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 1237 products
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01/2014 - 01/13/2014
Secchi, P.; Vantini, S.; Zanini, P.
Hierarchical Independent Component Analysis: a multi-resolution non-orthogonal data-driven basis | Abstract | | We introduce a new method, named HICA (Hierarchical Independent Component Analysis), suited to the dimensional reduction and the multi-resolution analysis of high dimensional and complex data. HICA solves a Blind Source
Separation problem by integrating Treelets with Independent Component Analysis and provides a multi-scale non-orthogonal data-driven basis apt
to meaningful data representations in reduced spaces. We describe some theoretical properties of HICA and we test the method on synthetic data. Finally, we apply HICA to the analysis of EEG traces. |
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67/2013 - 12/24/2013
Canuto, C.; Simoncini, V.; Verani, M.
On the decay of the inverse of matrices that are sum of Kronecker products | Abstract | | Decay patterns of
matrix inverses have recently attracted considerable interest,
due to their relevance in numerical analysis,
and in applications requiring matrix
function approximations.
In this paper we analyze the decay pattern of the inverse of banded matrices in the form
$S=M otimes I_n + I_n otimes M$ where $M$ is tridiagonal, symmetric and positive definite, $I_n$ is the identity matrix, and $ otimes$ stands for the Kronecker product.
It is well known that the inverses of banded matrices exhibit an exponential
decay pattern away from the main diagonal. However, the entries in $S^{-1}$
show a non-monotonic decay, which is not caught
by classical bounds. By using an alternative expression for $S^{-1}$, we
derive computable upper bounds that
closely capture the actual behavior of its entries. We also show that similar estimates
can be obtained when $M$ has a larger bandwidth, or when the sum of Kronecker
products involves two different matrices.
Numerical experiments illustrating the new bounds are also reported.
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66/2013 - 12/19/2013
Tricerri, P.; Dede ,L; Quarteroni, A.; Sequeira, A.
Numerical validation of isotropic and transversely isotropic constitutive models for healthy and unhealthy cerebral arterial tissues | Abstract | | This paper deals with the validation of constitutive models for healthy and unhealthy cerebral arterial tissues by means of numerical simulations of static inflation tests on a cylindrical geometry representing a specimen of anterior cerebral artery. The healthy arterial tissue is described by means of isotropic and transversely isotropic models. In particular, we validate a transversely isotropic multi-mechanism law, specifically proposed for the cerebral arterial tissue, for which the recruitment of the collagen fibers occurs at finite strains. Moreover, we consider numerical simulations of unhealthy cerebral arterial tissues by taking into account the mechanical weakening of the vessel wall that occurs during early development stages of cerebral aneurysms. We study the effects of the mechanical degradation on kinematic quantities of interest, namely the stresses distribution, that are commonly related to the progressive degradation of the arterial tissue by simulating static inflation tests for both isotropic and transversely isotropic models,
including the multi-mechanism law. |
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65/2013 - 12/17/2013
Ambrosi, D.; Ciarletta, P.
Plasticity in passive cell mechanics | Abstract | | A sufficiently large load applied to a living cell for a sufficiently long time produces a
deformation which is not entirely recoverable by passive mechanisms. This kind of plastic behavior is well documented by experiments but is still sel
dom investigated in terms of mechanical theories.
Here we discuss a finite visco-elasto-plastic
model where the rest elongation of the cell evolves in time as a function of the dissipated energy at a microstructural level. The theoretical predictions of the proposed model reproduce, also in quantitative terms, the passive mechanics of optically stretched cells. |
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64/2013 - 12/16/2013
Ciarletta, P; Ambrosi, D.; Maugin, G.A.
Mass transport in morphogenetic processes: a second gradient theory for volumetric growth and material remodeling | Abstract | | In this work, we derive a novel thermomechanical theory for growth and remodeling of biological materials in morphogenetic processes. This second gradient hyperelastic theory is the first attempt to describe both volumetric growth and mass transport phenomena in a single-phase continuum model, where both stress- and shape-dependent growth regulations can be investigated. The diffusion of biochemical species (e.g. morphogens, growth factors, migration signals) inside the material is driven by configurational forces, enforced in the balance equations and in the set of constitutive relations. Mass transport is found to depend both on first- and on second-order material connections, possibly withstanding a chemotactic behavior with respect to diffusing molecules. We find that the driving forces of mass diffusion can be written in terms of covariant material derivatives reflecting, in a purely geometrical manner, the presence of a (first-order) torsion and a (second-order) curvature.
Thermodynamical arguments show that the Eshelby stress and hyperstress tensors drive the rearrangement of the first- and second-order material inhomogeneities, respectively. In particular, an evolution law is proposed for the first-order transplant, extending a well-known
result for inelastic materials. Moreover, we define the first stress-driven evolution law of the second-order transplant in function of the completely material Eshelby hyperstress.
The theory is applied to two biomechanical examples, showing how an Eshelbian coupling
can coordinate volumetric growth, mass transport and internal stress state, both in physio-
logical and pathological conditions. Finally, possible applications of the proposed model are
discussed for studying the unknown regulation mechanisms in morphogenetic processes, as
well as for an optimizing scaffold architecture in regenerative medicine and tissue engineering. |
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63/2013 - 12/05/2013
Pigoli, D.; Menafoglio, A.; Secchi, P.
Kriging prediction for manifold-valued random field | Abstract | | The statistical analysis of data belonging to Riemannian manifolds is becoming increasingly important in many applications, such as shape analysis, diffusion tensor imaging and the analysis of covariance matrices. In many cases, data are spatially distributed but it is not trivial to take into account spatial dependence in the analysis because of the non linear geometry of the manifold. This work proposes a solution to the problem of spatial prediction for manifold valued data, with a particular focus on the case of positive definite symmetric matrices. Under the hypothesis that the dispersion of the observations on the manifold is not too large, data can be projected on a suitably chosen tangent space, where an additive model can be used to describe the relationship between response variable and covariates. Thus, we generalize classical kriging prediction, dealing with the spatial dependence in this tangent space, where well established Euclidean methods can be used. The proposed kriging prediction is applied to the matrix field of covariances between temperature and precipitation in Quebec, Canada. |
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62/2013 - 12/01/2013
Arioli, G.; Koch, H.
Existence and stability of traveling pulse solutions of the FitzHugh-Nagumo equation | Abstract | | The FitzHugh-Nagumo model is a reaction-diffusion equation describing the propagation of electrical signals in nerve axons and other biological tissues. One of the model parameters is the ratio epsilon of two time scales, which takes values between 0.001 and 0.1 in typical simulations of nerve axons. Based on the existence of a (singular) limit at epsilon = 0, it has been shown that the FitzHugh-Nagumo equation admits a stable traveling pulse solution for sufficiently small epsilon > 0. In this paper we prove the existence of such a solution for epsilon = 0.01. We consider both circular axons and axons of infinite length. Our method is non-perturbative and should apply to a wide range of other parameter values.
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61/2013 - 11/30/2013
Antonietti, P.F.; Sarti, M.; Verani, M.
Multigrid algorithms for hp-Discontinuous Galerkin discretizations of elliptic problems | Abstract | | We present W-cycle multigrid algorithms for the solution of the linear system of equations arising from a wide class of hp-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in multigrid analysis, we define a smoothing and an approximation property, which are used to prove the uniform convergence of the W-cycle scheme with respect to the granularity of the grid and the number of levels. The dependence of the convergence rate on the polynomial approximation degree p is also tracked, showing that the contraction factor of the scheme deteriorates with increasing p. A discussion on the effects of employing inherited or non-inherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results. |
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