Giuseppina Settanni

Morphological changes due to colorectal cancer propagation by an abnormal crypt fission is an interesting application arising in medicine and biology, that we try to analyse by a differential equations model coupled with a discrete crypt fission model. A colonic crypt can slowly change its shape in three steps: growth, bifurcation and fission. Fission is a rare event in a normal tissue, however if transit cells are unable to differentiate, due to an activation of Wnt signaling, then it may become fast and uncontrolled and cause a formation of aberrant crypt foci (ACF), defined as clusters of aberrant crypts. The differential equation system is composed by a convective diffusive equation for transit cell density and an elliptic equation for cell pressure, both defined on a manifold. The discrete crypt fission model acts in a set of adjacent crypts in order to investigate the ACF dynamics, due to a differentiation block. By using a Galerkin finite element method we solve numerically t...

We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order Finite Difference Schemes. We describe a code to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are proposed to emphasize the behaviour of the proposed algorithm.

Magnetic particle imaging (MPI) is a new medical imaging technique capable of recovering the distribution of superparamagnetic particles from their measured induced signals [1]. In literature there are two main MPI reconstruction techniques: measurement-based (MB) and x-space (XS). In the first approach the unknown magnetic particles concentration is reconstructed in the harmonic-space using a System Function (SF), describing the relation between particle positions and the signal response [2, 3]. The second approach requires the knowledge of the field free point (FFP) exact position and velocity at all time steps during the scanning process [4, 5]. The x-space method is based on the assumption of ideal magnetic field shapes used for spatial encoding (selection field), and for signal excitation (focus-drive field). The realization of human size devices with an open geometry requires specific calibration procedures related to the methods used in the reconstruction phase. One of the advantages of open bore scanners would be an easier open access to the patient, especially in interventional scenarios with simultaneous and real-time scanning processes. In this case of geometry configurations with larger FOV, the exact velocity gridding for x-space MPI could be difficult to achieve during the whole scanning process. Hence, our proposal is an innovative technique named hybrid x-space (HXS) resulting from the combination of the measurement-based and the classical x-space approach, reducing and optimizing the calibration time by a compressive sensing technique using circulant matrices.

In this talk we will examine some recent results obtained in the solution of Sturm-Liouville problems and singularly perturbed BVPs. In both cases the original problem is a second order ODE which is discretized approximating each derivative by means of different high order finite difference schemes. Several numerical tests will be proposed to show the effectiveness of the proposed approach.

We report on the progress achieved in the numerical simulation of self-adjoint multiparameter spectral problems for ordinary differential equations. We describe how to obtain a discrete problem by means of High Order Finite Difference Schemes and discuss its numerical solution. Based on this approach, we also define a recursive algorithm to compute approximations of the parameters by means of the solution of a set of problems converging to the original one.

The numerical solution of second order ordinary differential equations with initial conditions is here approached by approximating each derivative by means of a set of finite difference schemes of high order. The stability properties of the obtained methods are discussed. Some numerical tests, reported to emphasize pros and cons of the approach, motivate possible choices on the use of these formulae. c © 2010 European Society of Computational Methods in Sciences and Engineering

We investigate the numerical solution of regular and singular Sturm-Liouville problems by means of finite difference schemes of high order. In particular, a set of differ- ence schemes is used to approximate each derivative independently so to obtain an algebraic problem corresponding to the original continuous differential equation. The endpoints are treated depending on their classification and in case of limit points, no boundary condi- tion is required. Several numerical tests are finally reported on equispaced grids show the convergence properties of the proposed approach. c ⃝ 2011 European Society of Computational Methods in Sciences and Engineering

We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in concrete. Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy.

We propose a simple and quite efficient code to solve singular perturbation problems when the perturbation parameter ǫ is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses highly variable stepsize to fit the boundary regions with relatively few points. An extensive numerical test section shows the effectiveness of the proposed technique on linear problems. c 2009 European Society of Computational Methods in Sciences and Engineering

We focus on the solution of multiparameter spectral problems, and in particular on some strategies to compute coarse approximations of selected eigenparameters depending on the number of oscillations of the associated eigenfunctions. Since the computation of the eigenparameters is crucial in codes for multiparameter problems based on finite differences, we herein present two strategies. The first one is an iterative algorithm computing solutions as limit of a set of decoupled problems (much easier to solve). The second one solves problems depending on a parameter \(\sigma \in [0,1]\), that give back the original problem only when \(\sigma =1\). We compare the strategies by using well known test problems with two and three parameters.

Optimal control problems arise in many applications and need suitable numerical methods to obtain a solution. The indirect methods are an interesting class of methods based on the Pontryagin’s minimum principle that generates Hamiltonian Boundary Value Problems (BVPs). In this paper, we review some general-purpose codes for the solution of BVPs and we show their efficiency in solving some challenging optimal control problems.

In this work we propose a novel application of Partial Differential Equations (PDEs) inpainting techniques to two medical contexts. The first one concerning recovering of concentration maps for superparamagnetic nanoparticles, used as tracers in the framework of Magnetic Particle Imaging. The analysis is carried out by two set of simulations, with and without adding a source of noise, to show that the inpainted images preserve the main properties of the original ones. The second medical application is related to recovering data of corneal elevation maps in ophthalmology. A new procedure consisting in applying the PDEs inpainting techniques to the radial curvature image is proposed. The images of the anterior corneal surface are properly recovered to obtain an approximation error of the required precision. We compare inpainting methods based on second, third and fourth-order PDEs with standard approximation and interpolation techniques.

The solution of second order singular perturbation BVPs is one of the most challenging ODE problems [4]. Several codes for BVPs have been specialized in order to solve the most difficult problems (see, for example, COLMOD and ACDC written by Cash and Wright [5, 6]). In a recent paper [1] we proposed a simple and quite efficient code to solve linear singular perturbation problems when the perturbation parameter is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses ...

We propose a simple and quite efficient code to solve singular perturbation problems when the perturbation parameter epsilon is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses highly variable stepsize to fit the boundary regions with relatively few points. An extensive numerical test section shows the effectiveness of the proposed technique on linear problems.

We investigate the numerical solution of regular and singular Sturm-Liouville problems by means of finite difference schemes of high order. In particular, a set of difference schemes is used to approximate each derivative independently so to obtain an algebraic problem corresponding to the original continuous differential equation. The endpoints are treated depending on their classification and in case of limit points, no boundary condition is required. Several numerical tests are finally reported on equispaced grids show the convergence properties of the proposed approach. © 2011 European Society of Computational Methods in Sciences and Engineering.

ABSTRACT We discuss the solution of regular and singular Sturm–Liouville problems by means of High Order Finite Difference Schemes. We describe a method to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are considered to emphasize the behavior of a code based on the proposed algorithm. The methods solve any regular or singular Sturm–Liouville problem, providing high accuracy and computational efficiency thanks to the powerful strategy of stepsize variation.

In this note we show how a simple stepsize variation strategy improves the solution algorithm of regular Sturm‐Liouville problems. We suppose the eigenvalue problem is approximated by variable stepsize finite difference schemes and the obtained algebraic eigenvalue problem is solved by a matrix method estimating the first eigenvalues and eigenvectors of sparse matrices. The variable stepsize strategy is based on an equidistribution of the error (approximated by two methods with different orders). The ...

ABSTRACT We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in saturated porous media such as concrete. Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy.

The present work shoots for the presentation of a code able to solve secondorder ordinary differential equations that arise from different applications in physics, chemistry and engineering. Apart the restriction of the second order, we consider initial and boundary value problems, and also eigenvalue problems of differential equations, in particular we examine Sturm-Liouville problems. Chapter 1 begins with a theoretical description of second order differential equations and eigenvalue problems. Theorems of existence and ...

Abstract: We propose a simple and quite efficient code to solve singular perturbation problems when the perturbation parameter e is very small. The code is based on generalized upwind methods of order ranging from 4 to 10 and uses highly variable stepsize to fit the boundary regions with relatively few points. An extensive numerical test section shows the effectiveness of the proposed technique on linear problems. c© 2009 European Society of Computational Methods in Sciences and Engineering Keywords: Two-point ...