Probability Theory

Klenke, Achim

doi:10.1007/978-1-4471-5361-0

Cited by 248 publications

(171 citation statements)

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“…For a general introduction readers are referred to standard monographs on this topic, for instance [23,24,28]. For the measure theoretical background see [1,5].…”

Section: Preliminariesmentioning

confidence: 99%

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Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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Abstract. This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with timeirregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable.The main result consists of precise bounds for the discretization error with respect to the L p (Ω; R d )-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

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“…For a general introduction readers are referred to standard monographs on this topic, for instance [23,24,28]. For the measure theoretical background see [1,5].…”

Section: Preliminariesmentioning

confidence: 99%

“…Another well-known lemma considers the limiting behaviour of sequences of sets under probability measure (see Theorem 2.7 in [24]). …”

Section: Theorem 22 (Burkholder-davis-gundy Inequality)mentioning

confidence: 99%

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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“…Since, due to assumption (A1), the corresponding constant does not depend on ε, the sequence (u ε k ) k∈N is bounded in L 2 (∂D) for P-a.e. ω ∈ Γ implying its uniform integrability, see, e.g., [41]. As we already know the pointwise convergence u ε k (x, ω) → u * (x), x ∈ D, as k → ∞, an application of Egorov's theorem yields convergence in L 1 (∂D) so that the assertion follows by the triangle inequality and taking the limit k → ∞ inside the integration in (61).…”

Section: 2mentioning

confidence: 94%

From Feynman–Kac formulae to numerical stochastic homogenization in electrical impedance tomography

Piiroinen¹,

Simón

2016

Ann. Appl. Probab.

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Abstract. In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman-Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman-Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the development of new scalable stochastic numerical homogenization schemes.

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“…Systems with deterministic energy sources can be considered as a special case of the MMSE receiver described in (15) and (16). With a deterministic energy source, there will be no silent symbols; thus, the system equation can be expressed as …”

Section: Best-effort Sensing With Stochastic Energy Sourcesmentioning

confidence: 99%

On the asymptotic equivalence between stochastic and deterministic energy sources in energy harvesting sensing systems

Yang

2015

J Wireless Com Network

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In this paper, we seek answer to the question: can a wireless sensing system with energy harvesting power supplies perform as well as the one with conventional power supplies? Conventional sensing systems with deterministic energy sources usually employ uniform sampling. However, due to the stochastic nature of the energy harvested from the ambient environment, uniform sampling is usually infeasible for energy harvesting sensing systems. We thus propose a simple best-effort sensing scheme, which defines a set of equally spaced candidate sensing instants. At a given candidate sensing instant, the sensor will perform sensing if there is sufficient energy available, and it will remain silent otherwise. It is analytically shown that the percentage of silent candidate sensing instants goes to zero as time increases, if and only if the average energy harvesting rate is no less than the average energy consumption rate. Therefore, the difference between the best-effort sensing policy and the uniform sensing policy diminishes as time evolves. The theoretical results are then used to guide the design of a practical sensing system that monitors a time-varying event. Both analysis and simulations show that the energy harvesting system with the best-effort sensing scheme can asymptotically achieve the same mean squared error (MSE) performance as the one with uniform sensing and deterministic energy sources. Therefore, we provide a positive answer to the question by establishing the asymptotic equivalence between stochastic and deterministic energy sources, from both theoretical and practical aspects.

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Probability Theory

Cited by 248 publications

References 0 publications

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

From Feynman–Kac formulae to numerical stochastic homogenization in electrical impedance tomography

On the asymptotic equivalence between stochastic and deterministic energy sources in energy harvesting sensing systems

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