Applied Stochastic Differential Equations

Särkkä, Simo; Solin, Arno

doi:10.1017/9781108186735

Cited by 267 publications

(305 citation statements)

References 103 publications

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“…This standard route implies a heavy computational effort, in particular when we want to study the heat transport for several bath configurations, frequency gradients and chain parameters. It is possible to circumvent this difficulty and find ensemble averages like x n x m , x n p m , p n p m (second order moments) without integrating any SDE [48]. The idea is to impose the condition d ··· dt = 0 for all the second order moments and linearize the dynamical equations of the system around equilibrium.…”

Section: Calculation Of the Stationary Heat Currentsmentioning

confidence: 99%

Asymmetric heat transport in ion crystals

et al. 2019

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We numerically demonstrate heat rectification for linear chains of ions in trap lattices with graded trapping frequencies, in contact with thermal baths implemented by optical molasses. To calculate the local temperatures and heat currents we find the stationary state by solving a system of algebraic equations. This approach is much faster than the usual method that integrates the dynamical equations of the system and averages over noise realizations. * miguelangel.simon@ehu.eus † jg.muga@ehu.es

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Section: Calculation Of the Stationary Heat Currentsmentioning

confidence: 99%

Asymmetric heat transport in ion crystals

et al. 2019

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“…and F (d) exp = −1/ d (because the exponential covariance function has an exact LTI SDE representation [15]). Similarly, the rest of the model matrices are given in terms of the Kronecker products of the submodel matrices.…”

Section: The Corresponding Continuous State Space Modelmentioning

confidence: 99%

“…where f k is the M dimensional state,Ã = exp(F ∆t) and Q = P ∞ −ÃP ∞Ã T . The stationary state covariance P ∞ is straightforward to calculate for most common kernel functions [15], and in the exponential kernel case is P ∞ = diag(σ 2 1 , . .…”

Section: Returning To Discrete State Space Formmentioning

confidence: 99%

“…Given that exp(F Whilst we have derived this model for the exponential kernel, the framework outlined above can be followed regardless of the kernel choice, as long as it can be written in state space form. This is possible for most commonly used kernel functions [15,17]. An intuitive way to proceed now is to alter the kernel in Eq.…”

Section: Returning To Discrete State Space Formmentioning

confidence: 99%

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Unifying Probabilistic Models for Time-frequency Analysis

Wilkinson

Andersen

Reiss

et al. 2019

ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In audio signal processing, probabilistic time-frequency models have many benefits over their non-probabilistic counterparts. They adapt to the incoming signal, quantify uncertainty, and measure correlation between the signal's amplitude and phase information, making time domain resynthesis straightforward. However, these models are still not widely used since they come at a high computational cost, and because they are formulated in such a way that it can be difficult to interpret all the modelling assumptions. By showing their equivalence to Spectral Mixture Gaussian processes, we illuminate the underlying model assumptions and provide a general framework for constructing more complex models that better approximate real-world signals. Our interpretation makes it intuitive to inspect, compare, and alter the models since all prior knowledge is encoded in the Gaussian process kernel functions. We utilise a state space representation to perform efficient inference via Kalman smoothing, and we demonstrate how our interpretation allows for efficient parameter learning in the frequency domain.Index Termsprobabilistic time-frequency analysis, Gaussian processes, state space models * Corresponding author: w.j.wilkinson@qmul.ac.uk. † AS acknowledges funding from the Academy of Finland (308640).

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“…where v k is the measurement noise vector. Combined together, Equations 3 and 8 represent a SSM with continuoustime dynamics and discrete-time measurements known as the continuous-discrete state-space model (see [24], pg170 or [25], pg93)ẋ (t) = A c x(t) + B c f (t),…”

mentioning

confidence: 99%

A Gaussian process latent force model for joint input-state estimation in linear structural systems

Nayek

Chakraborty

Narasimhan

2019

Mechanical Systems and Signal Processing

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The problem of combined state and input estimation of linear structural systems based on measured responses and a priori knowledge of structural model is considered. A novel methodology using Gaussian process latent force models is proposed to tackle the problem in a stochastic setting. Gaussian process latent force models (GPLFMs) are hybrid models that combine differential equations representing a physical system with data-driven non-parametric Gaussian process models. In this work, the unknown input forces acting on a structure are modelled as Gaussian processes with some chosen covariance functions which are combined with the mechanistic differential equation representing the structure to construct a GPLFM. The GPLFM is then conveniently formulated as an augmented stochastic state-space model with additional states representing the latent force components, and the joint input and state inference of the resulting model is implemented using Kalman filter. The augmented state-space model of GPLFM is shown as a generalization of the class of input-augmented state-space models, is proven observable, and is robust compared to conventional augmented formulations in terms of numerical stability. The hyperparameters governing the covariance functions are estimated using maximum likelihood optimization based on the observed data, thus overcoming the need for manual tuning of the hyperparameters by trial-and-error. To assess the performance of the proposed GPLFM method, several cases of state and input estimation are demonstrated using numerical simulations on a 10-dof shear building and a 76-storey ASCE benchmark office tower. Results obtained indicate the superior performance of the proposed approach over conventional Kalman filter based approaches.

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Applied Stochastic Differential Equations

Cited by 267 publications

References 103 publications

Asymmetric heat transport in ion crystals

Asymmetric heat transport in ion crystals

Unifying Probabilistic Models for Time-frequency Analysis

A Gaussian process latent force model for joint input-state estimation in linear structural systems

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