Yonah Conjungo Taumhas, David Labeurthre, Francois Madiot, Olga Mula, Tommaso Taddei
In this paper, we consider the inverse problem of state estimation of nuclear power fields in a power plant from a limited number of observations of the neutron flux. For this, we use the Parametrized Background Data Weak approach. The method combines the observations with a parametrized PDE model for the behavior of the neutron flux. Since, in general, even the most sophisticated models cannot perfectly capture reality, an inevitable model error is made. We investigate the impact of the model error in the power reconstruction when we use a diffusion model for the neutron flux, and assume that the true physics are governed by a neutron transport model.
{"title":"Impact of physical model error on state estimation for neutronics applications","authors":"Yonah Conjungo Taumhas, David Labeurthre, Francois Madiot, Olga Mula, Tommaso Taddei","doi":"10.1051/proc/202373158","DOIUrl":"https://doi.org/10.1051/proc/202373158","url":null,"abstract":"In this paper, we consider the inverse problem of state estimation of nuclear power fields in a power plant from a limited number of observations of the neutron flux. For this, we use the Parametrized Background Data Weak approach. The method combines the observations with a parametrized PDE model for the behavior of the neutron flux. Since, in general, even the most sophisticated models cannot perfectly capture reality, an inevitable model error is made. We investigate the impact of the model error in the power reconstruction when we use a diffusion model for the neutron flux, and assume that the true physics are governed by a neutron transport model.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135058791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we investigate from a broad perspective the reduction of degrees of freedom through serendipity techniques for polytopal methods compatible with Hilbert complexes. We first establish an abstract framework that, given two complexes connected by graded maps, identifies a set of properties enabling the transfer of the homological and analytical properties from one complex to the other. This abstract framework is designed having in mind discrete complexes, with one of them being a reduced version of the other, such as occurring when applying serendipity techniques to numerical methods. We then use this framework as an overarching blueprint to design a serendipity DDR complex. Thanks to the combined use of higher-order reconstructions and serendipity, this complex compares favorably in terms of degrees of freedom (DOF) count to all the other polytopal methods previously introduced and also to finite elements on certain element geometries. The gain resulting from such a reduction in the number of DOFs is numerically evaluated on two model problems: a magnetostatic model, and the Stokes equations.
{"title":"Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method","authors":"Daniele Antonio Di Pietro, Jérôme Droniou","doi":"10.1051/m2an/2022067","DOIUrl":"https://doi.org/10.1051/m2an/2022067","url":null,"abstract":"In this work we investigate from a broad perspective the reduction of degrees of freedom through serendipity techniques for polytopal methods compatible with Hilbert complexes. We first establish an abstract framework that, given two complexes connected by graded maps, identifies a set of properties enabling the transfer of the homological and analytical properties from one complex to the other. This abstract framework is designed having in mind discrete complexes, with one of them being a reduced version of the other, such as occurring when applying serendipity techniques to numerical methods. We then use this framework as an overarching blueprint to design a serendipity DDR complex. Thanks to the combined use of higher-order reconstructions and serendipity, this complex compares favorably in terms of degrees of freedom (DOF) count to all the other polytopal methods previously introduced and also to finite elements on certain element geometries. The gain resulting from such a reduction in the number of DOFs is numerically evaluated on two model problems: a magnetostatic model, and the Stokes equations.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136053140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze a divergence based first order system least squares method applied to a second order elliptic model problem with homogeneous boundary conditions. We prove optimal convergence in the L 2 (Ω) norm for the scalar variable. Numerical results confirm our findings.
{"title":"Optimal convergence rates in <i>L</i><sup>2</sup> for a first order system least squares finite element method","authors":"Maximilian Bernkopf, Jens Markus Melenk","doi":"10.1051/m2an/2022026","DOIUrl":"https://doi.org/10.1051/m2an/2022026","url":null,"abstract":"We analyze a divergence based first order system least squares method applied to a second order elliptic model problem with homogeneous boundary conditions. We prove optimal convergence in the L 2 (Ω) norm for the scalar variable. Numerical results confirm our findings.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the behavior in a large time regime of an explicit tamed Euler-Maruyama scheme applied to a class of ergodic Ito stochastic differential equations with one-sided Lipschitz continuous drift coefficient and bounded globally Lipschitz diffusion coefficient. Our first main contribution is to prove moments for the numerical scheme, which, on the one hand, are uniform with respect to the time-step size, and which, on the other hand, may not be uniform but have at most polynomial growth with respect to time. Our second main contribution is to apply this result to obtain weak error estimates to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process, as a function of the time-step size and of the time horizon. The explicit tamed Euler scheme is shown to be computationally effective for the approximation of the invariant distribution: even if the moment bounds and error estimates are not proved to be uniform with respect to time, the obtained polynomial growth results in a marginal increase in the upper bound of the computational cost. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for stochastic differential equations with non-globally Lipschitz coefficients using an explicit tamed Euler-Maruyama scheme.
{"title":"Approximation of the invariant distribution for a class of ergodic SDEs with one-sided Lipschitz continuous drift coefficient using an explicit tamed Euler scheme","authors":"Charles-Edouard Bréhier","doi":"10.1051/ps/2023017","DOIUrl":"https://doi.org/10.1051/ps/2023017","url":null,"abstract":"We study the behavior in a large time regime of an explicit tamed Euler-Maruyama scheme applied to a class of ergodic Ito stochastic differential equations with one-sided Lipschitz continuous drift coefficient and bounded globally Lipschitz diffusion coefficient. Our first main contribution is to prove moments for the numerical scheme, which, on the one hand, are uniform with respect to the time-step size, and which, on the other hand, may not be uniform but have at most polynomial growth with respect to time. Our second main contribution is to apply this result to obtain weak error estimates to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process, as a function of the time-step size and of the time horizon. The explicit tamed Euler scheme is shown to be computationally effective for the approximation of the invariant distribution: even if the moment bounds and error estimates are not proved to be uniform with respect to time, the obtained polynomial growth results in a marginal increase in the upper bound of the computational cost. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for stochastic differential equations with non-globally Lipschitz coefficients using an explicit tamed Euler-Maruyama scheme.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study tail probabilities of superexponential infinite divisible distributions as well as tail probabilities of suprema of Lévy processes with superexponential marginal distributions over compact intervals.
研究了紧区间上的超指数无限可分分布的尾概率，以及具有超指数边际分布的lsamvy过程的上点尾概率。
{"title":"On the asymptotic behaviour of superexponential Levy processes","authors":"Patrik Albin, Mattias Sunden","doi":"10.1051/ps/2023015","DOIUrl":"https://doi.org/10.1051/ps/2023015","url":null,"abstract":"We study tail probabilities of superexponential infinite divisible distributions as well as tail probabilities of suprema of Lévy processes with superexponential marginal distributions over compact intervals.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135500807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Adrien Beguinet, Virginie Ehrlacher, Roberta Flenghi, Maria Fuente, Olga Mula, Agustin Somacal
Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantly challenged, and can potentially spoil the final quality of the approximated solution due to the convergence to bad local minima, and bad generalization capabilities. In this paper, we present an exhaustive numerical study of the merits and limitations of these schemes when solutions exhibit low-regularity, and compare performance with respect to more benign cases when solutions are very smooth. As a support for our study, we consider singularly perturbed convection-diffusion problems where the regularity of solutions typically degrades as certain multiscale parameters go to zero.
{"title":"Deep learning-based schemes for singularly perturbed convection-diffusion problems","authors":"Adrien Beguinet, Virginie Ehrlacher, Roberta Flenghi, Maria Fuente, Olga Mula, Agustin Somacal","doi":"10.1051/proc/202373048","DOIUrl":"https://doi.org/10.1051/proc/202373048","url":null,"abstract":"Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantly challenged, and can potentially spoil the final quality of the approximated solution due to the convergence to bad local minima, and bad generalization capabilities. In this paper, we present an exhaustive numerical study of the merits and limitations of these schemes when solutions exhibit low-regularity, and compare performance with respect to more benign cases when solutions are very smooth. As a support for our study, we consider singularly perturbed convection-diffusion problems where the regularity of solutions typically degrades as certain multiscale parameters go to zero.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135058788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce the Markov Stochastic Block Model (MSBM): a growth model for community-based networks where node attributes are assigned through a Markovian dynamic. We rely on HMMs' literature to design prediction methods that are robust to local clustering errors. We focus specifically on the link prediction and collaborative filtering problems and we introduce a new model selection procedure to infer the number of hidden clusters in the network. Our approaches for reliable prediction in MSBMs are not algorithm-dependent in the sense that they can be applied using your favourite clustering tool. In this paper, we use a recent SDP method to infer the hidden communities and we provide theoretical guarantees. In particular, we identify the relevant signal-to-noise ratio (SNR) in our framework and we prove that the misclassification error decays exponentially fast with respect to this SNR.
{"title":"Reliable temporal prediction in the Markov stochastic block model","authors":"Quentin Duchemin","doi":"10.1051/ps/2022019","DOIUrl":"https://doi.org/10.1051/ps/2022019","url":null,"abstract":"We introduce the Markov Stochastic Block Model (MSBM): a growth model for community-based networks where node attributes are assigned through a Markovian dynamic. We rely on HMMs' literature to design prediction methods that are robust to local clustering errors. We focus specifically on the link prediction and collaborative filtering problems and we introduce a new model selection procedure to infer the number of hidden clusters in the network. Our approaches for reliable prediction in MSBMs are not algorithm-dependent in the sense that they can be applied using your favourite clustering tool. \u0000In this paper, we use a recent SDP method to infer the hidden communities and we provide theoretical guarantees. In particular, we identify the relevant signal-to-noise ratio (SNR) in our framework and we prove that the misclassification error decays exponentially fast with respect to this SNR.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"4 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76024685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we introduce a stochastic growth-fragmentation model for the expansion of the network of filaments ( mycelium ) of a filamentous fungus. In this model, each individual is described by a discrete type e∈{0,1} indicating whether the individual corresponds to an internal or terminal segment of filament, and a continuous trait x≥0 corresponding to the length of this segment. The length of internal segments cannot grow, while the length of terminal segments increases at a deterministic speed. Both types of individuals/segments branch according to a type-dependent mechanism. After constructing the stochastic bi-type growth-fragmentation process, we analyse the corresponding mean measure. We show that its ergodic behaviour is governed by the maximal eigenelements. In the long run, the total mass of the mean measure increases exponentially fast while the type-dependent density in trait converges to an explicit distribution at some exponential speed. We then obtain a law of large numbers that relates the long term behaviour of the stochastic process to the limiting distribution. The model we consider depends on only 3 parameters and all the quantities needed to describe this asymptotic behaviour are explicit, which paves the way for parameter inference based on data collected in lab experiments.
{"title":"Ergodic behaviour of a multi-type growth-fragmentation process modelling the mycelial network of a filamentous fungus","authors":"M. Tomašević, Vincent Bansaye, A. Véber","doi":"10.1051/ps/2022013","DOIUrl":"https://doi.org/10.1051/ps/2022013","url":null,"abstract":"In this work, we introduce a stochastic growth-fragmentation model for the expansion of the network of filaments ( mycelium ) of a filamentous fungus. In this model, each individual is described by a discrete type e∈{0,1} indicating whether the individual corresponds to an internal or terminal segment of filament, and a continuous trait x≥0 corresponding to the length of this segment. The length of internal segments cannot grow, while the length of terminal segments increases at a deterministic speed. Both types of individuals/segments branch according to a type-dependent mechanism. After constructing the stochastic bi-type growth-fragmentation process, we analyse the corresponding mean measure. We show that its ergodic behaviour is governed by the maximal eigenelements. In the long run, the total mass of the mean measure increases exponentially fast while the type-dependent density in trait converges to an explicit distribution at some exponential speed. We then obtain a law of large numbers that relates the long term behaviour of the stochastic process to the limiting distribution. The model we consider depends on only 3 parameters and all the quantities needed to describe this asymptotic behaviour are explicit, which paves the way for parameter inference based on data collected in lab experiments.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"12 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85709965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rafal Marcin Lochowski, Witold Marek Bednorz, Rafał Martynek
For a general cad Lévy process $X$ on a separable Banach space $V$ we estimate values of $inf_{cge0} cbr{ psi(c)+ inf_{Yin{cal A}_{X}(c)}E TTV Y{left[0,Tright]}{}}$, where ${cal A}_{X}(c)$ is the family of processes on $V$ adapted to the natural filtration of $X$, a.s. approximating paths of $X$ uniformly with accuracy $c$, $psi$ is a penalty function with polynomial growth and $TTV Y{left[0,Tright]}{}$ denotes the total variation of the process $Y$ on the interval $[0,T]$. Next, we apply obtained estimates in three specific cases: Brownian motion with drift on $R$, standard Brownian motion on $R^{d}$ and a symmetric $alpha$-stable process ($alphain(1,2)$) on $R$.
{"title":"On optimal uniform approximation of Lévy processes on Banach spaces with finite variation processes","authors":"Rafal Marcin Lochowski, Witold Marek Bednorz, Rafał Martynek","doi":"10.1051/ps/2022011","DOIUrl":"https://doi.org/10.1051/ps/2022011","url":null,"abstract":"For a general cad Lévy process $X$ on a separable Banach\u0000space $V$ we estimate values of $inf_{cge0} cbr{ psi(c)+ inf_{Yin{cal A}_{X}(c)}E TTV Y{left[0,Tright]}{}}$,\u0000where ${cal A}_{X}(c)$ is the family of processes on $V$ adapted to\u0000the natural filtration of $X$, a.s. approximating paths of $X$ uniformly with accuracy $c$, $psi$ is a penalty function with polynomial growth and\u0000$TTV Y{left[0,Tright]}{}$ denotes the total variation of the process\u0000$Y$ on the interval $[0,T]$. Next, we apply obtained estimates in\u0000three specific cases: Brownian motion with drift on $R$, standard\u0000Brownian motion on $R^{d}$ and a symmetric $alpha$-stable process\u0000($alphain(1,2)$) on $R$.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"16 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84903444","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The usual stochastic order and the likelihood ratio order between probability distributions on the real line are reviewed in full generality. In addition, for the distribution of a random pair (X,Y), it is shown that the conditional distributions of Y, given X = x, are increasing in x with respect to the likelihood ratio order if and only if the joint distribution of (X,Y) is totally positive of order two (TP2) in a certain sense. It is also shown that these three types of constraints are stable under weak convergence, and that weak convergence of TP2 distributions implies convergence of the conditional distributions just mentioned.
{"title":"On stochastic orders and total positivity\u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 \u0000 ","authors":"L. Duembgen, Alexandre Mösching","doi":"10.1051/ps/2023005","DOIUrl":"https://doi.org/10.1051/ps/2023005","url":null,"abstract":"The usual stochastic order and the likelihood ratio order between probability distributions on the real line are reviewed in full generality. In addition, for the distribution of a random pair (X,Y), it is shown that the conditional distributions of Y, given X = x, are increasing in x with respect to the likelihood ratio order if and only if the joint distribution of (X,Y) is totally positive of order two (TP2) in a certain sense. It is also shown that these three types of constraints are stable under weak convergence, and that weak convergence of TP2 distributions implies convergence of the conditional distributions just mentioned.","PeriodicalId":51249,"journal":{"name":"Esaim-Probability and Statistics","volume":"34 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2022-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80152912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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