensional approximation space to the parametric solution-manifold, i.e. the set of solutions of the PDE for all parameter values, is empirically constructed. In the online stage, where (an almost) real-time query takes place, the solution to the parametrized PDE is obtained by a Galerkin projection onto the low-dimensional approximation space. Additionally, thanks to the Galerkin setting, a posteriori error estimates can be constructed in order to certify the error committed through the model reduction by means of rigorous upper bounds.
An introduction to isogeometric analysis
Rafael Vázquez (IMATI – CNR)
Isogeometric analysis (IGA) is an innovative technique introduced to improve the interaction between Computer Aided Design (CAD) software and partial differential equations solvers. IGA can be simply defined as a Galerkin method in which the computational domain is given by standard CAD functions, such as B-splines or NURBS, and the discrete solution is computed with the same kind functions. Apart from working with a CAD geometry, isogeometric methods may also benefit of the higher continuity of splines with respect to the standard finite element method.
The course will start from the definition and properties of B-splines and NURBS, to see how they are used to generate geometries in CAD software. After that, we will see how the same functions are used in isogeometric methods. The course will cover both theoretical aspects and computational aspects of IGA, paying special attention to the differences and similarities with the finite element method.
Conferences
Mathematical methods in image processing and computer vision
Antonio Baeza (Universidad de Valencia)
Image processing and computer vision aim to understand the reality from the information contained in images and videos. A myriad of different problems are involved in such a goal, and many different areas of mathematics have naturally found applications to the analysis and solution of these problems, to such an extent that it is actually hard to find a field in mathematics with no applications to image processing and computer vision. The goal of the talk is to give an overview of some of the most active computer vision problems, ranging from simple methods for image processing to complex models for image analysis and understanding, as well as the leading mathematical techniques used for their solution.
Modeling and optimization techniques with applications in bio-processes and bio-systems
Eva Balsa-Canto (Bioprocess Engineering Group, IIM-CSIC, Vigo, Spain)
Computer-aided simulation and model-based optimization offer a powerful, rational and systematic way to improve bio-process understanding or performance. In recent decades there has been a growing interest in the development of rigorous models, based on first principles, that enable not only to perform experiments in silico, but to design and to optimise operation policies.
However several problems have to be faced mostly related to i) insufficient a priori knowledge to deduce the right model structure; ii) insufficient measurement capabilities to estimate model unknowns; iii) the complexity of bio-processes that include physical, chemical and biological phenomena on a wide range of time and space scales and iv) the complexity of the associated optimization problems due mainly to multi-modality.
In this talk a number of examples taken from the food and biotechnology industry will be used to illustrate how those problems emerge and to present some alternatives to tackle them. Special emphasis will be paid to the model identification loop, involving parameter estimation, identifiability analyses and model based experimental design; to model simulation techniques, including accurate and efficient reduced order modelling approaches and to the use of global optimization methods.
To finish with, all elements will be combined to design and implement a real-time optimization architecture, which is able to assure high operational stability, process reproducibility and optimal operation.
New tools from mathematics describing metastatic spreading in cancer
Florence Hubert (Université d’Aix-Marseille)
Classification of cancer as localized or metastatic disease remains the mainstay for determining the best therapeutic strategy to be undertaken at bedside. Despite major progresses in imaging methods, these technologies still cannot detect metastatic lesions under 108 cells in patients, or occult metastases are now considered as a real issue in therapeutic care of cancer patient. Developing a mathematical tool for better identify non detectable metastasis status, would help the clinicians to decide the most adequate therapeutic strategy, even if no metastasis is yet made detectable at bedside. We have developed some mathematical models providing new metastatic indexes (indication on the number and size of metastatic lesions throughout time) that could enable us to better predict the risk of invasive cancer, even when it does not show with standard imaging techniques.
Our mathematical model is based on a small amount of parameters related for instance to tumor aggressiveness and vascularization that we need to calibrate thanks to preclinical or clinical observations. After a presentation of the different models that we developped, we will present you the potential of such models through in particular, the outcome of a preclinical study that we performed in Marseille.
Workshop
An introduction to GPU computing for numerical simulation
José Miguel Mantas (Universidad de Granada)
Graphics Processing Units (GPUs) have proved to be a powerful accelerator for intensive numerical computations. The massive parallelism of these platforms makes it possible to achieve dramatic runtime reductions over a standard CPU in many numerical applications at a very affordable price. Moreover, several programming environments, such as NVIDIA’s Compute Unified Device Architecture (CUDA) have shown a high efectiveness in the mapping of numerical algorithms to GPUs. This workshop provides an introduction to the development of CUDA programs for numerical simulation using CUDA C/C++, the most popular GPU programming toolkit. An overview of CUDA programming will be illustrated through the CUDA implementation of several simple numerical methods for PDEs. These CUDA implementations will be studied and run on modern GPU-based platforms.