Numerical methods for hyperbolic SPDEs: a Wiener chaos approach

Kalpinelli, Evangelia A.; Frangos, Nikos E.; Yannacopoulos, A. N.

doi:10.1007/s40072-013-0019-x

Cited by 5 publications

(9 citation statements)

References 17 publications

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“…Theorem (Kalpinelli et al and Lototsky and Rozovskii) If

z \in double-struckH

and z α = E [ z H α ] ∈ V , then

z = \sum_{α \in frakturJ} z_{α} H_{α},

and

E {(‖‖, z)}_{V}^{2} = \sum_{α \in J} {(‖‖, z_{α})}_{V}^{2}

.…”

Section: Numerical Schemesmentioning

confidence: 99%

“…Now, as a PC Galerkin approximation of , as is discussed in Hou et al and Kalpinelli et al and references therein, we seek a solution

ũ false(t false) \in L_{2}^{γ}

, for t ∈ [0, T ] and γ as is given in Theorem , in the following form

\begin{matrix} array ũ (t) = \sum_{α \in {frakturJ}_{frakturf}} ũ_{α} (t) H_{α}, \end{matrix}

where the

ũ_{α} false(t false) \in {\overset{H}{true}}^{γ}

α \in J_{f}

, satisfies the following estimate

\frac{\partial ũ_{α} false(t false)}{\partial t} = A ũ_{α} false(t false) + G_{α} false(ũ false(t false) false) + f_{α} false(t false), ũ_{α} false(0 false) = u_{α, 0}, α \in J_{f},

in which

f_{α} (t) : = \{\begin{matrix} m_{j} false(t false), & i f H_{α} = ξ_{j}, 0 ⩽ j ⩽ n, \\ 0, & otherwise, \end{matrix}

where m j and n are introduced in and , respectively. The coupled and deterministic propagators are obtained through the Galerkin projection upon substituting the approximations , into the governing equation .…”

Section: Numerical Schemesmentioning

confidence: 99%

“…Also note that by considering assumptions of Theorems 2.3 and 3.1, the PC expansion of u(t), which is represented in (9), exists. Now, as a PC Galerkin approximation of (1), as is discussed in Hou et al 21 and Kalpinelli et al 29 and references therein, we seek a solutionũ(t) ∈ L 2 , for t ∈ [0, T] and as is given in Theorem 2.3, in the following form…”

Section: Pc Galerkin Approximationmentioning

confidence: 99%

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Convergence of a method based on the exponential integrator and Fourier spectral discretization for stiff stochastic PDEs

Asgari

Hosseini

2018

Math Methods in App Sciences

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In this article, we propose and analyze some efficient numerical methods for the solution of a class of stiff stochastic partial differential equations (SPDEs) with additive noise. First, we apply the polynomial chaos (PC) expansion to treat the randomness, which results to a set of deterministic stiff PDEs. Then this system of PDEs is discretized in space by the Fourier pseudo‐spectral method, enabling us to use the FFT algorithm to reduce computational cost efficiently. To overcome the instability in discretizing time component of resulting system of ordinary differential equations (ODEs), we apply the exponential time differencing (ETD), integrating factor Runge‐Kutta (IFRK) and modified exponential time differencing Runge‐Kutta (ETDRKB) methods. We also analyze the strong error bound of numerical approximation in a framework of Lipschitz nonlinearities for the method based on the ETDRKB scheme. Some numerical experiments are included to demonstrate the performance and convergence of methods. Numerical results show that the method based on the ETDRKB scheme is more efficient for stiff SPDEs.

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“…Theorem (Kalpinelli et al and Lototsky and Rozovskii) If

z \in double-struckH

and z α = E [ z H α ] ∈ V , then

z = \sum_{α \in frakturJ} z_{α} H_{α},

and

E {(‖‖, z)}_{V}^{2} = \sum_{α \in J} {(‖‖, z_{α})}_{V}^{2}

.…”

Section: Numerical Schemesmentioning

confidence: 99%

“…Now, as a PC Galerkin approximation of , as is discussed in Hou et al and Kalpinelli et al and references therein, we seek a solution

ũ false(t false) \in L_{2}^{γ}

, for t ∈ [0, T ] and γ as is given in Theorem , in the following form

\begin{matrix} array ũ (t) = \sum_{α \in {frakturJ}_{frakturf}} ũ_{α} (t) H_{α}, \end{matrix}

where the

ũ_{α} false(t false) \in {\overset{H}{true}}^{γ}

α \in J_{f}

, satisfies the following estimate

\frac{\partial ũ_{α} false(t false)}{\partial t} = A ũ_{α} false(t false) + G_{α} false(ũ false(t false) false) + f_{α} false(t false), ũ_{α} false(0 false) = u_{α, 0}, α \in J_{f},

in which

f_{α} (t) : = \{\begin{matrix} m_{j} false(t false), & i f H_{α} = ξ_{j}, 0 ⩽ j ⩽ n, \\ 0, & otherwise, \end{matrix}

Section: Numerical Schemesmentioning

confidence: 99%

Section: Pc Galerkin Approximationmentioning

confidence: 99%

See 1 more Smart Citation

Convergence of a method based on the exponential integrator and Fourier spectral discretization for stiff stochastic PDEs

Asgari

Hosseini

2018

Math Methods in App Sciences

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show abstract

“…These operators are the main operators of an infinite dimensional stochastic calculus of variations called the Malliavin calculus [50]. Classes of elliptic and evolution stochastic differential equations (SDEs) that involve operators of the Malliavin calculus within white noise framework were recently studed in [46,40,44,38,48,58]. In [42,43] it was proved that the Malliavin derivative indicates the rate of change in time between the ordinary product and the Wick product.…”

mentioning

confidence: 99%

“…The chaos expansion methodology is a very useful technique for solving many types of SDEs [40,44,45]. The main statistical properties of the solution, its mean, variance, higher moments, can be calculated from the formulas involving only the coefficients of the chaos expansion representation [46,58]. Moreover, numerical methods for SDEs and uncertainty quantification based on the polynomial chaos approach have become very popular in recent years.…”

mentioning

confidence: 99%

The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

Levajković¹,

Mena²,

Tuffaha³

2016

EECT

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We consider the stochastic linear quadratic optimal control problem for state equations of the Itô-Skorokhod type, where the dynamics are driven by strongly continuous semigroup. We provide a numerical framework for solving the control problem using a polynomial chaos expansion approach in white noise setting. After applying polynomial chaos expansion to the state equation, we obtain a system of infinitely many deterministic partial differential equations in terms of the coefficients of the state and the control variables. We set up a control problem for each equation, which results in a set of deterministic linear quadratic regulator problems. Solving these control problems, we find optimal coefficients for the state and the control. We prove the optimality of the solution expressed in terms of the expansion of these coefficients compared to a direct approach. Moreover, we apply our result to a fully stochastic problem, in which the state, control and observation operators can be random, and we also consider an extension to state equations with memory noise.

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Brownian motion and stochastic calculus

Zhang

Karniadakis

2017

Numerical Methods for Stochastic Partial Differential Equations With White Noise

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Numerical methods for hyperbolic SPDEs: a Wiener chaos approach

Cited by 5 publications

References 17 publications

Convergence of a method based on the exponential integrator and Fourier spectral discretization for stiff stochastic PDEs

Convergence of a method based on the exponential integrator and Fourier spectral discretization for stiff stochastic PDEs

The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach

Brownian motion and stochastic calculus

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