Using parametric model order reduction for inverse analysis of large nonlinear cardiac simulations

Predictive high-fidelity finite element simulations of human cardiac mechanics commonly require a large number of structural degrees of freedom. Additionally, these models are often coupled with lumped-parameter models of hemodynamics. High computational demands, however, slow down model calibration and therefore limit the use of cardiac simulations in clinical practice. As cardiac models rely on several patient-specific parameters, just one solution corresponding to one specific parameter set does not at all meet clinical demands. Moreover, while solving the nonlinear problem, 90% of the computation time is spent solving linear systems of equations. We propose to reduce the structural dimension of a monolithically coupled structure-Windkessel system by projection onto a lower-dimensional subspace. We obtain a good approximation of the displacement field as well as of key scalar cardiac outputs even with very few reduced degrees of freedom, while achieving considerable speedups. For subspace generation, we use proper orthogonal decomposition of displacement snapshots. Following a brief comparison of subspace interpolation methods, we demonstrate how projection-based model order reduction can be easily integrated into a gradient-based optimization. We demonstrate the performance of our method in a real-world multivariate inverse analysis scenario. Using the presented projection-based model order reduction approach can significantly speed up model personalization and could be used for many-query tasks in a clinical setting

An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier-Stokes Equations

We discuss model reduction for a particular class of quadratic-bilinear (QB) descriptor systems. The main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QBODEs [Benner et al. '16] to a class of descriptor systems in an efficient and reliable way. Recently, it has been shown in the case of linear or bilinear systems that a direct extension of interpolation-based model reduction techniques to descriptor systems, without any modifications, may lead to poor reduced-order systems. Therefore, for the analysis, we aim at transforming the considered QB descriptor system into an equivalent QBODE system by means of projectors for which standard model reduction techniques for QBODEs can be employed, including aforementioned interpolation scheme. Subsequently, we discuss related computational issues, thus resulting in a modified algorithm that allows us to construct \emph{near}--optimal reduced-order systems without explicitly computing the projectors used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier-Stokes equations

Een schatkamer vol oude problemen

Op 2 april werd tijdens een feestelijke bijeenkomst in museum NEMO het boek De Pythagoras Code officieel gepresenteerd. Het boek bevat een bloemlezing uit 50 jaargangen Pythagoras, met de bekende 'Denkertjes' alsmede puzzels en dionigma's. Het rakelt oude problemen op, zoals het 3n+1 vermoeden, bevat een selectie van prijsvragen en een leuk hoofdstukje 'Geschiedenis' over een aatal bekende wiskundigen. Wil Schilders doet verslag

Platform Wiskunde Nederland : plannen en terugblik

In het najaar van 2010 werd het Platform Wiskunde Nederland opgericht door het Koninklijk Wiskundig Genootschap (KWG) en de Nederlandse Vereniging vanWiskundeleraren (NVvW), ondersteund door de wiskunde-instituten van de Nederlandse universiteiten (inclusief Eurandom en het Freudenthal Instituut), de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), het Centrum Wiskunde & Informatica (CWI), en de stichtingen Thomas Stieltjes en Compositio. PWN is opgericht als centrale plaats binnen de Nederlandse wiskunde van waaruit alle zaken die voor de wiskunde van belang zijn op efficiënte wijze behartigd kunnen worden. In het eerste jaar is PWN als landelijke organisatie neergezet, en zijn de commissies aan de slag gegaan. Hiermee is een solide basis gelegd voor het tweede jaar. Wil Schilders, directeur van het PWN, blikt terug op het eerste jaar en licht de plannen voor de toekomst toe

Reduced order modelling of RLC-networks using an SVD-Laguerre based method

With interconnect increasingly contributing to the electrical behaviour of integrated circuits, both by higher frequencies and smaller dimensions, it becomes increasingly important to incorporate its behaviour into simulations of ICs. This can be done rather elegantly by summarizing interconnect behaviour into a compact or reduced order model which is then co-simulated with the circuit. A similar approach can be used in the case of more conventional printed circuit boards. The SVDLaguerre algorithm proposed by Knockaert and De Zutter [4] can be used for this purpose. In this paper, we describe an e#cient implementation of the algorithm for multiple inputs, and show how the mathematical reduced order models can be translated into realizable circuit elements

Krylov subspace methods in the electronic industry

Summary. Krylov subspace methods are well-known for their nice properties, but they have to be implemented with care. In this article the mathematical consequences encountered during implementation of Krylov subspace methods in an existing layout-simulator are discussed. Briefly, the representation in a circuit is visited and two methods to avoid parts of the redundancy are drawn

Error bounds for reduction of multi-port resistor networks

The interconnect layouts of chips can be modeled by large resistor networks. In order to be able to speed up simulations of such large networks, reduction techniques are applied to reduce the size of the networks. For some class of networks, an existing reduction strategy does not provide sufficient reduction in terms of the number of resistors appearing in the final network. In this paper we propose an approach for obtaining a further reduction in the amount of resistors. The suggested approach improves sparsity of the conductance matrix by neglecting resistors which do not contribute significantly to the behavior of the circuit. Explicit error bounds, which give an opportunity to control the errors due to approximation, have been derived. Numerical examples show that the suggested approach appears promising for multi-terminal resistor networks and, in combination with the existing reduction strategy, leads to better reduction

Order reduction techniques for coupled multi-domain electromagnetic based models

This work presents a comprehensive flow able to efficiently generate reduced order models for realistic, hierarchy aware, Electromagnetic (EM) based models. Knowledge of the structure of the problem is explicitly exploited using domain partitioning and novel electromagnetic connector modeling techniques to generate a hierarchically coupled representation. This enables the efficient use of structure preserving block model order reduction techniques to generate block-wise compressed models that satisfy overall requirements, and provide cheap evaluation and simulation accurate approximations of the complete EM behaviour