Regularized shadowing-based data assimilation method for imperfect models and its comparison to the weak constraint 4DVar method

We consider a data assimilation problem for imperfect models. We propose a novel shadowing-based data assimilation method that takes model error into account following the Levenberg-Marquardt regularization approach. We illuminate how the proposed shadowing-based method is related to the weak constraint 4DVar method both analytically and numerically. We demonstrate that the shadowing-based method respects the distribution of the data mismatch, while the weak constraint 4DVar does not, which becomes even more pronou

On shadowing methods for data assimilation

Combining orbits from a model of a (chaotic) dynamical system with measured data to arrive at an improved estimate of the state of a physical system is known as data assimilation. This thesis deals with various algorithms for data assimilation. These algorithms are based on shadowing. Shadowing is a concept from the theory of dynamical systems. When a dynamical system has the property that an exact orbit of the dynamical system is located in a neighborhood of each pseudo-orbit, then this exact orbit shadows the pseudo-orbit. Shadowing can be used to show that a numerical solution of a dynamical system is located in a neighborhood of an exact solution. Shadowing refinement is a numerical technique in which an improved approximation to an exact solution is found from a pseudo-orbit. It is possible to use a shadowing refinement technique for data assimilation. Starting from observations, Newton's method is applied to approximate a zero of a cost operator, where the cost operator assigns costs to deviations from model solutions. The algorithms of Chapter 2 are based on a numerical time-dependent split between stable and unstable directions. The algorithm uses time-dependent projections onto the unstable subspace determined by using Lyapunov exponents and Lyapunov vectors. A shadowing algorithm is used in the unstable subspace, while synchronization is used in the stable subspace. The method is further extended to include parameter estimation and to some cases where only partial observations are available. Chapter 3 discusses data assimilation for imperfect models. Through regularization according to the Levenberg-Marquardt method, imperfections in the model are considered. It also describes how the shadowing method compares, both analytically and numerically, with the weak constraint 4DVar method and shows that the shadowing method is consistent with the measurement error distribution, which is not the case for the weak constraint 4DVar method. This effect is particularly evident when there are fewer observations. Moreover, when there are few observations, they have a smaller impact on unobserved variables in the shadowing method than in the weak constraint 4DVar method. Chapter 4 extends the method of Chapter 2 to other cases of partial observations, in a similar way to Chapter 3. Local convergence to a solution manifold is proved and a lower bound on an algorithmic time step is provided. Numerical experiments with the Lorenz-'63 and Lorenz-'96 models show convergence of the algorithm and further show that the method compares favorably with the weak constraint 4DVar method and another shadowing method called pseudo-orbit data assimilation. Chapter 5 further develops the method of the previous chapters. The algorithm is extended to an ensemble of states for estimating uncertainties of the algorithm, based on the concept of indistinguishable states. The chapter also includes some proofs on uniqueness, accuracy and consistency of the algorithm. The algorithm is applied to an imperfect model to show how the unmodeled components of the model can be estimated using the data assimilation algorithm

Shadowing-based data assimilation method for partially observed models

In this article we develop further an algorithm for data assimilation based upon a shadowing refinement technique [de Leeuw et al., SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 2446-2477] to take partial observations into account. Our method is based on a regularized Gauss-Newton method. We prove local convergence to the solution manifold and provide a lower bound on the algorithmic time step. We use numerical experiments with the Lorenz 63 and Lorenz 96 models to illustrate convergence of the algorithm and show that the results compare favorably with a variational technique --- weak-constraint four-dimensional variational method --- and a shadowing technique-pseudo-orbit data assimilation. Numerical experiments show that a preconditioner chosen based on a cost function allows the algorithm to find an orbit of the dynamical system in the vicinity of the true solution

Observation of WWWWWW Production in pppp Collisions at s\sqrt s =13 TeV with the ATLAS Detector

International audienceThis Letter reports the observation of $WWW$ production and a measurement of its cross section using 139 fb$^{-1}$ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from $WWW$ production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive $WWW$ production cross section is measured to be $820 \pm 100\,\text{(stat)} \pm 80\,\text{(syst)}$ fb, approximately 2.6 standard deviations from the predicted cross section of $511 \pm 18$ fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy

Observation of WWWWWW Production in pppp Collisions at s\sqrt s =13 TeV with the ATLAS Detector

International audienceThis Letter reports the observation of $WWW$ production and a measurement of its cross section using 139 fb$^{-1}$ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from $WWW$ production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive $WWW$ production cross section is measured to be $820 \pm 100\,\text{(stat)} \pm 80\,\text{(syst)}$ fb, approximately 2.6 standard deviations from the predicted cross section of $511 \pm 18$ fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy

Observation of WWWWWW Production in pppp Collisions at s\sqrt s =13 TeV with the ATLAS Detector

International audienceThis Letter reports the observation of $WWW$ production and a measurement of its cross section using 139 fb$^{-1}$ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from $WWW$ production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive $WWW$ production cross section is measured to be $820 \pm 100\,\text{(stat)} \pm 80\,\text{(syst)}$ fb, approximately 2.6 standard deviations from the predicted cross section of $511 \pm 18$ fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy

Observation of WWWWWW Production in pppp Collisions at s\sqrt s =13 TeV with the ATLAS Detector

International audienceThis Letter reports the observation of $WWW$ production and a measurement of its cross section using 139 fb$^{-1}$ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from $WWW$ production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive $WWW$ production cross section is measured to be $820 \pm 100\,\text{(stat)} \pm 80\,\text{(syst)}$ fb, approximately 2.6 standard deviations from the predicted cross section of $511 \pm 18$ fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy

Observation of WWWWWW Production in pppp Collisions at s\sqrt s =13 TeV with the ATLAS Detector

International audienceThis Letter reports the observation of $WWW$ production and a measurement of its cross section using 139 fb$^{-1}$ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from $WWW$ production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive $WWW$ production cross section is measured to be $820 \pm 100\,\text{(stat)} \pm 80\,\text{(syst)}$ fb, approximately 2.6 standard deviations from the predicted cross section of $511 \pm 18$ fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy

Observation of WWWWWW Production in pppp Collisions at s\sqrt s =13 TeV with the ATLAS Detector

International audienceThis Letter reports the observation of $WWW$ production and a measurement of its cross section using 139 fb$^{-1}$ of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from $WWW$ production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive $WWW$ production cross section is measured to be $820 \pm 100\,\text{(stat)} \pm 80\,\text{(syst)}$ fb, approximately 2.6 standard deviations from the predicted cross section of $511 \pm 18$ fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy

Observation of WWW Production in pp Collisions at p=13 TeV with the ATLAS Detector ffi s

This Letter reports the observation of WWW production and a measurement of its cross section using 139  fb^{-1} of proton-proton collision data recorded at a center-of-mass energy of 13 TeV by the ATLAS detector at the Large Hadron Collider. Events with two same-sign leptons (electrons or muons) and at least two jets, as well as events with three charged leptons, are selected. A multivariate technique is then used to discriminate between signal and background events. Events from WWW production are observed with a significance of 8.0 standard deviations, where the expectation is 5.4 standard deviations. The inclusive WWW production cross section is measured to be 820±100 (stat)±80 (syst)  fb, approximately 2.6 standard deviations from the predicted cross section of 511±18  fb calculated at next-to-leading-order QCD and leading-order electroweak accuracy