Partial functional quantization and generalized bridges

Corlay, Sylvain

doi:10.3150/12-bej504

Cited by 6 publications

(10 citation statements)

References 27 publications

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“…Therefor, we need the following formulation of Gronwall's inequality for integrable functions. The result might be deduced from more general formulations, see [6,38,39].…”

Section: The Stochastic Stokes Equationsmentioning

confidence: 95%

Optimal distributed and tangential boundary control for the unsteady stochastic Stokes equations

Benner

Trautwein

2020

ESAIM: COCV

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We consider a control problem constrained by the unsteady stochastic Stokes equations with nonhomogeneous boundary conditions in connected and bounded domains. In this paper, controls are defined inside the domain as well as on the boundary. Using a stochastic maximum principle, we derive necessary and sufficient optimality conditions such that explicit formulas for the optimal controls are derived. As a consequence, we are able to control the stochastic Stokes equations using distributed controls as well as boundary controls in a desired way.

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“…Therefor, we need the following formulation of Gronwall's inequality for integrable functions. The result might be deduced from more general formulations, see [6,38,39].…”

Section: The Stochastic Stokes Equationsmentioning

confidence: 95%

Optimal distributed and tangential boundary control for the unsteady stochastic Stokes equations

Benner

Trautwein

2020

ESAIM: COCV

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“…Convergence issues are discussed in a more general multidimensional setting in the appendix. The approximation (26) has the advantage of re-casting an infinite dimensional problem in finite-dimensional terms. We can view the solution of (26) as a functionX…”

Section: Series Expansion Approximation Of Sdementioning

confidence: 99%

“…We choose a set {σ j } of sigma points to represent the joint distribution of the state and the random coefficients {Z i } in (26). Each sigma point can be thought of as a vector of dimension n + d × N ,…”

Section: The Series Expansion Filtermentioning

confidence: 99%

“…Ideas of this type were first explored by Wong and Zakai [24]. Similar ideas have been explored in [25], [26] in the context of variance reduction for Monte-Carlo simulation, and in [27] in the context of parameter estimation. In this framework, one can interpret the approximation we use as the image of an N × d-dimensional standard normal distribution under a nonlinear transform.…”

Section: Introductionmentioning

confidence: 96%

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Series Expansion Approximations of Brownian Motion for Non-Linear Kalman Filtering of Diffusion Processes

Lyons

Särkkä

Storkey

2014

IEEE Trans. Signal Process.

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In this paper, we describe a novel application of sigma-point methods to continuous-discrete filtering. The nonlinear continuous-discrete filtering problem is often computationally intractable to solve. Assumed density filtering methods attempt to match statistics of the filtering distribution to some set of more tractable probability distributions. Filters such as these are usually decompose the problem into two sub-problems. The first of these is a prediction step, in which one uses the known dynamics of the signal to predict its state at time t k+1 given observations up to time t k . In the second step, one updates the prediction upon arrival of the observation at time t k+1 .The aim of this paper is to describe a novel method that improves the prediction step. We decompose the Brownian motion driving the signal in a generalised Fourier series, which is truncated after a number of terms. This approximation to Brownian motion can be described using a relatively small number of Fourier coefficients, and allows us to compute statistics of the filtering distribution with a single application of a sigmapoint method.Assumed density filters that exist in the literature usually rely on discretisation of the signal dynamics followed by iterated application of a sigma point transform (or a limiting case thereof). Iterating the transform in this manner can lead to loss of information about the filtering distribution in highly non-linear settings. We demonstrate that our method is better equipped to cope with such problems. * S.M.J Lyons and A.J. Storkey are with the department of Informatics, Edinburgh University.

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“…attaining values in some function spaces, therefore called functional quantiation, see e.g. [10,7,8,9,14,15,5,12,1] and references therein for a selection, and their applications to numerical probability, see e.g. [16].…”

Section: Introductionmentioning

confidence: 99%

How complex is a random picture?

Аурзада

Lifshits

2019

Journal of Complexity

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We study the amount of information that is contained in "random pictures", by which we mean the sample sets of a Boolean model. To quantify the notion "amount of information", two closely connected questions are investigated: on the one hand, we study the probability that a large number of balls is needed for a full reconstruction of a Boolean model sample set. On the other hand, we study the quantization error of the Boolean model w.r.t. the Hausdorff distance as a distortion measure. Keywords: Boolean model; functional quantization; high resolution quantization; information based complexity; metric entropy. 2010 Mathematics Subject Classification: 94A29, 60D05, secondary: 52A22.problem, the following random variable is crucial. Moreover, we believe that it may be of independent interest. Define the effective number of balls visible in the picture asIn other words, with K balls one can reproduce the black picture S exactly as with the original N balls.We are interested in the upper tail of K, i.e. P[K ≥ n] when n → ∞. This means we study the probability that one needs many balls in order to reconstruct the picture S. In particular, we would like to understand when one can "save balls" w.r.t. the original Poisson number of balls N . To make this more precise, note that clearly K ≤ N , and soWe would like to show that the upper tail of K is thinner, i.e. for some a > 1 P[K ≥ n] = exp (−a · n log n · (1 + o(1))) , n → ∞.It turns out that this question is non-trivial and interesting. The answer depends on the dimension d, on the type of norm used, and on the distribution of the radii L(R 1 ). Boolean models are fundamental objects in stochastic geometry and have a large range of applications, [4,17]. However, to the knowledge of the authors, until recently mostly the average of observables of Boolean models are studied. Often this plays a role when estimating parameters of the model in applications. On the contrary, the present paper deals with rare events, i.e. with large deviation probabilities.As mentioned above, the upper tail of the random variable K is an essential ingredient for solving the so-called quantization problem, which we recall now. Let an arbitrary norm ||.|| be fixed on R d . Let d H denote the corresponding Hausdorff distance between the closed subsets of R d . We define the respective quantization error for pictures byHere, the sets C are called codebooks and the upper index (q) stands for "quantization". The idea is that the "analog" signal S should be encoded by an element A ∈ C. This incurs an error, d H (A, S), measured in Hausdorff distance. Losely speaking, D (q) is then the minimal average error over all codebooks C of a size not exceeding e r . We are interested in letting r → ∞, that is, the size of the codebooks grows; and we would like to understand the rate of decay of the corresponding quantization error.

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Partial functional quantization and generalized bridges

Cited by 6 publications

References 27 publications

Optimal distributed and tangential boundary control for the unsteady stochastic Stokes equations

Optimal distributed and tangential boundary control for the unsteady stochastic Stokes equations

Series Expansion Approximations of Brownian Motion for Non-Linear Kalman Filtering of Diffusion Processes

How complex is a random picture?

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