15 research outputs found
Fast Collocation methods for Volterra Integral equations of convolution type
No full textIn this paper we present fast discrete collocation methods forVolterra integral equations of Hammerstein type, where the Laplace
transform of the kernel is known a priori. To compute the numerical solution over N time steps, the constructed methods require
O(N log(N )) operations, O(log(N )) memory and preserve the order of accuracy of the corresponding exact collocation methods.
The numerical experiments confirm the expected accuracy and the computational cost
Fast Runge-Kutta methods for nonlinear convolution systems of Volterra Integral equations
No full textIn this paper fast implicit and explicit Runge–Kutta methods for systems of Volterra
integral equations of Hammerstein type are constructed. The coefficients of the methods
are expressed in terms of the values of the Laplace transform of the kernel. These
methods have been suitably constructed in order to be implemented in an efficient way,
thus leading to a very low computational cost both in time and in space. The order of
convergence of the constructed methods is studied. The numerical experiments confirm
the expected accuracy and computational cost
Efficient numerical methods for Volterra integral equations of Hammerstein type
No full textVolterra integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, engineering. It is known that the numerical treatment of VIEs has an high computational cost, due mainly to the computation of the ``lag term'' or ``tail term'' which contains the past history of the phenomenon. Since it depends on time, the ``lag term'' has to be computed for each time step and its cost increases when time passes. Among the Volterra equations, the Hammerstein type ones, are particularly interesting for the applications.
The aim of this thesis is the construction of numerical methods for VIEs of Hammerstein type which produce accurate solution at a low computational cost and ``catch'' the qualitative behaviour of the exact solution.
The study developed has been concerned at first with the construction and analysis of efficient methods for the numerical treatment of VIEs of Hammerstein type where the Laplace transform of the kernel rather than the convolution kernel itself is a priori known. This is not an anomalous or restricting situation, as a matter of fact these kind of problems arise in chemical absorption kinetics in the determination of non reflecting boundary conditions, and in general in situations when Laplace transform tecnique are used to reduce systems of ordinary or partial differential equations in VIEs.
It is known that a classical numerical method for computing the numerical solution of such equations over Nt time steps requires O(N2t) operations and O(Nt) memory space.
In this thesis we construct two classes of fast numerical methods based on collocation and Runge-Kutta formulas respectively. These methods have a computational cost of O(NtlogNt) operations, O(logNt) memory requirement and they have an high order of accuracy. In both cases the knowledge of the Laplace transform of the kernel and the convolution nature of the kernel itself are exploited in order to obtain a fast computation of the lag term. This is possible by using an opportune inverse Laplace transform approximation formula for computing the kernel evaluations.
The fast numerical methods constructed in this thesis tend to the corresponding classical methods when the inverse Laplace transfrom approximation formula is exact. The convergence analysis of the fast collocation and Runge-Kutta methods shows that their order of convergence coincides with the order of the corresponding classical methods.
We also analyse the stability properties of the fast Runge-Kutta methods with respect to test equations.
We prove that the stability regions depend on the approxiamation of the inverse Laplace transform and that the stability properties of the classical Runge-Kutta methods are obtained when the error of the inverse Laplace transform approximation formula tends to zero.
The numerical experiments on some significant problems taken from the ``Test Set'' collection project confirm the expected accuracy, computational cost and the stability properties of the constructed methods.
The second part of the thesis concerns with the numerical treatment of problems of SIS epidemic diffusion with periodic immigration flow. The mathematical model of such problems is represented by an Hammerstein type VIE with convolution kernel.
We consider problems caracterized by the relapse of the epidemic which implies that the VIE has an asymptotically periodic solution.
It is clear that an efficient numerical method has to reproduce the asymptotically periodic solution whenever applied to equations that show this behaviour.
For this reason we analyse the discrete Volterra equation (DVE) corresponding to such VIEs and we prove a theorem which establishes the existence and the uniqueness of the asymptotically periodic solution of the DVE.
Moreover we consider SIS epidemic models with periodic immigration flow and constant contact rate. Also in this case we prove, for the DVE corresponding to the problem, the existence and the uniqueness of the asymptotically periodic solution when the DVE satisfies some significant hypothesis depending only on its kernel and forcing term.
In order to analyse if the existing numerical methods satisfy these conditions, that is if they are AP-stable, we consider the class of θ-methods and we prove that they are AP-stable if the integration step satisfies an inequality depending only on some parameters that are characteristic of the problem
High performance parallel numerical methods for Volterra equations with weakly singular kernels
No full textNon-stationary discrete time waveform relaxation methods for Abel systems of Volterra integral equations using fractional linear multistep formulae are introduced. Fully parallel discrete waveform relaxation methods having an optimal convergence rate are constructed. A significant expression of the error is proved, which allows us to estimate the number of iterations needed to satisfy a prescribed tolerance and allows us to identify the problems where the optimal methods offer the best performance. The numerical experiments confirm the theoretical expectations
Asymptotic periodicity of nonlinear discrete Volterra equations and applications
No full textSufficient conditions for the asymptotic periodicity of solutions of nonlinear discrete Volterra equations of Hammerstein type are obtained. Such results are applied to analyze the property of a class of numerical methods to preserve the asymptotic periodicity of the analytical solution of Volterra integral equations
EFFICIENT NUMERICAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS OF HANMERSTEIN TYPE
No full textVolterra integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, engineering. It is known that the numerical treatment of VIEs has an high computational cost, due mainly to the computation of the ``lag term'' or ``tail term'' which contains the past history of the phenomenon. Since it depends on time, the ``lag term'' has to be computed for each time step and its cost increases when time passes. Among the Volterra equations, the Hammerstein type ones, are particularly interesting for the applications. The aim of this thesis is the construction of numerical methods for VIEs of Hammerstein type which produce accurate solution at a low computational cost and ``catch'' the qualitative behaviour of the exact solution. The study developed has been concerned at first with the construction and analysis of efficient methods for the numerical treatment of VIEs of Hammerstein type where the Laplace transform of the kernel rather than the convolution kernel itself is a priori known. This is not an anomalous or restricting situation, as a matter of fact these kind of problems arise in chemical absorption kinetics in the determination of non reflecting boundary conditions, and in general in situations when Laplace transform tecnique are used to reduce systems of ordinary or partial differential equations in VIEs. It is known that a classical numerical method for computing the numerical solution of such equations over Nt time steps requires O(N^2t) operations and O(Nt) memory space. In this thesis we construct two classes of fast numerical methods based on collocation and Runge-Kutta formulas respectively. These methods have a computational cost of O(NtlogNt) operations, O(logNt) memory requirement and they have an high order of accuracy. In both cases the knowledge of the Laplace transform of the kernel and the convolution nature of the kernel itself are exploited in order to obtain a fast computation of the lag term. This is possible by using an opportune inverse Laplace transform approximation formula for computing the kernel evaluations. The fast numerical methods constructed in this thesis tend to the corresponding classical methods when the inverse Laplace transfrom approximation formula is exact. The convergence analysis of the fast collocation and Runge-Kutta methods shows that their order of convergence coincides with the order of the corresponding classical methods. We also analyse the stability properties of the fast Runge-Kutta methods with respect to test equations. We prove that the stability regions depend on the approxiamation of the inverse Laplace transform and that the stability properties of the classical Runge-Kutta methods are obtained when the error of the inverse Laplace transform approximation formula tends to zero. The numerical experiments on some significant problems taken from the ``Test Set'' collection project confirm the expected accuracy, computational cost and the stability properties of the constructed methods. The second part of the thesis concerns with the numerical treatment of problems of SIS epidemic diffusion with periodic immigration flow. The mathematical model of such problems is represented by an Hammerstein type VIE with convolution kernel. We consider problems caracterized by the relapse of the epidemic which implies that the VIE has an asymptotically periodic solution. It is clear that an efficient numerical method has to reproduce the asymptotically periodic solution whenever applied to equations that show this behaviour. For this reason we analyse the discrete Volterra equation (DVE) corresponding to such VIEs and we prove a theorem which establishes the existence and the uniqueness of the asymptotically periodic solution of the DVE. Moreover we consider SIS epidemic models with periodic immigration flow and constant contact rate. Also in this case we prove, for the DVE corresponding to the problem, the existence and the uniqueness of the asymptotically periodic solution when the DVE satisfies some significant hypothesis depending only on its kernel and forcing term. In order to analyse if the existing numerical methods satisfy these conditions, that is if they are AP-stable, we consider the class of θ-methods and we prove that they are AP-stable if the integration step satisfies an inequality depending only on some parameters that are characteristic of the problem
MicroRNA-423-5p promotes autophagy in cancer cells and is increased in serum from hepatocarcinoma patients treated with sorafenib
No full textHepatocellular carcinoma (HCC) is the third cause of cancer-related deaths worldwide. Sorafenib is the only approved drug for patients with advanced HCC but has shown limited activity. microRNAs (miRs) have been involved in several neoplasms including HCC suggesting their use or targeting as good tools for HCC treatment. The purpose of this study was to identify novel approaches to sensitize HCC cells to sorafenib through miRs. miR-423-5p was validated as positive regulator of autophagy in HCC cell lines by transient transfection of miR and anti-miR molecules. miR-423-5p expression level was evaluated by real-time polymerase chain reaction (PCR) in sera collected from 39 HCC patients before and after treatment with sorafenib. HCC cells were cotreated with sorafenib and miR-423-5p and the effects on cell cycle, apoptosis, and autophagy were evaluated. Secretory miR-423-5p was upregulated both in vitro and in vivo by sorafenib treatment and its increase was correlated with response to therapy since 75% of patients in which an increase of secretory miR423-5p was found were in partial remission or stable disease after 6 moths from the beginning of therapy. HCC cells transfected with miR-423-5p showed an increase of cell percentage in S-phase of cell cycle paralleled by a similar increase of autophagic cells evaluated at both fluorescence activated cell sorter (FACS) and transmission electron microscopy. Our results suggest the miR423-5p can be used as a useful tool to predict response to sorafenib in HCC patients and is involved in autophagy regulation in HCC cells
MOLECULAR EVALUATION OF ZNF224 MRNA EXPRESSION IN CML PATIENTS AS A NOVEL DETERMINANT OF TKI RESPONSIVENESS
No full textThe transcription factor Wilms’ tumor gene 1, WT1, is implicated both in normal developmental processes and in the generation of a variety of solid tumors and hematological malignancies. WT1 is highly expressed in leukemia cells and its overexpression is associated with a poor response to therapy. Recently the Krüppel-like zinc-finger protein, ZNF224 was identified as a novel WT1-interacting factor involved in WT1 transcriptional regulation. ZNF224 itself could be modulated by cytosine arabinoside (ara-C), a drug widely used in the treatment of myeloid leukemia and that ZNF224 overexpression increases susceptibility to apoptosis of Ph+ K562 cell lines. In our retrospective analysis we evaluated the relative expression of ZNF224 mRNA in 30 adult patients with BCR-ABL–positive chronic phase chronic myeloid leukaemia (CP-CML) as a determinant of imatinib sensitivity.
Methods: Response to tyrosine kinase inhibitor (TKI) imatinib is assessed with standardized real quantitative polymerase chain reaction and/or cytogenetics at 3, 6, and 12 months. Response to the therapy was classified as optimal, warning, and failure, according to the recent ELN criteria. We compared the ZNF224 expression at diagnosis with molecular response over the first 12 month of imatinib therapy. Sample have been selected, for retrospective analysis, for them interim molecular results a 12 month, showing 15 patients in optimal response (OR), 10 patients in a warning response (WR) and 5 patients in failure response (FR). 5 healthy donors (HDs) were included to the study. All patients signed informed consent in accordance with the Declaration of Helsinki. RT-qPCR results were normalized by the expression of ABL mRNA (Normalized mRNA copy Number: NCN).Results:ZNF224 mRNA were significantly up-regulated in PB samples at diagnosis of patients with OR compared to patients with WR/FR, (1.13±0.76 vs 0.62±0.25 NCN,respectively; p=0.05). Interesting the ZNF224 mRNA expression in HDs was significantly higher (2.11±0.98 NCN vs OR patients, p=0.05 and WR/FR patients; p=0.0005). The treatment for 12 month with imatinib increase the ZNF224 expression in both CML categories (2.91±1.72 NCN in OR and1.77±1.52 NCN in WR/FR; p=0.05).
Conclusions:We observed that the OR patients express a significantly higher number of copies of the ZNF224 transcript than WR/FR. Furthermore, in both
groups of patients at diagnosis, ZNF224 protein levels are lower than those after therapy with TKI at 12 month
