arXiv subjects

Antoine Tambue

Publications and source records attributed to Antoine Tambue.

At least 37 records · Page 2Linked to original sources

arXiv2020

Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs

In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the scheme toward the mild solution of the PDE under polynomial growth condition of the nonlinearity. Our convergence rate are obtain for smooth and non-smooth initial data and is similar to that of the autonomous case. Our convergence result for smooth initial data is very important in numerical analysis. For instance, it is one step forward in approximating non-autonomous stochastic partial differential equations by the finite element method. In addition, we provide realistic conditions on the nonlinearity, appropriated to achieve optimal convergence rate without logarithmic reduction by exploiting the smooth properties of the two parameters evolution operator.

math.NA↗

arXiv2019

Strong Convergence of the Linear Implicit Euler Method for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise under more relaxed conditions. The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. This extends the current results in the literature to not necessary self-adjoint operator with more general boundary conditions. As a consequence key part of the proof does not rely on the spectral decomposition of the linear operator. We achieve optimal convergence orders which depend on the regularity of the noise and the initial data. In particular, for multiplicative noise we achieve optimal order $\mathcal{O}(h^2+Δt^{1/2})$ and for additive noise, we achieve optimal order $\mathcal{O}(h^2+Δt)$. In contrast to current work in the literature, where the optimal convergence orders are achieved for additive noise by incorporating further regularity assumptions on the nonlinear drift function, our optimal convergence orders are obtained under only the standard Lipschitz condition of the nonlinear drift term. Numerical experiments to sustain our theoretical results are provided.

math.NA↗

arXiv2019

Strong convergence of the backward Euler approximation for the finite element discretization of semilinear parabolic SPDEs with non-global Lipschitz drift driven by additive noise

This paper deals with the backward Euler method applied to semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. The SPDE is discretized in space by the finite element method and in time by the backward Euler. We consider a larger class of nonlinear drift functions, which are of Nemytskii type and polynomial of any odd degree with negative leading term, instead of only dealing with the special case of stochastic Allen-Chan equation as in the up to date literature.Moreover our linear operator is of second order and not necessarily self-adjoint, therefore makes estimates more challenging than in the case of self-adjoint operator. We prove the strong convergence of our fully discrete schemes toward the mild solution and results indicate how the convergence rates depend on the regularities of the initial data and the noise. In particular, for trace class noise, we achieve convergence order $\mathcal{O} (h^{2}+Δt^{1-ε})$, where $ε>0$ is positive number, small enough. We also provide numerical experiments to illustrate our theoretical results.

math.NA↗

arXiv2019

Convergence of the Two Point Flux Approximation and a novel fitted Two-Point Flux Approximation method for pricing options

In this paper, we deal with numerical approximations for solving the Black-Scholes Partial Differential Equation (PDE). This PDE is well known to be degenerated. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called fitted Two-Point Flux Approximation (TPFA).The fitted Two-Point Flux Approximation (TPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero in order to handle the degeneracy of the PDE and the TPFA method is used on the rest of space domain. This combination yields our novel fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support our theoretical results are provided.

math.NA↗

arXiv2019

A fitted L-Multi-point Flux Approximation method for pricing options

In this paper, we introduce a special kind of finite volume method called Multi-Point Flux Approximation method (MPFA) to price European and American options in two dimensional domain. We focus on the L-MPFA method for space discretization of the diffusion term of Black-Scholes operator. The degeneracy of the Black Scholes operator is tackled using the fitted finite volume method. This combination of fitted finite volume method and L-MPFA method coupled to upwind methods gives us a novel scheme called the fitted L-MPFA method. Numerical experiments show the accuracy of the novel fitted L-MPFA method comparing to well known schemes for pricing options.

math.NA↗

arXiv2019

Optimal strong convergence rates of some Euler-type timestepping schemes for the finite element discretization SPDEs driven by additive fractional Brownian motion and Poisson random measure

In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac 12$ and Poisson random measure, more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time, more precisely linear implicit method, exponential integrator and exponential Rosenbrock scheme are used for time discretisation. In contract to the current literature in the field for SPDE driven only by fBm, our linear operator is not necessary self-adjoint and optimal strong convergence rates have been achieved for SPDE driven only by fBm and SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains an optimal convergence rate of order $\mathcal{O}(h^2+Δt)$ for the exponential integrator and implicit schemes (linear operator $A$ self-adjoint for implicit). Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven with fBm noise.

math.NA↗

arXiv2019

A Fitted Multi-Point Flux Approximation Method for Pricing two options

In this paper, we develop novel numerical methods based on the Multi-Point Flux Approximation (MPFA) method to solve the degenerated partial differential equation (PDE) arising from pricing two-assets options. The standard MPFA is used as our first method and is coupled with a fitted finite volume in our second method to handle the degeneracy of the PDE and the corresponding scheme is called fitted MPFA method. The convection part is discretized using the upwinding methods (first and second order) that we have derived on non uniform grids. The time discretization is performed with $θ$- Euler methods. Numerical simulations show that our new schemes can be more accurate than the current fitted finite volume method proposed in the literature.

math.NA↗

arXiv2018

Magnus-type integrator for the finite element discretization of semilinear parabolic non-autonomous SPDEs driven by additive noise

In this paper, we investigate a numerical approximation of a general second order semilinear parabolic non-autonomous stochastic partial differential equation (SPDE) driven by additive noise. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature while the non-autonomous case is not yet well understood. We discretize the non-autonomous SPDE in space by the finite element method and in time by the Magnus-type integrator. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $L^2$ norm. Appropriate assumptions on the drift term and the noise allow to achieve optimal convergence order in time greater than $1/2$, without any logarithmic reduction of convergence order in time. In particular, for trace class noise, we achieve optimal convergence orders $\mathcal{O}\left(h^{2-ε}+Δt\right)$, where $ε$ is a positive number small enough. Numerical simulations are provided to illustrate our theoretical results.

math.NA↗

arXiv2018

Magnus-type Integrator for the Finite Element Discretization of Semilinear Parabolic non-Autonomous SPDEs Driven by multiplicative noise

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE driven by multiplicative noise by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $L^2$ norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order $\mathcal{O}\left(h^2\left(1+\max(0,\ln\left(t_m/h^2\right)\right)+Δt^{1/2}\right)$. Numerical simulations to illustrate our theoretical finding are provided.

math.NA↗

arXiv2018

Convergence analysis of the Magnus-Rosenbrock type method for the finite element discretization of semilinear non-autonomous parabolic PDEs with nonsmooth initial data

This paper aims to investigate a full numerical approximation of non-autonomous semilnear parabolic partial differential equations (PDEs) with nonsmooth initial data. Our main interest is on such PDEs where the nonlinear part is stronger than the linear part, also called reactive dominated transport equations. For such equations, many classical numerical methods lose their stability properties. We perform the space and time discretizations respectively by the finite element method and an exponential integrator. We obtain a novel explicit, stable and efficient scheme for such problems called Magnus-Rosenbrock method. We prove the convergence of the fully discrete scheme toward the exact solution. The result shows how the convergence orders in both space and time depend on the regularity of the initial data. In particular, when the initial data belongs to the domain of the family of the linear operator, we achieve convergence orders $\mathcal{O}\left(h^{2}+Δt^{2-ε}\right)$, for an arbitrarily small $ε>0$. Numerical simulations to illustrate our theoretical result are provided.

math.NA↗

arXiv2018

Strong convergence of the tamed and the semi-tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition

We consider the explicit numerical approximations of stochastic differential equations (SDEs) driven by Brownian process and Poisson jump. It is well known that under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution of such SDEs without jumps, while implicit Euler method converges but requires much computational efforts. We investigate the strong convergence, the linear and nonlinear exponential stabilities of tamed Euler and semi-tamed methods for stochastic differential equation driven by Brownian process and Poisson jumps, both in compensated and non compensated forms. We prove that under non-global Lipschitz condition and superlinearly growing drift term, these schemes converge strongly with the standard one-half order. Numerical simulations to substain the theoretical results are provided.

math.NA↗

arXiv2016

A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise

We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDE) driven by additive space-time noise. We introduce a new modified scheme using a linear functional of the noise with a semi--implicit Euler--Maruyama method in time and in space we analyse a finite element method (although extension to finite differences or finite volumes would be possible). We prove convergence in the root mean square $L^{2}$ norm for a diffusion reaction equation and diffusion advection reaction equation. We present numerical results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation. We see from both the analysis and numerics that the proposed scheme has better convergence properties than the standard semi--implicit Euler--Maruyama method.

math.NA↗

arXiv2016

Localized numerical impulses solutions in diffuse neural networks modeled by the complex fractional Ginzburg-Landau equation

We investigate localized wave solutions in a network of Hindmarsh-Rose neural model taking into account the long-range diffusive couplings. We show by a specific analytical technique that the model equations in the infrared limit (wave number $k\rightarrow 0$) can be governed by the complex fractional Ginzburg-Landau (CFGL) equation. According to the stiffness of the system, we propose both the semi and the linearly implicit Riesz fractional finite-difference schemes to solve efficiently the CFGL equation. The obtained fractional numerical solutions for the nerve impulse reveal localized short impulse properties. We also show the equivalence between the continuous CFGL and the discrete Hindmarsh-Rose models for relatively large network.

math.NA↗

arXiv2016

Localized modulated wave solutions in diffusive glucose-insulin systems

We investigate intercellular insulin dynamics in an array of diffusively coupled pancreatic islet \b{eta}-cells. The cells are connected via gap junction coupling, where nearest neighbor interactions are included. Through the multiple scale expansion in the semi-discrete approximation, we show that the insulin dynamics can be governed by the complex Ginzburg-Landau equation. The localized solutions of this equation are reported. The results suggest from the biophysical point of view that the insulin propagates in pancreatic islet \b{eta}-cells using both temporal and spatial dimensions in the form of localized modulated waves.

q-bio.TO↗

arXiv2015

Stability of the semi-tamed and tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition

Under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution, while Euler implicit method converges but requires much computational efforts. Tamed scheme was first introduced in [2] to overcome this failure of the standard explicit method. This technique is extended to SDEs driven by Poisson jump in [3] where several schemes were analyzed. In this work, we investigate their nonlinear stability under non-global Lipschitz and their linear stability. Numerical simulations to sustain the theoretical results are also provided.

math.NA↗

arXiv2015

An exponential Integrator for finite volume discretization of nonlinear parabolic differential equation

We consider the numerical approximation of a general second order semi--linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media which is fundamental in many geo-engineering applications, including oil and gas recovery from subsurface. Using the finite volume with two-point flux approximation on regular mesh combined with exponential time differencing of order one (ETD1) for temporal discretization, we derive the $L^{2}$ estimate under the assumption that the non linear term is locally Lipschitz. Numerical simulations to sustain the theoretical results are provided.

math.NA↗

arXiv2015

Weak convergence for a stochastic exponential integrator and finite element discretization of SPDE for multiplicative \& additive noise

We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler scheme when the linear operator is self adjoint and provide preliminaries results toward the full weak convergence rate for non-self-adjoint linear operator. Key part of the proof does not rely on Malliavin calculus. Depending of the regularity of the noise and the initial solution, we found that in some cases the rate of weak convergence is twice the rate of the strong convergence. Our convergence rate is in agreement with some numerical results in two dimensions.

math.NA↗

arXiv2013

A Stochastic Delay Model for Pricing Debt and Equity: Numerical Techniques and Applications

In the accompanied paper [14], a delayed nonlinear model for pricing corporate liabilities was developed. Using self-financed strategy and duplication we were able to derive two Random Partial Differential Equations (RPDEs) describing the evolution of debt and equity values of the corporate in the last delay period interval. In this paper, we provide numerical techniques to solve our delayed nonlinear model along with the corresponding RPDEs modeling the debt and equity values of the corporate. Using financial data from some firms, we compare numerical solutions from both our nonlinear model and classical Merton model [7] to the real corporate data. From this comparison, it comes up that in corporate finance the past dependence of the firm value process may be an important feature and therefore should not be ignored.

math.NA↗