286 research outputs found
A multi-resolution, non-parametric, Bayesian framework for identification of spatially-varying model parameters
This paper proposes a hierarchical, multi-resolution framework for the
identification of model parameters and their spatially variability from noisy
measurements of the response or output. Such parameters are frequently
encountered in PDE-based models and correspond to quantities such as density or
pressure fields, elasto-plastic moduli and internal variables in solid
mechanics, conductivity fields in heat diffusion problems, permeability fields
in fluid flow through porous media etc. The proposed model has all the
advantages of traditional Bayesian formulations such as the ability to produce
measures of confidence for the inferences made and providing not only
predictive estimates but also quantitative measures of the predictive
uncertainty. In contrast to existing approaches it utilizes a parsimonious,
non-parametric formulation that favors sparse representations and whose
complexity can be determined from the data. The proposed framework in
non-intrusive and makes use of a sequence of forward solvers operating at
various resolutions. As a result, inexpensive, coarse solvers are used to
identify the most salient features of the unknown field(s) which are
subsequently enriched by invoking solvers operating at finer resolutions. This
leads to significant computational savings particularly in problems involving
computationally demanding forward models but also improvements in accuracy. It
is based on a novel, adaptive scheme based on Sequential Monte Carlo sampling
which is embarrassingly parallelizable and circumvents issues with slow mixing
encountered in Markov Chain Monte Carlo schemes
Cell Detection by Functional Inverse Diffusion and Non-negative Group SparsityPart II: Proximal Optimization and Performance Evaluation
In this two-part paper, we present a novel framework and methodology to
analyze data from certain image-based biochemical assays, e.g., ELISPOT and
Fluorospot assays. In this second part, we focus on our algorithmic
contributions. We provide an algorithm for functional inverse diffusion that
solves the variational problem we posed in Part I. As part of the derivation of
this algorithm, we present the proximal operator for the non-negative
group-sparsity regularizer, which is a novel result that is of interest in
itself, also in comparison to previous results on the proximal operator of a
sum of functions. We then present a discretized approximated implementation of
our algorithm and evaluate it both in terms of operational cell-detection
metrics and in terms of distributional optimal-transport metrics.Comment: published, 16 page
Scalable CAIM Discretization on Multiple GPUs Using Concurrent Kernels
CAIM(Class-Attribute InterdependenceMaximization) is one of the stateof-
the-art algorithms for discretizing data for which classes are known. However, it
may take a long time when run on high-dimensional large-scale data, with large number
of attributes and/or instances. This paper presents a solution to this problem by
introducing a GPU-based implementation of the CAIM algorithm that significantly
speeds up the discretization process on big complex data sets. The GPU-based implementation
is scalable to multiple GPU devices and enables the use of concurrent
kernels execution capabilities ofmodernGPUs. The CAIMGPU-basedmodel is evaluated
and compared with the original CAIM using single and multi-threaded parallel
configurations on 40 data sets with different characteristics. The results show great
speedup, up to 139 times faster using 4 GPUs, which makes discretization of big
data efficient and manageable. For example, discretization time of one big data set is
reduced from 2 hours to less than 2 minute
Inference on Riemannian Manifolds: Regression and Stochastic Differential Equations
Statistical inference for manifolds attracts much attention because of its power of working with more general forms of data or geometric objects. We study regression and stochastic differential equations on manifolds from the intrinsic point of view.
Firstly, we are able to provide alternative parametrizations for data that lie on Lie group in the problem of fitting a regression model, by mapping this space intrinsically onto its Lie algebra, while we explore the behaviour of fitted values when this base point is chosen differently. Due to the nature of our data in the application of soft tissue artefacts, we employ two correlation structures, namely Matern and quasi-periodic correlation functions when using the generalized least squares, and show that some patterns of the residuals are removed.
Secondly, we construct a generalization of the Ornstein-Uhlenbeck process on the cone of covariance matrices SP(n) endowed with two popular Riemannian metrics, namely Log-Euclidean (LE) and Affine-Invariant (AI) metrics. We show that the Riemannian Brownian motion on SP(n) has infinite explosion time as on the Euclidean space and establish the calculation for the horizontal lifts of smooth curves. Moreover, we provide Bayesian inference for discretely observed diffusion processes of covariance matrices associated with either the LE or the AI metrics, and present a novel diffusion bridge sampling method using guided proposals when equipping SP(n) with the AI metric. The estimation algorithms are illustrated with an application in finance, together with a goodness-of-fit test comparing models associated with different metrics. Furthermore, we explore the multivariate volatility models via simulation study, in which covariance matrices in the models are assumed to be unobservable
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