The generation of structured grids on bounded domains is a crucial issue in the development of nu... more The generation of structured grids on bounded domains is a crucial issue in the development of numerical models for solving differential problems. In particular, the representation of the given computational domain through a regular parameterization allows us to define a univalent mapping, which can be computed as the solution of an elliptic problem, equipped with suitable Dirichlet boundary conditions. In recent years, Physics-Informed Neural Networks (PINNs) have been proved to be a powerful tool to compute the solution of Partial Differential Equations (PDEs) replacing standard numerical models, based on Finite Element Methods and Finite Differences, with deep neural networks; PINNs can be used for predicting the values on simulation grids of different resolutions without the need to be retrained. In this work, we exploit the PINN model in order to solve the PDE associated to the differential problem of the parameterization on both convex and non-convex planar domains, for which ...
In this note we consider the use of Euler–Maclaurin methods for the solution of canonical Hamilto... more In this note we consider the use of Euler–Maclaurin methods for the solution of canonical Hamiltonian problems. As a subclass of multi-derivative Runge–Kutta methods, these integrators cannot be symplectic, however they turn out to be conjugate symplectic. The numerical solutions provided by a conjugate symplectic integrator essentially share the same qualitative long time behavior as those yielded by a symplectic integrator. This aspect, along with an efficient evaluation of the derivatives, suggests that Euler–Maclaurin methods could play an interesting role in the context of geometric integration.
Monthly Notices of the Royal Astronomical Society, 2019
The energy transfer among the components in a gas determines its fate. Especially at low temperat... more The energy transfer among the components in a gas determines its fate. Especially at low temperatures, inelastic collisions drive the cooling and the heating mechanisms. In the early Universe as well as in zero- or low-metallicity environments the major contribution comes from the collisions among atomic and molecular hydrogen, also in its deuterated version. This work shows some updated calculations of the H2 cooling function based on novel collisional data which explicitly take into account the reactive pathway at low temperatures. Deviations from previous calculations are discussed and a multivariate data analysis is performed to provide a fit depending on both the gas temperature and the density of the gas.
This paper presents new hybrid mesh selection strategies for boundary value problems implemented ... more This paper presents new hybrid mesh selection strategies for boundary value problems implemented in the code TOM. Originally the code was proposed for the numerical solution of stiff or singularly perturbed problems. The code has been now improved with the introduction of three classes of mesh selection strategies, that can be used for different categories of problems. Numerical experiments show that the mesh selection and, in the nonlinear case, the strategy for solving the nonlinear equations are determinant for the good behaviour of a general purpose code. The possibility to choose the mesh selection should be considered for all general purposes codes to make them suitable for wider classes of problems.
The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov sche... more The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.
2015 5th International Workshop on Magnetic Particle Imaging (IWMPI), 2015
Magnetic particle imaging (MPI) is a new medical imaging technique capable of recovering the dist... more 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.
Journal of Computational and Applied Mathematics, 2005
We present a hybrid mesh selection strategy for use in codes for the numerical solution of two-po... more We present a hybrid mesh selection strategy for use in codes for the numerical solution of two-point boundary value problems. This new mesh strategy is based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on a standard estimate of the local discretisation error. We have implemented this algorithm in the well
We devise a variable precision floating-point arithmetic by exploiting the framework provided by ... more We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix $${\textcircled {1}}$$ 1 . The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.
Optimal control problems arise in many applications and need suitable numerical methods to obtain... more 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 article, we present a new strategy to determine an unmanned aerial vehicle trajectory tha... more In this article, we present a new strategy to determine an unmanned aerial vehicle trajectory that minimizes its flight time in presence of avoidance areas and obstacles. The method combines classical results from optimal control theory, i.e. the Euler-Lagrange Theorem and the Pontryagin Minimum Principle, with a continuation technique that dynamically adapts the solution curve to the presence of obstacles. We initially consider the two-dimensional path planning problem and then move to the three-dimensional one, and include numerical illustrations for both cases to show the efficiency of our approach.
The generation of structured grids on bounded domains is a crucial issue in the development of nu... more The generation of structured grids on bounded domains is a crucial issue in the development of numerical models for solving differential problems. In particular, the representation of the given computational domain through a regular parameterization allows us to define a univalent mapping, which can be computed as the solution of an elliptic problem, equipped with suitable Dirichlet boundary conditions. In recent years, Physics-Informed Neural Networks (PINNs) have been proved to be a powerful tool to compute the solution of Partial Differential Equations (PDEs) replacing standard numerical models, based on Finite Element Methods and Finite Differences, with deep neural networks; PINNs can be used for predicting the values on simulation grids of different resolutions without the need to be retrained. In this work, we exploit the PINN model in order to solve the PDE associated to the differential problem of the parameterization on both convex and non-convex planar domains, for which ...
In this note we consider the use of Euler–Maclaurin methods for the solution of canonical Hamilto... more In this note we consider the use of Euler–Maclaurin methods for the solution of canonical Hamiltonian problems. As a subclass of multi-derivative Runge–Kutta methods, these integrators cannot be symplectic, however they turn out to be conjugate symplectic. The numerical solutions provided by a conjugate symplectic integrator essentially share the same qualitative long time behavior as those yielded by a symplectic integrator. This aspect, along with an efficient evaluation of the derivatives, suggests that Euler–Maclaurin methods could play an interesting role in the context of geometric integration.
Monthly Notices of the Royal Astronomical Society, 2019
The energy transfer among the components in a gas determines its fate. Especially at low temperat... more The energy transfer among the components in a gas determines its fate. Especially at low temperatures, inelastic collisions drive the cooling and the heating mechanisms. In the early Universe as well as in zero- or low-metallicity environments the major contribution comes from the collisions among atomic and molecular hydrogen, also in its deuterated version. This work shows some updated calculations of the H2 cooling function based on novel collisional data which explicitly take into account the reactive pathway at low temperatures. Deviations from previous calculations are discussed and a multivariate data analysis is performed to provide a fit depending on both the gas temperature and the density of the gas.
This paper presents new hybrid mesh selection strategies for boundary value problems implemented ... more This paper presents new hybrid mesh selection strategies for boundary value problems implemented in the code TOM. Originally the code was proposed for the numerical solution of stiff or singularly perturbed problems. The code has been now improved with the introduction of three classes of mesh selection strategies, that can be used for different categories of problems. Numerical experiments show that the mesh selection and, in the nonlinear case, the strategy for solving the nonlinear equations are determinant for the good behaviour of a general purpose code. The possibility to choose the mesh selection should be considered for all general purposes codes to make them suitable for wider classes of problems.
The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov sche... more The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.
2015 5th International Workshop on Magnetic Particle Imaging (IWMPI), 2015
Magnetic particle imaging (MPI) is a new medical imaging technique capable of recovering the dist... more 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.
Journal of Computational and Applied Mathematics, 2005
We present a hybrid mesh selection strategy for use in codes for the numerical solution of two-po... more We present a hybrid mesh selection strategy for use in codes for the numerical solution of two-point boundary value problems. This new mesh strategy is based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on a standard estimate of the local discretisation error. We have implemented this algorithm in the well
We devise a variable precision floating-point arithmetic by exploiting the framework provided by ... more We devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix $${\textcircled {1}}$$ 1 . The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.
Optimal control problems arise in many applications and need suitable numerical methods to obtain... more 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 article, we present a new strategy to determine an unmanned aerial vehicle trajectory tha... more In this article, we present a new strategy to determine an unmanned aerial vehicle trajectory that minimizes its flight time in presence of avoidance areas and obstacles. The method combines classical results from optimal control theory, i.e. the Euler-Lagrange Theorem and the Pontryagin Minimum Principle, with a continuation technique that dynamically adapts the solution curve to the presence of obstacles. We initially consider the two-dimensional path planning problem and then move to the three-dimensional one, and include numerical illustrations for both cases to show the efficiency of our approach.
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