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Register a new userAdaptive PBDW approach to state estimation: noisy observations; user-defined update spaces
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We provide a number of extensions and further interpretations of the Parameterized-Background Data-Weak (PBDW) formulation, a real-time and in-situ Data Assimilation (DA) framework for physical systems modeled by parametrized Partial Differential Equations (PDEs), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. Given $M$ noisy measurements of the state, PBDW seeks an approximation of the form $u^{star} = z^{star} + eta^{star}$, where the emph{background} $z^{star}$ belongs to a $N$-dimensional emph{background space} informed by a parameterized mathematical model, and the emph{update} $eta^{star}$ belongs to a $M$-dimensional emph{update space} informed by the experimental observations. The contributions of the present work are threefold: first, we extend the adaptive formulation proposed in [T Taddei, M2AN, 51(5), 1827-1858] to general linear observation functionals, to effectively deal with noisy observations; second, we consider an user-defined choice of the update space, to improve convergence with respect to the number of measurements; third, we propose an emph{a priori} error analysis for general linear functionals in the presence of noise, to identify the different sources of state estimation error and ultimately motivate the adaptive procedure. We present results for two synthetic model problems in Acoustics, to illustrate the elements of the methodology and to prove its effectiveness. We further present results for a synthetic problem in Fluid Mechanics to demonstrate the applicability of the approach to vector-valued fields.
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We present an error analysis and further numerical investigations of the Parameterized-Background Data-Weak (PBDW) formulation to variational Data Assimilation (state estimation), proposed in [Y Maday, AT Patera, JD Penn, M Yano, Int J Numer Meth Eng, 102(5), 933-965]. The PBDW algorithm is a state estimation method involving reduced models. It aims at approximating an unknown function $u^{rm true}$ living in a high-dimensional Hilbert space from $M$ measurement observations given in the form $y_m = ell_m(u^{rm true}),, m=1,dots,M$, where $ell_m$ are linear functionals. The method approximates $u^{rm true}$ with $hat{u} = hat{z} + hat{eta}$. The emph{background} $hat{z}$ belongs to an $N$-dimensional linear space $mathcal{Z}_N$ built from reduced modelling of a parameterized mathematical model, and the emph{update} $hat{eta}$ belongs to the space $mathcal{U}_M$ spanned by the Riesz representers of $(ell_1,dots, ell_M)$. When the measurements are noisy {--- i.e., $y_m = ell_m(u^{rm true})+epsilon_m$ with $epsilon_m$ being a noise term --- } the classical PBDW formulation is not robust in the sense that, if $N$ increases, the reconstruction accuracy degrades. In this paper, we propose to address this issue with an extension of the classical formulation, {which consists in} searching for the background $hat{z}$ either on the whole $mathcal{Z}_N$ in the noise-free case, or on a well-chosen subset $mathcal{K}_N subset mathcal{Z}_N$ in presence of noise. The restriction to $mathcal{K}_N$ makes the reconstruction be nonlinear and is the key to make the algorithm significantly more robust against noise. We {further} present an emph{a priori} error and stability analysis, and we illustrate the efficiency of the approach on several numerical examples.
This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations and corresponding sparse approximations of lower-dimensional tensor components are determined adaptively. A principal obstruction to a simultaneous control of rank growth and accuracy turns out to be the fact that the underlying elliptic operator is an isomorphism only between spaces that are not endowed with cross norms. Therefore, as central part of this scheme, we devise a method for preconditioning low-rank tensor representations of operators. Under standard assumptions on the data, we establish convergence to the solution of the continuous problem with a guaranteed error reduction. Moreover, for the case that the solution exhibits a certain low-rank structure and representation sparsity, we derive bounds on the computational complexity, including in particular bounds on the tensor ranks that can arise during the iteration. We emphasize that such assumptions on the solution do not enter in the formulation of the scheme, which in fact is shown to detect them automatically. Our findings are illustrated by numerical experiments that demonstrate the practical efficiency of the method in high spatial dimensions.
In this paper, we give estimators of the frequency, amplitude and phase of a noisy sinusoidal signal with time-varying amplitude by using the algebraic parametric techniques introduced by Fliess and Sira-Ramirez. We apply a similar strategy to estimate these parameters by using modulating functions method. The convergence of the noise error part due to a large class of noises is studied to show the robustness and the stability of these methods. We also show that the estimators obtained by modulating functions method are robust to large sampling period and to non zero-mean noises.
Attribute editing has become an important and emerging topic of computer vision. In this paper, we consider a task: given a reference garment image A and another image B with target attribute (collar/sleeve), generate a photo-realistic image which combines the texture from reference A and the new attribute from reference B. The highly convoluted attributes and the lack of paired data are the main challenges to the task. To overcome those limitations, we propose a novel self-supervised model to synthesize garment images with disentangled attributes (e.g., collar and sleeves) without paired data. Our method consists of a reconstruction learning step and an adversarial learning step. The model learns texture and location information through reconstruction learning. And, the models capability is generalized to achieve single-attribute manipulation by adversarial learning. Meanwhile, we compose a new dataset, named GarmentSet, with annotation of landmarks of collars and sleeves on clean garment images. Extensive experiments on this dataset and real-world samples demonstrate that our method can synthesize much better results than the state-of-the-art methods in both quantitative and qualitative comparisons.
We consider the inverse problem of estimating the spatially varying pulse wave velocity in blood vessels in the brain from dynamic MRI data, as it appears in the recently proposed imaging technique of Magnetic Resonance Advection Imaging (MRAI). The problem is formulated as a linear operator equation with a noisy operator and solved using a conjugate gradient type approach. Numerical examples on experimental data show the usefulness and advantages of the developed algorithm in comparison to previously proposed methods.
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