Resolving the details of an object from coarse-scale measurements is a classical problem in applied mathematics. This problem is usually formulated as extrapolating the Fourier transform of the object from a bounded region to the entire space, that is, in terms of performing extrapolation in frequency. This problem is ill-posed unless one assumes that the object has some additional structure. When the object is compactly supported, then it is well-known that its Fourier transform can be extended to the entire space. However, it is also well-known that this problem is severely ill-conditioned. In this work, we assume that the object is known to belong to a collection of compactly supported functions and, instead performing extrapolation in frequency to the entire space, we study the problem of extrapolating to a larger bounded set using dilations in frequency and a single Fourier multiplier. This is reminiscent of the refinement equation in multiresolution analysis. Under suitable conditions, we prove the existence of a worst-case optimal multiplier over the entire collection, and we show that all such multipliers share the same canonical structure. When the collection is finite, we show that any worst-case optimal multiplier can be represented in terms of an Hermitian matrix. This allows us to introduce a fixed-point iteration to find the optimal multiplier. This leads us to introduce a family of multipliers, which we call \Sigma-multipliers, that can be used to perform extrapolation in frequency. We establish connections between \Sigma-multipliers and multiresolution analysis. We conclude with some numerical experiments illustrating the practical consequences of our results.

"Chemical Shift Imaging (CSI) or Chemical Shift Encoded Magnetic Resonance Imaging (CSE-MRI) enables the quantification of different chemical species in the human body, and it is one of the most widely used imaging modalities used to quantify fat in the human body. Although there have been substantial improvements in the design of signal acquisition protocols and the development of a variety of methods for the recovery of parameters of interest from the measured signal, it is still challenging to obtain a consistent and reliable quantification over the entire field of view. In fact, there are still discrepancies in the quantities recovered by different methods, and each exhibits a different degree of sensitivity to acquisition parameters such as the choice of echo times. Some of these challenges have their origin in the signal model itself. In particular, it is non-linear, and there may be different sets of parameters of interest compatible with the measured signal. For this reason, a thorough analysis of this model may help mitigate some of the remaining challenges, and yield insight into novel acquisition protocols. In this work, we perform an analysis of the signal model underlying CSI, focusing on finding suitable conditions under which recovery of the parameters of interest is possible. We determine the sources of non-identifiability of the parameters, and we propose a reconstruction method based on smooth non-convex optimization under convex constraints that achieves exact local recovery under suitable conditions. A surprising result is that the concentrations of the chemical species in the sample may be identifiable even when other parameters are not. We present numerical results illustrating how our theoretical results may help develop novel acquisition techniques, and showing how our proposed recovery method yields results comparable to the state-of-the-art. "

Large-scale nonlinear dynamical systems often involve several interacting components, and parameters must be estimated from experimental data. While identifying key components in a complex dynamical system enables model reduction to accelerate simulations, care must be taken when accounting for the uncertainty in the parameters used to find such reduced models. In this talk, we present a data-driven method for model reduction using l1-regularization that involves minimal parameterization, that is interpretable, and that generates a family of models which trade-off model complexity for estimation error. Then, we introduce Tikhonov regularization under moment constraints as form of a maximum a posteriori (MAP) estimate for a probability density on unknown parameters induced by a suitable Bayesian nonparametric method. We show that the MAP can be computed by solving a finite-dimensional problem which, in several cases of interest, is convex, unconstrained, and has a smooth objective function. Combining both, we are able to both find interpretable reduced models, and quantify the uncertainty in their parameters. We present experimental results showing that, on one hand, our model reduction method selects reduced models with good extrapolation properties, which is an important consideration in practical applications; on the other, that using Tikhonov regularization under moment constraints may be an efficient approach to quantify the uncertainty on model parameters.

Estimation of cardiovascular model parameters from electronic health records (EHRs) poses a significant challenge primarily due to lack of identifiability. Structural non-identifiability arises when a manifold in the space of parameters is mapped to a common output, while practical non-identifiability can result due to limited data, model misspecification or noise corruption. To address the resulting ill-posed inverse problem, optimization-based or Bayesian inference approaches typically use regularization, thereby limiting the possibility of discovering multiple solutions. In this study, we use inVAErt networks, a neural network-based, data-driven framework for enhanced digital twin analysis of stiff dynamical systems. We demonstrate the flexibility and effectiveness of inVAErt networks in the context of physiological inversion of a six-compartment lumped‐parameter haemodynamic model from synthetic data to real data with missing components. This article is part of the theme issue ‘Uncertainty quantification for healthcare and biological systems (Part 2)’.

Publisher:  Computer Methods in Applied Mechanics and Engineering  Link>

ABSTRACT

Sparse signal recovery has become one of the preferred methods to recover signals from a set of incomplete linear measurements. This is due both to its appealing computational properties, as it involves solving a convex optimization problem, and its rigorous justification by the theory of Compressed Sensing. When the underlying signal is not sparse, but it is instead a sparse combination of elementary building blocks called atoms, the signal can be recovered by minimizing the atomic norm, i.e., the gauge associated to the convex hull of the atomic set. Although this approach has been successfully used in several applications, there is an implicit geometric constraint in this approach: only the atoms that are exposed points of the convex hull will be selected to represent the solution to atomic norm minimization. This can be an issue when the representation of the underlying signal is sparse when using all the atoms, but dense when using exposed ones. In this work, we propose an approach based on lifting that allows us to promote representations using atoms that are not exposed. Our method is based on convex optimization, preserving many of the computational benefits of atomic norm minimization. We present phase diagrams derived from a suitable signal model showing the benefits of using our approach.

Resolving the fine-scale details of a signal from coarse-scale measurements is a classical problem in signal processing. This problem is usually formulated in terms of extrapolation in frequency, i.e., as extrapolating the Fourier transform of the signal from a set of low-frequencies to a larger set. An approach to perform extrapolation in frequency is to use a multiplier, or a filter, that minimizes a suitable approximation error metric over a known collection of signals. However, one of the drawbacks of this approach is that this multiplier is not able to exploit the relations between the signals in the collection. In this work, we propose a formulation that is translation-invariant, finding both the optimal multipliers and the optimal centering for the signals in the collection. A consequence of our formulation is that the optimal centering does not correspond to a usual choice such as the center of mass. We perform numerical experiments supporting our claims.