arXiv subjects

Antoine Tambue

Publications and source records attributed to Antoine Tambue.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Stochastic Krylov based Techniques for Large- Scale Log-Determinant Estimation

Estimating the logarithm of the determinant of large sparse positive definite symmetric matrices is an important task in numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. In this work, we introduce two scalable and efficient methods for large-scale log-determinant termed the Optimal Stochastic Arnoldi with Incomplete Orthogonalization Procedure (OSA-IOP) and the Optimal Stochastic Lanczos Quadrature (OSLQ). The OSA-IOP approach extends the Incomplete Orthogonalization Procedure (IOP), originally developed for matrix exponential functions for exponential time stepping integrators, to compute the action of the matrix algorithm on a vector. We observe that combining IOP with a randomized Hutch++ algorithm, the OSA-IOP significantly reduces computational cost while maintaining high accuracy. The OSLQ method estimates log-determinants by coupling Lanczos quadrature with Hutch++ and controlled orthogonalization, leveraging Krylov subspaces as efficient quadrature mechanisms to approximate quadratic forms involving the matrix logarithm. We derive error bounds for both methods. Extensive numerical experiments on large-scale sparse matrices from real-world applications demonstrate the accuracy, robustness, and scalability of the proposed approaches.

math.NA↗

arXiv2026

Pathwise convergence of a linearization scheme for stochastic differential-algebraic equations under the local Lipschitz coefficients

The paper deals with the numerical treatment of index-1 stochastic differential-algebraic equations (SDAEs) with nonlinear coefficients that satisfy the local Lipschitz and the Khasminskii conditions. The key challenge here is the presence of a singular and non-autonomous matrix in the equation, which makes the numerical method challenging to analyze. To tackle this challenge, we develop a more general numerical method using a local linearization technique. More precisely, we use the Taylor expansion to decompose locally the drift component of the SDAEs in linear and nonlinear parts. The linear part is approximated implicitly and must resolve the singularity issue of each time step, while the nonlinear part is approximated explicitly. This method is fascinating due to the fact that it is efficient in high dimension. We prove that this novel numerical method converges in the pathwise sense with rate $\frac{1}{2}-ε$, for arbitrary $ε>0$. The implementation of this novel numerical method is also carried out to verify our theoretical result.

math.NA↗

arXiv2026

Novel technique based on Léja Points Approximation for Log-determinant Estimation of Large matrices

The computation of the Log-determinant of large, sparse, symmetric positive definite (SPD) matrices is essential in many scientific computational fields such as numerical linear algebra and machine learning. In low dimensions, Cholesky is preferred, but in high dimensions, its computation may be prohibitive due to memory limitation. To circumvent this, Krylov subspace techniques have proven to be efficient but may be computationally expensive due to the required orthogonalization processes. In this paper, we introduce a novel technique to estimate the Log-determinant of a matrix using Léja points, where the implementation is only based on matrix multiplications and a rough estimation of eigenvalue bounds of the matrix. By coupling Léja points interpolation with a randomized algorithm called Hutch++, we achieve substantial reductions in computational complexity while preserving significant accuracy compared to the stochastic Lanczos quadrature. We establish the approximation errors of the matrix function together with multiplicative error bounds for the approximations obtained by this method. The effectiveness and scalability of the proposed method on both large sparse synthetic matrices (maximum likelihood in Gaussian Markov Random fields) and large-scale real-world matrices are confirmed through numerical experiments.

math.NA↗

arXiv2025

Strong convergence of a semi tamed scheme for stochastic differential algebraic equation under non-global Lipschitz coefficients

We are investigating the first strong convergence analysis of a numerical method for stochastic differential algebraic equations (SDAEs) under a non-global Lipschitz setting. It is well known that the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDEs) when at least one of the coefficients grows superlinearly. The problem becomes more challenging in the case of stochastic differential-algebraic equations (SDAEs) due to the singularity of the matrix. To address this, we build a new scheme called the semi-implicit tamed method for SDAEs and provide its strong convergence result under non-global Lipschitz setting. In other words, the linear component of the drift term is approximated implicitly, whereas its nonlinear component is tamed and approximated explicitly. We show that this method strongly converges with order $\frac{1}{2}$ to the exact solution. To prove this strong convergence result, we first derive an equivalent scheme, that we call the dual tamed scheme, which is more suitable for mathematical analysis and is associated with the inherent stochastic differential equation obtained by eliminating the constraints from the original SDAEs. To demonstrate the effectiveness of the proposed scheme, numerical simulations are performed, confirming that the theoretical findings are consistent with the numerical results.

math.NA↗

arXiv2025

Pathwise convergence of a novel numerical scheme based on semi-implicit method for stochastic differential-algebraic equations with non-global Lipschitz coefficients

This paper delves into the well-posedness and the numerical approximation of non-autonomous stochastic differential algebraic equations (SDAEs) with nonlinear local Lipschitz coefficients that satisfy the more general monotonicity condition called Khasminskii condition. The key challenge is the presence of a singular matrix which makes the numerical integration hard and heavy. To address this challenge, we propose a novel numerical scheme based on semi-implicit method for the drift component of the SDAEs. More precisely we split the drift term as the sum of a linear term and a nonlinear term. The linear part is approximated implicitly, while the nonlinear part is approximated explicitly. The linear component's role is to handle the singularity issues during the numerical integration without the resolution of nonlinear algebraic equations in the constraint equations. This novel scheme is therefore very efficient for SDAEs in high dimension that come after the spatial discretisation of stochastic partial differential algebraic equations (SPDAEs). To prove the pathwise convergence of our novel scheme, we first derive a equivalent scheme called dual scheme, suitable for mathematical analysis and linked to the inherent stochastic differential equation resulting from the elimination of constraints in the initial SDAEs. We prove that our novel scheme converges to the exact solution with rate $\frac{1}{2}-ε$, for arbitrary $ε>0$ in the pathwise sense. Numerical simulations are performed to demonstrate the efficiency of the scheme in high dimension and to show that our theoretical results are in agreement with numerical experiments.

math.NA↗

arXiv2024

Strong convergence of some Magnus-type schemes for the finite element discretization of non-autonomous parabolic SPDEs driven by additive fractional Brownian motion and Poisson random measure

The aim of this work is to provide the strong convergence results of numerical approximations of a general second order non-autonomous semilinear stochastic partial differential equation (SPDE) driven simultaneously by an additive fractional Brownian motion (fBm) with Hurst parameter H \in (1/2,1) and a Poisson random measure, more realistic in modelling real world phenomena. Approximations in space are performed by the standard finite element method and in time by the stochastic Magnus-type integrator or the linear semi-implicit Euler method. We investigate the mean-square errors estimates of our fully discrete schemes and the results show how the convergence orders depend on the regularity of the initial data and the driven processes. To the best of our knowledge, these two schemes are the first numerical methods to approximate the non-autonomous semilinear stochastic partial differential equation (SPDE) driven simultaneously by an additive fractional Brownian motion with Hurst parameter H and a Poisson random measure.

math.NA↗

arXiv2024

Existence and uniqueness for the solutions of non-autonomous stochastic differential algebraic equations with locally Lipschitz coefficients

In this paper, we study the well-posedness and regularity of non-autonomous stochastic differential algebraic equations (SDAEs) with nonlinear, locally Lipschitz and monotone (2) coefficients of the form (1). The main difficulty is the fact that the operator A(.) is non-autonomous, i.~e. depends on t and the matrix $A(t)$ is singular for all $t\in \left[0,T\right]$. Our interest is in SDAE of index-1. This means that in order to solve the problem, we can transform the initial SDAEs into an ordinary stochastic differential equation with algebraic constraints. Under appropriate hypothesizes, the main result establishes the existence and uniqueness of the solution in $\mathcal{M}^p(\left[0, T\right], \mathbb{R}^n)$, $p\geq 2$, $p\in \mathbb{N}$. Several strong estimations and regularity results are also provided. Note that, in this paper, we use various techniques such as Itô's lemma, Burkholder-Davis-Gundy inequality, and Young inequality.

math.PR↗

arXiv2022

Approximation of homogenized coefficients in deterministic homogenization and convergence rates in the asymptotic almost periodic setting

For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the asymptotic almost periodic setting, and we show that the rates of convergence for the zero order approximation, are near optimal. The results obtained constitute a step towards the numerical implementation of results from the deterministic homogenization theory beyond the periodic setting. To illustrate this, numerical simulations based on finite volume method are provided to sustain our theoretical results.

math.AP↗

arXiv2022

Strong convergence of an fractional exponential integrator scheme for the finite element discretization of time-fractional SPDE driven by standard and fractional Brownian motions

The aim of this work is to provide the first strong convergence result of numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order $α\in(\frac 12; 1)$ and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter $H\in(\frac 12, 1)$, more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results and perform the spatial discretization using the finite element and the temporal discretization using a fractional exponential integrator scheme. We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter $H$.

math.NA↗

arXiv2021

A novel high dimensional fitted scheme for stochastic optimal control problems

Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton-Jacobi-Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ($ n\geq 3$). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme. We show that matrices resulting from spatial discretization and temporal discretization are M--matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.

math.NA↗

arXiv2021

Strong convergence rates of an exponential integrator and finite elements method for time-fractional SPDEs driven by Gaussian and non-Gaussian noises

In this work, we provide the first strong convergence result of numerical approximation of a general second order semilinear stochastic fractional order evolution equation involving a Caputo derivative in time of order $α\in(\frac 34, 1)$ and driven by Gaussian and non-Gaussian noises simultaneously more useful in concrete applications. The Gaussian noise considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Lévy process. The linear operator is not necessary self-adjoint. The fractional stochastic partial differential equation is discretized in space by the finite element method and in time by a variant of the exponential integrator scheme. We investigate the mean square error estimate of our fully discrete scheme and the result shows how the convergence orders depend on the regularity of the initial data and the power of the fractional derivative.

math.NA↗

arXiv2020

Strong convergence analysis of the stochastic exponential Rosenbrock scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

In this paper, we consider the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme (SERS) based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square $L^2$ norm. Numerical experiments to sustain theoretical results are provided.

math.NA↗

arXiv2020

Optimal strong convergence rates of numerical methods for semilinear parabolic SPDE driven by Gaussian noise and Poisson random measure

This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena. The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems. We prove the strong convergence of the fully discrete schemes toward the mild solution. The results reveal how convergence orders depend on the regularity of the noise and the initial data.In addition, we exceed the classical orders $1/2$ in time and $1$ in space achieved in the literature when dealing with SPDE driven by Poisson measure with less regularity assumptions on the nonlinear drift function. In particular, for trace class multiplicative Gaussian noise we achieve convergence order $\mathcal{O}(h^2+Δt^{1/2})$.For additive trace class Gaussian noise and an appropriate jump function, we achieve convergence order $\mathcal{O}(h^2+Δt)$. Numerical experiments to sustain the theoretical results are provided.

math.NA↗

arXiv2020

A note on exponential Rosenbrock-Euler method for the finite element discretization of a semilinear parabolic partial differential equation

In this paper 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. Using finite element method for space discretization and the exponential Rosenbrock-Euler method for time discretization, we provide a rigorous convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solution. This is in contrast to very restrictive assumptions made in the literature, where the authors have considered only approximation in time so far in their convergence proofs. The optimal orders of convergence in space and in time are achieved for smooth and nonsmooth initial solution.

math.NA↗

arXiv2020

Strong Convergence of a Stochastic Rosenbrock-type Scheme 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. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretise the SPDE in space by the finite element method and propose a new scheme in time appropriate for such equations, called stochastic Rosenbrock-Type scheme, which is based on the local linearisation of the semi-discrete problem obtained after space discretisation. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise. Our convergence rates are in agreement with results in the literature. Numerical experiments to sustain our theoretical results are provided.

math.NA↗

arXiv2020

Strong Convergence of the Linear Implicit Euler Method for the Finite Element Discretization of Semilinear non-Autonomous SPDEs Driven by Multiplicative or Additive Noise

This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than autonomous SPDEs while modeling real world phenomena. Numerical approximations for autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet well understood. The non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the time depending, not necessarily self-adjoint linear operators with the corresponding semi group and provide the strong convergence result of the fully discrete scheme toward the exact solution in the root-mean-square $L^2$ norm. The results indicate how the converge order depends on the regularity of the initial solution and the noise. In particular, for multiplicative trace class noise we achieve convergence order $\mathcal{O}(h^{2-ε}+Δt^{1/2})$ and for additive noise with trace class, we achieve convergence order $\mathcal{O}(h^{2-ε}+Δt^{1-ε})$, for an arbitrarily small $ε>0$. Numerical experiments to sustain our theoretical results are provided.

math.NA↗

arXiv2020

A fitted finite volume method for stochastic optimal control Problems

In this article, we provide a numerical method based on fitted finite volume method to approximate the Hamilton-Jacobi-Bellman (HJB) equation coming from stochastic optimal control problems. The computational challenge is due to the nature of the HJB equation, which may be a second-order degenerated partial differential equation coupled with optimization. In the work, we discretize the HJB equation using the fitted finite volume method and show that matrix resulting from spatial discretization is an M-matrix. The optimization problem is solved at every time step using iterative method. Numerical results are presented to show the robustness of the fitted finite volume numerical method comparing to the standard finite difference method.

math.NA↗

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↗