MOX Reports
The preprint collection of the Laboratory for Modeling and Scientific Computation MOX. It mainly contains works on numerical
analysis and mathematical modeling applied to engineering problems. MOX web site is mox.polimi.it
Found 1238 products
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44/2015 - 09/20/2015
Antonietti, P.F.; Houston, P.; Smears, I.
A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-Version Discontinuous Galerkin methods | Abstract | | In this article, we consider the derivation of hp-optimal spectral bounds for a class of domain decomposition preconditioners based on the Schwarz framework for discontinuous Galerkin finite element approximations of second-order elliptic partial differential equations. In particular, we improve the bounds derived in our earlier article [P.F. Antonietti and P. Houston, J. Sci. Comput., 46(1):124–149, 2011] in the sense that the resulting bound on the condition number of the preconditioned system is not only explicit with respect to the coarse and fine mesh sizes H and h, respectively, and the fine mesh polynomial degree p, but now also explicit with respect to the polynomial degree q employed for the coarse grid solver. More precisely, we show that the resulting spectral bounds are of order p2H/(qh) for the hp-version of the discontinuous Galerkin method. |
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43/2015 - 09/09/2015
Deparis, S.; Forti, D.; Gervasio, P.; Quarteroni, A.
INTERNODES: an accurate interpolation-based method for coupling the Galerkin solutions of PDEs on subdomains featuring non-conforming interfaces | Abstract | | We are interested in the approximation of partial differential equations on domains decomposed into two (or several) subdomains featuring non-conforming interfaces. The non-conformity may be due to different meshes and/or different polynomial degrees used from the two sides, or even to a geometrical mismatch. Across each interface, one subdomain is identified as master and the other as slave. We consider Galerkin methods for the discretization (such as finite element or spectral element methods) that make use of two interpolants for transferring information across the interface: one from master to slave and another one from slave to master. The former is used to ensure continuity of the primal variable (the problem solution), while the latter for the dual variable (the normal flux). In particular, since the dual variable is expressed in weak form, we first compute a strong representation of the dual variable from the slave side, interpolate it, transform the interpolated quantity back into weak form and assign it to the master side. In case of slightly non-matching geometries, we use a radial-basis function interpolant instead of Lagrange interpolant.
We name the proposed method INTERNODES (INTERpolation for NOnconforming DEcompositionS). It can be regarded as an alternative to the mortar element method and it is much simpler to implement in a numerical code. We show on two dimensional problems that by using the Lagrange interpolation we obtain at least as good convergence results as with the mortar element method with any order of polynomials. When using low order polynomials, the radial-basis interpolant leads to the same convergence properties as the Lagrange interpolant. We conclude with a comparison between INTERNODES and a standard conforming approximation in a three dimensional case. |
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42/2015 - 09/07/2015
Brugiapaglia, S.; Nobile, F.; Micheletti, S.; Perotto, S.
A theoretical study of COmpRessed SolvING for advection-diffusion-reaction problems | Abstract | | We present a theoretical analysis of the CORSING (COmpRessed SolvING) method for the numerical approximation of partial differential equations based on compressed sensing. In particular, we show that the best s-term approximation of the weak solution of a PDE with respect to an orthonormal system of N trial functions, can be recovered via a Petrov-Galerkin approach using m ? N orthonormal test functions. This recovery is guaranteed if the local a-coherence associated with the bilinear form and the selected trial and test bases fulfills suitable decay properties. The fundamental tool of this analysis is the restricted inf-sup property, i.e., a combination of the classical inf-sup condition and the well-known restricted isometry property of compressed sensing. |
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41/2015 - 08/28/2015
Quarteroni, A.; Veneziani, A.; Vergara, C.
Geometric multiscale modeling of the cardiovascular system, between theory and practice | Abstract | | This review paper addresses the so called geometric multiscale approach for the numerical simulation of blood flow problems, from its origin (that we can identify in 1997) to our days. By this approach the blood fluid-dynamics in the whole circulatory system is described mathematically by means of heterogeneous featuring different degree of detail and different geometric dimension that interact together through appropriate interface coupling conditions.
Our review starts with the introduction of the stand-alone problems, namely the 3D fluid-structure interaction problem,
its reduced representation by means of 1D models, and the so-called lumped parameters (aka 0D) models, where only the dependence on time survives.
We then address specific methods for stand-alone models when the available boundary data are not enough to ensure the mathematical well posedness. These so-called ``defective problems'' naturally arise in practical applications of clinical relevance but also because of the interface coupling of heterogeneous problems that are generated by the geometric multiscale process.
We also describe specific issues related to th boundary treatment of reduced models, particularly relevant to the multiscale coupling.
Next, we detail the most popular numerical
algorithms for the solution of these problems.
Finally, we review some of the most representative works - from different research groups - which addressed the geometric multiscale approach in the past years.
A proper treatment of the different scales relevant to the hemodynamics and their interplay is essential for the accuracy of numerical simulations and eventually for their clinical impact. This paper aims at providing a state-of-the-art picture of these topics, where the gap between theory and practice demands rigorous mathematical models to be reliably filled. |
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40/2015 - 08/24/2015
Patelli, A.S.; Dedè, L.; Lassila, T.; Bartezzaghi, A.; Quarteroni, A.
Isogeometric approximation of cardiac electrophysiology models on surfaces: an accuracy study with application to the human left atrium | Abstract | | We consider Isogeometric Analysis in the framework of the Galerkin method for the spatial approximation of cardiac electrophysiology models defined on NURBS surfaces; specifically, we perform a numerical comparison between basis functions of degree p ? 1 and globally C^k-continuous, with k = 0 or p ? 1, to find the most accurate approximation of a propagating front with the minimal number of degrees of freedom.We show that B-spline basis functions of degree p ? 1, which are C^(p?1)-continuous capture accurately the front velocity of the transmembrane potential even with moderately refined meshes; similarly, we show that,for accurate tracking of curved fronts, high-order continuous B-spline basis functions should be used. Finally, we apply Isogeometric Analysis to an idealized human left atrial geometry described by NURBS with physiologically sound fiber directions and anisotropic conductivity tensor to demonstrate that the numerical scheme retains its favorable approximation properties also in a more realistic setting. Keywords: Isogeometric Analysis, cardiac electrophysiology, surface PDEs, high-order approximation |
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39/2015 - 08/10/2015
Guglielmi, A.; Ieva, F.; Paganoni, A.M.; Quintana, F.A.
A semiparametric Bayesian joint model for multiple mixed-type outcomes: an Application to Acute Myocardial Infarction | Abstract | | We propose a Bayesian semiparametric regression model to represent mixed-type multiple outcomes concerning patients affected by Acute Myocardial Infarction. Our approach is motivated by data coming from the ST-Elevation Myocardial Infarction(STEMI) Archive, a multi-center observational prospective clinical study planned as part of the Strategic Program of Lombardy, Italy. We specifically consider a joint model for a variable measuring treatment time and in-hospital and 60-day survival indicators.
One of our motivations is to understand how the various hospitals differ in terms of the variety of information collected as part of the study. We are particularly interested in using the available data to detect differences across hospitals. In order to do so we postulate a semiparametric random effects model that incorporates dependence on a location indicator that is used to explicitly differentiate among hospitals in or outside the city of Milano. The model is based on the two parameter Poisson-Dirichlet prior, also known as the Pitman-Yor process prior. We discuss the resulting posterior inference, including sensitivity analysis, and a comparison with the particular sub-model arising when a Dirichlet process prior is assumed. |
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38/2015 - 07/16/2015
Grasso, M.; Menafoglio, A.; Colosimo, B.M.; Secchi, P.
Using Curve Registration Information for Profile Monitoring | Abstract | | The quality characteristics in manufacturing processes are often represented in terms of spatially or time ordered data, called “profiles”, which are characterized by amplitude and phase variability. In this context, curve registration plays a key role, as it allows separating the two kinds of between-profiles variability, and to reduce any undesired inflation of the natural phase variability. In the mainstream literature, registration warping functions are not generally considered in the monitoring process, even though this may cause a significant information loss. We propose a novel approach for profile monitoring, which combines the Functional Principal Component Analysis and the use of parametric warping functions. The key idea is to
jointly monitor the stability over time of the registered profiles (i.e., the information related to amplitude variability) and the registration coefficients (i.e., the information related to phase
variability). This allows improving the capability of detecting unnatural pattern modifications, thanks to a better characterization of the overall natural variability. The benefits of a proper management of functional data registration, together with the advantages over the most common approaches used in the literature, are demonstrated by means of Monte Carlo simulations. The proposed methodology is finally applied to a real industrial case study relying on a dataset acquired in waterjet cutting processes under different health conditions of
the machine tool. |
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37/2015 - 07/10/2015
Aletti, M.; Perotto, S.; Veneziani, A.
Educated bases for the HiMod reduction of advection-diffusion-reaction problems with general boundary conditions | Abstract | | Hierarchical Model (HiMod) reduction is a method introduced in cite{perotto:2008} to effectively solve
advection-diffusion-reaction (ADR) and fluid dynamics problems in pipes. The rationale of the method is to regard
the solution as a mainstream axial dynamics added by transverse components. The mainstream
component is approximated by finite elements as often done in classical 1D models (like the popular Euler equations for gasdynamics). However, the HiMod formulation includes also the transverse dynamics by a spectral expansion.
A few modes are expected to capture the transverse (somehow secondary) dynamics with a good level of approximation. This
drastically reduces the size of the discrete problem, yet preserving accuracy. The method is ``hierarchical'' since the selection of the number of transverse modes can be hierarchically and adaptively performed~cite{perotto:2013}.
We have previously considered only Dirichlet boundary conditions for the lateral walls of the pipe and the procedure was tested only in 2D domains. With an appropriate selection of the spectral basis functions,
here we extend our formulation to 3D problems with general boundary conditions, still pursuing an essential approach.
The modal basis functions fulfill by construction the (homogeneous) boundary conditions associated with the solution.
This is achieved
by solving a Sturm-Liouville eigenpair problem.
We analyze this approach and provide a convergence analysis for the numerical error in the case of a linear ADR problem in rectangles (2D) and slabs (3D). Numerical results confirm the theory. |
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