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Ryan Kurniawan - One of the best experts on this subject based on the ideXlab platform.
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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
Foundations of Computational Mathematics, 2020Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have, loosely speaking, been investigated since 2003 and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions. In the recent article (Conus et al. in Ann Appl Probab 29(2):653–716, 2019 ) this weak convergence problem has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome this weak convergence problem in the case of a class of time-discrete Euler-Type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild Itô-Type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.
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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
Foundations of Computational Mathematics, 2020Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89–117] for details. In the recent article [D. Conus, A. Jentzen & R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, arXiv:1408.1108] the weak convergence problem emerged from Debussche’s article has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome the problem emerged from Debussche’s article in the case of a class of time-discrete Euler-Type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are ap
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Weak convergence rates for Euler-Type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
arXiv: Probability, 2015Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In the recent article [D. Conus, A. Jentzen & R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, arXiv:1408.1108] the weak convergence problem emerged from Debussche's article has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome the problem emerged from Debussche's article in the case of a class of time-discrete Euler-Type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild It\^o Type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.
Arnulf Jentzen - One of the best experts on this subject based on the ideXlab platform.
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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
Foundations of Computational Mathematics, 2020Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have, loosely speaking, been investigated since 2003 and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions. In the recent article (Conus et al. in Ann Appl Probab 29(2):653–716, 2019 ) this weak convergence problem has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome this weak convergence problem in the case of a class of time-discrete Euler-Type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild Itô-Type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.
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Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
Foundations of Computational Mathematics, 2020Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89–117] for details. In the recent article [D. Conus, A. Jentzen & R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, arXiv:1408.1108] the weak convergence problem emerged from Debussche’s article has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome the problem emerged from Debussche’s article in the case of a class of time-discrete Euler-Type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are ap
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Strong convergence of full-discrete nonlinearity-truncated accelerated exponential Euler-Type approximations for stochastic Kuramoto-Sivashinsky equations
Communications in Mathematical Sciences, 2018Co-Authors: Martin Hutzenthaler, Arnulf Jentzen, Diyora SalimovaAbstract:This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-Type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly non-globally monotone nonlinearities such as stochastic Kuramoto-Sivashinsky equations. The main result of this article
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strong convergence of full discrete nonlinearity truncated accelerated exponential Euler Type approximations for stochastic kuramoto sivashinsky equations
arXiv: Probability, 2016Co-Authors: Martin Hutzenthaler, Arnulf Jentzen, Diyora SalimovaAbstract:This article introduces and analyzes a new explicit, easily implementable, and full discrete accelerated exponential Euler-Type approximation scheme for additive space-time white noise driven stochastic partial differential equations (SPDEs) with possibly non-globally monotone nonlinearities such as stochastic Kuramoto-Sivashinsky equations. The main result of this article proves that the proposed approximation scheme converges strongly and numerically weakly to the solution process of such an SPDE. Key ingredients in the proof of our convergence result are a suitable generalized coercivity-Type condition, the specific design of the accelerated exponential Euler-Type approximation scheme, and an application of Fernique's theorem.
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Weak convergence rates for Euler-Type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
arXiv: Probability, 2015Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In the recent article [D. Conus, A. Jentzen & R. Kurniawan, Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, arXiv:1408.1108] the weak convergence problem emerged from Debussche's article has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome the problem emerged from Debussche's article in the case of a class of time-discrete Euler-Type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp weak convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild It\^o Type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.
Banu Mermerkaya - One of the best experts on this subject based on the ideXlab platform.
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Oscillation and nonoscillation of half-linear Euler Type differential equations with different periodic coefficients
Open Mathematics, 2017Co-Authors: Adil Misir, Banu MermerkayaAbstract:Abstract In this paper, we compute explicitly the oscillation constant for certain half-linear second-order differential equations having different periodic coefficients. Our result covers known result concerning half-linear Euler Type differential equations with α—periodic positive coefficients. Additionally, our result is new and original in case that the least common multiple of these periods is not defined. We give an example and corollaries which illustrate cases that are solved with our result.
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critical oscillation constant for Euler Type half linear differential equation having multi different periodic coefficients
International Journal of Differential Equations, 2017Co-Authors: Adil Misir, Banu MermerkayaAbstract:We compute explicitly the oscillation constant for Euler Type half-linear second-order differential equation having multi-different periodic coefficients.
Maxime Hauray - One of the best experts on this subject based on the ideXlab platform.
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WASSERSTEIN DISTANCES FOR VORTICES APPROXIMATION OF Euler-Type EQUATIONS
Mathematical Models and Methods in Applied Sciences, 2009Co-Authors: Maxime HaurayAbstract:We establish the convergence of a vortex system towards equations similar to the 2D Euler equation in vorticity formulation. The only but important difference is that we use singular kernel of the Type x⊥/|x|α+1, with α < 1, instead of the Biot–Savard kernel x⊥/|x|2. This paper follows a previous work of Jabin and the author about the particles approximation of Vlasov equation in Ref. 13. Here we study a different mean-field equation, simplify the proofs and weaken non-physical initial conditions. The simplification is due to the introduction of the infinite Wasserstein distance. The results are obtained for L1 ∩ L∞ vorticities without any sign assumption, in the periodic setting, on the whole space and on the half space (with Neumann boundary conditions). A vortex-blob result is also given, that is valid for short times in the true vortex case.
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Approximation of Euler-Type equations by systems of vortices
2007Co-Authors: Maxime HaurayAbstract:We prove the weak convergence for any time of a system of quasi-vortex with positive and negative signs and without any trucature of the kernel to the solution of the Euler equation. Quasi-vortex means here that the kernel has a singularity in 1/|x|α with α ≤ 1 instead of diverging in 1/|x| near the origin. We also give some bounds on the force field for the true vortex case and explain why our technic fails in this case.
Ondřej Došlý - One of the best experts on this subject based on the ideXlab platform.
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Oscillation and non-oscillation of Euler Type half-linear differential equations
Journal of Mathematical Analysis and Applications, 2015Co-Authors: Ondřej Došlý, Michal VeselýAbstract:Abstract We investigate oscillatory properties of second order Euler Type half-linear differential equations whose coefficients are given by periodic functions and functions having mean values. We prove the conditional oscillation of these equations. In addition, we prove that the known oscillation constants for the corresponding equations with only periodic coefficients do not change in the studied more general case. The presented results are new for linear equations as well.
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Euler Type half linear differential equation with periodic coefficients
Abstract and Applied Analysis, 2013Co-Authors: Ondřej Došlý, Hana FunkovaAbstract:We investigate oscillatory properties of the perturbed half-linear Euler differential equation. We show that the results of the recent paper by O. Doslý and H. Funkova (2012) remain to hold when constants in perturbation terms are replaced by periodic functions.
