Fast reconstruction of 3D blood flows from Doppler ultrasound images and reduced models

This paper deals with the problem of building fast and reliable 3D reconstruction methods for blood flows for which partial information is given by Doppler ultrasound measurements. This task is of interest in medicine since it could enrich the available information used in the diagnosis of certain diseases which is currently based essentially on the measurements coming from ultrasound devices. The fast reconstruction of the full flow can be performed with state estimation methods that have been introduced in recent years and that involve reduced order models. One simple and efficient strategy is the so-called Parametrized Background Data-Weak approach (PBDW). It is a linear mapping that consists in a least squares fit between the measurement data and a linear reduced model to which a certain correction term is added. However, in the original approach, the reduced model is built a priori and independently of the reconstruction task (typically with a proper orthogonal decomposition or a greedy algorithm). In this paper, we investigate the construction of other reduced spaces which are built to be better adapted to the reconstruction task and which result in mappings that are sometimes nonlinear. We compare the performance of the different algorithms on numerical experiments involving synthetic Doppler measurements. The results illustrate the superiority of the proposed alternatives to the classical linear PBDW approach

An Adaptive Parareal Algorithm

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the improvement of the parallel efficiency of the parareal in time method. The parareal method is based on combining predictions made by a numerically inexpensive solver (with coarse physics and/or coarse resolution) with corrections coming from an expensive solver (with high-fidelity physics and high resolution). At convergence, the parareal algorithm provides a solution that has the fine solver's high-fidelity physics and high resolution In the classical version of parareal, the fine solver has a fixed high accuracy which is the major obstacle to achieve a competitive parallel efficiency. In this paper, we develop an adaptive variant of the algorithm that overcomes this obstacle. Thanks to this, the only remaining factor impacting performance becomes the cost of the coarse solver. We show both theoretically and in a numerical example that the parallel efficiency becomes very competitive when the cost of the coarse solver is small

State Estimation with Model Reduction and Shape Variability. Application to biomedical problems

We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems are posed on certain organs or portions of the body which inevitably involve morphological variations. If one wants to provide fast reconstruction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows to take this variability into account without needing any a priori knowledge on a parametrization of the geometrical variations. For this, we rely on morphometric techniques involving Multidimensional Scaling, and couple them with reconstruction algorithms that make use of reduced model spaces pre-computed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired from the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging

Homogenization in the energy variable for a neutron transport model

This article addresses the homogenization of linear Boltzmann equation when the optical parameters are highly heterogeneous in the energy variable. We employ the method of two-scale convergence to arrive at the homogenization result. In doing so, we show the induction of a memory effect in the homogenization limit. We also provide some numerical experiments to illustrate our results. In the Appendix, we treat a related case of the harmonic oscillator

TOWARDS A FULLY SCALABLE BALANCED PARAREAL METHOD: APPLICATION TO NEUTRONICS

In the search of new approaches for the efficient exploitation of large scale compu- tational platforms, the parallelization in the time direction for time dependent problems is a very promising approach. Among the existing methods in this frame, the parallel in time method, since its introduction in [10], has been developed in many ways that, altogether, allow to identify its pros and cons. Among the cons, the current approaches present in practice some efficiency limitations as regards correct scalings that “spoil” the huge potential of the idea. This article is a contribution towards overcoming this major obstruction by exploiting the idea that the numerical schemes to parallelize time could be coupled to other iterative numerical algorithms that are needed to solve the PDE. We present a parareal scheme in which these alternative iterations are truncated (i.e. not converged) during each parareal iteration but in which convergence is nevertheless achieved across the parareal iterations. In order to limit the use of too much memory necessitated by the recovery of these alternative iterations over the parareal iterations, we propose also a compression procedure via proper orthogonal decomposition. After a mathematical analysis of the convergence properties of this new approach, we present some numerical results dealing with the application of the scheme to ac- celerate the time-dependent neutron diffusion equation in a reactor core. The numerical results show a significant improvement of the performances with respect to the plain parareal algorithm, which is an important step towards making the parallelization in the time direction be a fully competitive option for the exploitation of massively parallel computers

MINARET or the quest towards the use of time-dependent neutron transport solvers for nuclear core calculations on a regular basis

International audienceThe present paper deals with the resolution of the time-dependent neutron transport equation that is involved in the field of nuclear safety studies. Through the presentation of the newly implemented kinetic module in the MINARET solver [24] (developed at CEA in the framework of the APOLLO3\registered $$ project), we aim first of all at presenting a brief and comprehensive overview of the most widespread resolution techniques employed nowadays in neutron transport industrial codes. Given that the main obstacle in the use of this type of accurate solver on a regular basis relies in the long computing times, MINARET has been used in the present work as a support to rigorously quantify the efficiency of the most common sequential and parallel acceleration techniques that are currently used in this field. An important part of the paper will be devoted to study the performances of an acceleration method that has never been considered before in the resolution of this equation, which is the parallelization of the time variable. In this regard, the parareal in time algorithm (a domain decomposition method for the time variable, [20]) has been implemented to explore its potentialities in this particular application

The generalized Empirical Interpolation Method: stability theory on Hilbert spaces with an application to the Stokes equation

International audienceThe Generalized Empirical Interpolation Method (GEIM) is an extension first presented in [1] of the classical empirical interpolation method (see [2], [3], [4]) where the evaluation at interpolating points is replaced by the evaluation at interpolating continuous linear functionals on a class of Banach spaces. As outlined in [1], this allows to relax the continuity constraint in the target functions and expand the application domain. A special effort has been made in this paper to understand the concept of stability condition of the generalized interpolant (the Lebesgue constant) by relating it in the first part of the paper to an inf-sup problem in the case of Hilbert spaces. In the second part, it will be explained how GEIM can be employed to monitor in real time physical experiments by combining the acquisition of measurements from the processes with their mathematical models (parameter-dependent PDE's). This idea will be illustrated through a parameter dependent Stokes problem in which it will be shown that the pressure and velocity fields can efficiently be reconstructed with a relatively low dimension of the interpolating spaces

Nonlinear reduced models for state and parameter estimation

State estimation aims at approximately reconstructing the solution $u$ to a parametrized partial differential equation from $m$ linear measurements, when the parameter vector $y$ is unknown. Fast numerical recovery methods have been proposed based on reduced models which are linear spaces of moderate dimension $n$ which are tailored to approximate the solution manifold $\mathcal{M}$ where the solution sits. These methods can be viewed as deterministic counterparts to Bayesian estimation approaches, and are proved to be optimal when the prior is expressed by approximability of the solution with respect to the reduced model. However, they are inherently limited by their linear nature, which bounds from below their best possible performance by the Kolmogorov width $d_m(\mathcal{M})$ of the solution manifold. In this paper we propose to break this barrier by using simple nonlinear reduced models that consist of a finite union of linear spaces $V_k$, each having dimension at most $m$ and leading to different estimators $u_k^*$. A model selection mechanism based on minimizing the PDE residual over the parameter space is used to select from this collection the final estimator $u^*$. Our analysis shows that $u^*$ meets optimal recovery benchmarks that are inherent to the solution manifold and not tied to its Kolmogorov width. The residual minimization procedure is computationally simple in the relevant case of affine parameter dependence in the PDE. In addition, it results in an estimator $y^*$ for the unknown parameter vector. In this setting, we also discuss an alternating minimization (coordinate descent) algorithm for joint state and parameter estimation, that potentially improves the quality of both estimators