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In this paper we propose and compare two methods to optimize the numerical computations for the diffusion term in a nonlocal formulation for a reaction-diffusion equation. The diffusion term is particularly computationally intensive due to the integral formulation, and thus finding a better way of computing its numerical approximation could be of i...
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... results are presented in Tables 1-4. In the first table for relatively small images (128 × 128 pixels) the parallel GPU implementation results are comparable to parallel CPU ones, while the sequential implementation results are already a few orders of magnitude worse than parallel ...Citations
... For the mathematical analysis of nonlocal models we refer to the book [1] and references therein. The behaviour of solutions to the nonlocal model (1)-(2) when one rescales the kernel J considering J(z) = 1 ε 3 J( z ε ) are studied in M. Bogoya and J. Gómez [4] while for the numerical solutions we refer to P. W. Bates, S. Brown and J. Han [3] and M. Craus and S.D. Paval [7]. ...
... Du [14]. Computations with several different higher-order time-stepping schemes are used in M. Craus and S.D. Paval [7], C. Moroşanu and A.-M. Moşneagu [11], while for the existence, estimate, uniqueness and regularity of a solution in Sobolev spaces W 1,2 p (Q) we refer to A. Miranville and C. Moroşanu [10]. ...
... 3. Numerical approximation. In this section we propose an explicit numerical scheme to approximate the unique solution of the nonlocal and nonlinear reactiondiffusion problem (1)-(2), based on the finite difference method (see also [3], [5], [7], [11], [12] and [14]). The problem is discretized on the two-dimensional domain Ω = [0, 1] × [0, 1] by using a uniform spatial step size h. ...
The main goal of this paper is to introduce and analyze a new nonlocal reaction-diffusion model with in-homogeneous Neumann boundary conditions. We prove the existence and uniqueness of a solution in the class \begin{document}$ C((0, T], L^\infty(\Omega)) $\end{document} and the dependence on the data. Proofs are based on the Banach fixed-point theorem. Our results extend the results already proven by other authors. A numerical approximating scheme and a series of numerical experiments are also presented in order to illustrate the effectiveness of the theoretical result. The overall scheme is explicit in time and does not need iterative steps; therefore it is fast.
... The study of the solutions of the non-v by the term (E 1 * v)(t, x) = Ω E 1 (x − y)v(t, y)dy. The second integral of (1) involves the given flux of individuals that enter or leave the set Ω by the sign of f (t, x), (t, x) ∈ (0, T ] × ∂Ω (the in-homogeneous Neumann boundary conditions) and the nonlinear term v(t, x) − v 3 (t, x) is the reaction term (see [5], [6], [7], [8], [10]- [13], [17]- [18], [21], [24] and [25]). Thus, the density v(t, x) verifies the equation (1) without any internal or external sources. ...
In this paper we study the existence and uniqueness of the solution in \begin{document}$ C((0, T], L^\infty(\Omega)) $\end{document} of a new nonlocal and nonlinear second-order anisotropic reaction-diffusion problem with in-homogeneous Neumann boundary conditions, generalizing other problems in the literature. Then, by using the finite difference method, we propose an explicit in time numerical scheme to approximate the unique solution of our problem. We also present some numerical simulations that come to show the performance of our theoretical model.
... One proves the existence and the uniqueness of solutions (Theorem 2.2 below) for the new mathematical problem in question, by considering the cubic nonlinearity S(t, x) − S 3 (t, x), as well as nonlinear diffusion coefficients. In order to approximate the unique solution of the nonlinear reaction-diffusion problems (1)-(5), an implicit-explicit (IMEX) numerical scheme is constructed (see [3], [5], [13], [14], [25], [35]) and, to illustrate the effectiveness of theory and applications equally, some numerical experiments in two dimensions are presented too. Our simulations highlight the new modeling capabilities introduced in this work via the nonlinear second-order anisotropic reaction-diffusion model (1)- (5). ...
... The nonlinear operator H in (12) depends on λ ∈ [0, 1] and its fixed point for λ = 1 are solutions of(13). ...
The paper is concerned with two main topics, as follows. In the first instance, a serious qualitative analysis is performed for a second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as in-homogeneous Neumann boundary conditions. The PDEs system is implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. Under certain hypothesis on the input data: \begin{document}$ S_0(x) $\end{document}, \begin{document}$ E_0(x) $\end{document}, \begin{document}$ I_0(x) $\end{document} \begin{document}$ R_0(x) $\end{document}, \begin{document}$ D_0(x) $\end{document}, \begin{document}$ f(t,x) $\end{document} and \begin{document}$ w_{_i}(t,x), i = 1,2,3,4,5 $\end{document}, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space \begin{document}$ W^{1,2}_p(Q) $\end{document}, extending the types already proven by other authors. The nonlinear second-order anisotropic reaction-diffusion model considered here is then particularized to monitor the spread of an epidemic infection.
The second topic refers to the construction of the IMEX numerical approximation schemes in order to compute the solution of the system of coupled PDEs. We also show several simulation examples highlighting the present model's capabilities.
... Particularly, the study of the solutions to the nonlocal problem (1) (or (2)-(4)), using the kernel Φ S1 (z) = 1 ε 3 Φ S1 z ε , can be found in [1], [2], [7], [13], [17], [33], [34], [37] and the numerical approximations in [4], [5], [11], [12], [13], [32]- [34] and [38]. The problem (1) (or (2)-(4)) is a nonlocal one due to the diffusion of the density S(t, x) that depends on all values of S through the convolution-like term (Φ S1 * S)(t, x) = Ω Φ S1 (x−y)S(t, y)dy (see [3] or [8]). ...
... One proves the existence and the uniqueness of solutions (Theorem 2.2 below) for the new mathematical problem in question, by considering the cubic nonlinearity S(t, x)−S 3 (t, x), as well as nonlinear diffusion coefficients. In order to approximate the unique solution of the nonlinear reaction-diffusion problems (1)-(5), an implicit-explicit (IMEX) numerical scheme is constructed (see [11], [12], [13], [17], [32]- [34]), [36], [40] and, to illustrate the effectiveness of theory and applications equally, some numerical experiments in two dimensions are presented too. Our simulations highlight the new modeling capabilities introduced in this work via the nonlocal and nonlinear second-order anisotropic reaction-diffusion model (1)- (5). ...
In our current paper we are following the results obtained by Pavăl et al. in [36] and study a nonlocal form of the system they propose. First we are performing a qualitative analysis for the equivalent non-local second-order system of coupled PDEs, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction. As in [36] our PDEs system is also implementing a SEIRD (Susceptible, Exposed, Infected, Recovered, Deceased) epidemic model. In order to be able to compare with the before mentioned results we use the same hypothesis on the input data: \begin{document}$ S_0(x) $\end{document}, \begin{document}$ E_0(x) $\end{document}, \begin{document}$ I_0(x) $\end{document}, \begin{document}$ R_0(x) $\end{document}, \begin{document}$ D_0(x) $\end{document}, \begin{document}$ f(t,x) $\end{document} and \begin{document}$ w_{_i}(t,x), i = 1,2,3,4,5 $\end{document}, and we prove the well-posedness of a classical solution in \begin{document}$ C((0,T],C(\Omega)) $\end{document}, extending the types already proven by other authors.
Secondly we construct the implicit-explicit (IMEX) numerical approximation scheme which allows to compute the solution of the system of coupled PDEs. The results are then compared with the ones obtained by [36].
... 1 Set m = 1 2 Initialize the unknown function v 1 with the input image to be segmented 3 while v m did not reach stable state do 4 Compute diffusion and reaction terms according to (41), (42) and respectively (46) 5 Evolve function v m in (47) to obtain v m+1 i,j 6 Increase m by 1 7 Segmented image is given by v m Figure 1 shows the segmentation results of our model for a brain CT scan image. The results are satisfactory even after only one iteration. ...
... Regarding time complexity, due to the integral formulation of NlD term in (41) and (42), the proposed algorithm is slower than the compared K-means or Chan-Vese counterparts. To obtain better performance results, regarding running time, we had to implement the program on parallel architectures such as CUDA [42]. ...
... Regarding time complexity, due to the integral formulation of NlD term in (41) and (42), the proposed algorithm is slower than the compared K-means or Chan-Vese counterparts. To obtain better performance results, regarding running time, we had to implement the program on parallel architectures such as CUDA [42]. Table 1 shows the time taken by a CUDA implementation for different input image sizes (total number of pixels being I * J). ...
In this paper we are addressing two main topics, as follows. First, a rigorous qualitative study is elaborated for a second-order parabolic problem, equipped with nonlinear anisotropic diffusion and cubic nonlinear reaction, as well as non-homogeneous Cauchy-Neumann boundary conditions. Under certain assumptions on the input data: f(t,x), w(t,x) and v0(x), we prove the well-posedness (the existence, a priori estimates, regularity, uniqueness) of a solution in the Sobolev space Wp1,2(Q), facilitating for the present model to be a more complete description of certain classes of physical phenomena. The second topic refers to the construction of two numerical schemes in order to approximate the solution of a particular mathematical model (local and nonlocal case). To illustrate the effectiveness of the new mathematical model, we present some numerical experiments by applying the model to image segmentation tasks.
