Ancestry-specific proteome-wide association studies (PWAS) based on genetically predicted protein expression can reveal complex disease etiology specific to certain ancestral groups. These studies require ancestry-specific models for protein expression as a function of SNP genotypes. In order to improve protein expression prediction in ancestral populations historically underrepresented in genomic studies, we propose a new penalized maximum likelihood estimator for fitting ancestry-specific joint protein quantitative trait loci models. Our estimator borrows information across ancestral groups, while simultaneously allowing for heterogeneous error variances and regression coefficients. We propose an alternative parameterization of our model which makes the objective function convex and the penalty scale invariant. To improve computational efficiency, we propose an approximate version of our method and study its theoretical properties. Our method provides a substantial improvement in protein expression prediction accuracy in individuals of African ancestry, and in a downstream PWAS analysis, leads to the discovery of multiple associations between protein expression and blood lipid traits in the African ancestry population.
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Multiscale Finite Element Methods (MsFEMs) are now well-established finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions that generate a suitable discretization space, and next perform a Galerkin approximation of the problem on that space. We investigate here how these approaches can be implemented in a non-intrusive way, in order to facilitate their dissemination within industrial codes or non-academic environments. We develop an abstract framework that covers a wide variety of MsFEMs for linear second-order partial differential equations. Non-intrusive MsFEM approaches are developed within the full generality of this framework, which may moreover be beneficial to steering software development and improving the theoretical understanding and analysis of MsFEMs.
We examine the last-iterate convergence rate of Bregman proximal methods - from mirror descent to mirror-prox and its optimistic variants - as a function of the local geometry induced by the prox-mapping defining the method. For generality, we focus on local solutions of constrained, non-monotone variational inequalities, and we show that the convergence rate of a given method depends sharply on its associated Legendre exponent, a notion that measures the growth rate of the underlying Bregman function (Euclidean, entropic, or other) near a solution. In particular, we show that boundary solutions exhibit a stark separation of regimes between methods with a zero and non-zero Legendre exponent: the former converge at a linear rate, while the latter converge, in general, sublinearly. This dichotomy becomes even more pronounced in linearly constrained problems where methods with entropic regularization achieve a linear convergence rate along sharp directions, compared to convergence in a finite number of steps under Euclidean regularization.
The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model problem consisting of the Poisson equation subject to non-standard boundary conditions. Namely, instead of classical boundary conditions, the model problem involves Dirichlet- and Neumann-type nonlocal boundary conditions. Variational formulations with strongly and weakly imposed inhomogeneous Dirichlet-type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non-uniform rational B-splines (NURBS). The attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods.
A non-intrusive model order reduction (MOR) method that combines features of the dynamic mode decomposition (DMD) and the radial basis function (RBF) network is proposed to predict the dynamics of parametric nonlinear systems. In many applications, we have limited access to the information of the whole system, which motivates non-intrusive model reduction. One bottleneck is capturing the dynamics of the solution without knowing the physics inside the "black-box" system. DMD is a powerful tool to mimic the dynamics of the system and give a reliable approximation of the solution in the time domain using only the dominant DMD modes. However, DMD cannot reproduce the parametric behavior of the dynamics. Our contribution focuses on extending DMD to parametric DMD by RBF interpolation. Specifically, a RBF network is first trained using snapshot matrices at limited parameter samples. The snapshot matrix at any new parameter sample can be quickly learned from the RBF network. DMD will use the newly generated snapshot matrix at the online stage to predict the time patterns of the dynamics corresponding to the new parameter sample. The proposed framework and algorithm are tested and validated by numerical examples including models with parametrized and time-varying inputs.
We propose a many-sorted modal logic for reasoning about knowledge in multi-agent systems. Our logic introduces a clear distinction between participating agents and the environment. This allows to express local properties of agents and global properties of worlds in a uniform way, as well as to talk about the presence or absence of agents in a world. The logic subsumes the standard epistemic logic and is a conservative extension of it. The semantics is given in chromatic hypergraphs, a generalization of chromatic simplicial complexes, which were recently used to model knowledge in distributed systems. We show that the logic is sound and complete with respect to the intended semantics. We also show a further connection of chromatic hypergraphs with neighborhood frames.
A full approximation scheme (FAS) nonlinear multigrid solver for two-phase flow and transport problems driven by wells with multiple perforations is developed. It is an extension to our previous work on FAS solvers for diffusion and transport problems. The solver is applicable to discrete problems defined on unstructured grids as the coarsening algorithm is aggregation-based and algebraic. To construct coarse basis that can better capture the radial flow near wells, coarse grids in which perforated well cells are not near the coarse-element interface are desired. This is achieved by an aggregation algorithm proposed in this paper that makes use of the location of well cells in the cell-connectivity graph. Numerical examples in which the FAS solver is compared against Newton's method on benchmark problems are given. In particular, for a refined version of the SAIGUP model, the FAS solver is at least 35% faster than Newton's method for time steps with a CFL number greater than 10.
This article introduces a new Neural Network stochastic model to generate a 1-dimensional stochastic field with turbulent velocity statistics. Both the model architecture and training procedure ground on the Kolmogorov and Obukhov statistical theories of fully developed turbulence, so guaranteeing descriptions of 1) energy distribution, 2) energy cascade and 3) intermittency across scales in agreement with experimental observations. The model is a Generative Adversarial Network with multiple multiscale optimization criteria. First, we use three physics-based criteria: the variance, skewness and flatness of the increments of the generated field that retrieve respectively the turbulent energy distribution, energy cascade and intermittency across scales. Second, the Generative Adversarial Network criterion, based on reproducing statistical distributions, is used on segments of different length of the generated field. Furthermore, to mimic multiscale decompositions frequently used in turbulence's studies, the model architecture is fully convolutional with kernel sizes varying along the multiple layers of the model. To train our model we use turbulent velocity signals from grid turbulence at Modane wind tunnel.
Purpose: Disease progression of retinal atrophy associated with AMD requires the accurate quantification of the retinal atrophy changes on longitudinal OCT studies. It is based on finding, comparing, and delineating subtle atrophy changes on consecutive pairs (prior and current) of unregistered OCT scans. Methods: We present a fully automatic end-to-end pipeline for the simultaneous detection and quantification of time-related atrophy changes associated with dry AMD in pairs of OCT scans of a patient. It uses a novel simultaneous multi-channel column-based deep learning model trained on registered pairs of OCT scans that concurrently detects and segments retinal atrophy segments in consecutive OCT scans by classifying light scattering patterns in matched pairs of vertical pixel-wide columns (A-scans) in registered prior and current OCT slices (B-scans). Results: Experimental results on 4,040 OCT slices with 5.2M columns from 40 scans pairs of 18 patients (66% training/validation, 33% testing) with 24.13+-14.0 months apart in which Complete RPE and Outer Retinal Atrophy (cRORA) was identified in 1,998 OCT slices (735 atrophy lesions from 3,732 segments, 0.45M columns) yield a mean atrophy segments detection precision, recall of 0.90+-0.09, 0.95+-0.06 and 0.74+-0.18, 0.94+-0.12 for atrophy lesions with AUC=0.897, all above observer variability. Simultaneous classification outperforms standalone classification precision and recall by 30+-62% and 27+-0% for atrophy segments and lesions. Conclusions: simultaneous column-based detection and quantification of retinal atrophy changes associated with AMD is accurate and outperforms standalone classification methods. Translational relevance: an automatic and efficient way to detect and quantify retinal atrophy changes associated with AMD.
Conformal prediction (CP) is a framework to quantify uncertainty of machine learning classifiers including deep neural networks. Given a testing example and a trained classifier, CP produces a prediction set of candidate labels with a user-specified coverage (i.e., true class label is contained with high probability). Almost all the existing work on CP assumes clean testing data and there is not much known about the robustness of CP algorithms w.r.t natural/adversarial perturbations to testing examples. This paper studies the problem of probabilistically robust conformal prediction (PRCP) which ensures robustness to most perturbations around clean input examples. PRCP generalizes the standard CP (cannot handle perturbations) and adversarially robust CP (ensures robustness w.r.t worst-case perturbations) to achieve better trade-offs between nominal performance and robustness. We propose a novel adaptive PRCP (aPRCP) algorithm to achieve probabilistically robust coverage. The key idea behind aPRCP is to determine two parallel thresholds, one for data samples and another one for the perturbations on data (aka "quantile-of-quantile" design). We provide theoretical analysis to show that aPRCP algorithm achieves robust coverage. Our experiments on CIFAR-10, CIFAR-100, and ImageNet datasets using deep neural networks demonstrate that aPRCP achieves better trade-offs than state-of-the-art CP and adversarially robust CP algorithms.
We consider parametrized linear-quadratic optimal control problems and provide their online-efficient solutions by combining greedy reduced basis methods and machine learning algorithms. To this end, we first extend the greedy control algorithm, which builds a reduced basis for the manifold of optimal final time adjoint states, to the setting where the objective functional consists of a penalty term measuring the deviation from a desired state and a term describing the control energy. Afterwards, we apply machine learning surrogates to accelerate the online evaluation of the reduced model. The error estimates proven for the greedy procedure are further transferred to the machine learning models and thus allow for efficient a posteriori error certification. We discuss the computational costs of all considered methods in detail and show by means of two numerical examples the tremendous potential of the proposed methodology.
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