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- ItemStatistical parametric maps for functional MRI experiments in R: The package fmri(Los Angeles : UCLA, 2011) Tabelow, K.; Polzehl, J.The purpose of the package fmri is the analysis of single subject functional magnetic resonance imaging (fMRI) data. It provides fMRI analysis from time series modeling by a linear model to signal detection and publication quality images. Specifically, it implements structural adaptive smoothing methods with signal detection for adaptive noise reduction which avoids blurring of activation areas. Within this paper we describe the complete pipeline for fMRI analysis using fmri. We describe data reading from various medical imaging formats and the linear modeling used to create the statistical parametric maps. We review the rationale behind the structural adaptive smoothing algorithms and explain their usage from the package fmri. We demonstrate the results of such analysis using two experimental datasets. Finally, we report on the usage of a graphical user interface for some of the package functions.
- ItemLow-rank tensor reconstruction of concentrated densities with application to Bayesian inversion(Dordrecht [u.a.] : Springer Science + Business Media B.V, 2022) Eigel, Martin; Gruhlke, Robert; Marschall, ManuelThis paper presents a novel method for the accurate functional approximation of possibly highly concentrated probability densities. It is based on the combination of several modern techniques such as transport maps and low-rank approximations via a nonintrusive tensor train reconstruction. The central idea is to carry out computations for statistical quantities of interest such as moments based on a convenient representation of a reference density for which accurate numerical methods can be employed. Since the transport from target to reference can usually not be determined exactly, one has to cope with a perturbed reference density due to a numerically approximated transport map. By the introduction of a layered approximation and appropriate coordinate transformations, the problem is split into a set of independent approximations in seperately chosen orthonormal basis functions, combining the notions h- and p-refinement (i.e. “mesh size” and polynomial degree). An efficient low-rank representation of the perturbed reference density is achieved via the Variational Monte Carlo method. This nonintrusive regression technique reconstructs the map in the tensor train format. An a priori convergence analysis with respect to the error terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback–Leibler divergence is derived. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a main motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity and degrees of perturbation of the transport to the reference density. The (superior) convergence is demonstrated in comparison to Monte Carlo and Markov Chain Monte Carlo methods.
- ItemVariable Step Mollifiers and Applications(Berlin ; Heidelberg : Springer, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections. © 2020, The Author(s).
- ItemDrift–diffusion simulation of S-shaped current–voltage relations for organic semiconductor devices(Dordrecht : Springer Science + Business Media B.V., 2020) Doan, Duy Hai; Fischer, Axel; Fuhrmann, Jürgen; Glitzky, Annegret; Liero, MatthiasWe present an electrothermal drift–diffusion model for organic semiconductor devices with Gauss–Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature-dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter–Gummel scheme. Using path-following techniques, we demonstrate that the model exhibits S-shaped current–voltage curves with regions of negative differential resistance, which were only recently observed experimentally. © 2020, The Author(s).
- ItemHigh order discretization methods for spatial-dependent epidemic models(Amsterdam [u.a.] : Elsevier Science, 2022) Takács, Bálint; Hadjimichael, YiannisIn this paper, an epidemic model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of partial-differential equations with integral terms. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different spatial and temporal discretizations are employed, and step-size restrictions for the discrete model’s positivity, monotonicity preservation, and population conservation are investigated. We provide sufficient conditions under which high-order numerical schemes preserve the stability of the computational process and provide sufficiently accurate numerical approximations. Computational experiments verify the convergence and accuracy of the numerical methods.
- ItemA function space framework for structural total variation regularization with applications in inverse problems(Bristol [u.a.] : Inst., 2018) Hintermüller, Michael; Holler, Martin; Papafitsoros, KostasIn this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semi-continuous envelope (relaxation) of a suitable TV type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted TV for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddle-point problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson log-likelihood data discrepancy terms. Finally, we provide proof-of-concept numerical examples where we solve the saddle-point problem for weighted TV denoising as well as for MR guided PET image reconstruction.
- ItemInverse learning in Hilbert scales(Dordrecht [u.a.] : Springer Science + Business Media B.V, 2023) Rastogi, Abhishake; Mathé, PeterWe study linear ill-posed inverse problems with noisy data in the framework of statistical learning. The corresponding linear operator equation is assumed to fit a given Hilbert scale, generated by some unbounded self-adjoint operator. Approximate reconstructions from random noisy data are obtained with general regularization schemes in such a way that these belong to the domain of the generator. The analysis has thus to distinguish two cases, the regular one, when the true solution also belongs to the domain of the generator, and the ‘oversmoothing’ one, when this is not the case. Rates of convergence for the regularized solutions will be expressed in terms of certain distance functions. For solutions with smoothness given in terms of source conditions with respect to the scale generating operator, then the error bounds can then be made explicit in terms of the sample size.
- ItemReconstruction of Quasi-Local Numerical Effective Models from Low-Resolution Measurements(New York, NY [u.a.] : Springer Science + Business Media B.V., 2020) Caiazzo, A.; Maier, R.; Peterseim, D.We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on coarse measurements. The approach is motivated by quasi-local numerical effective forward models that are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that an identification of the matrix representation related to these effective models is possible. On the one hand, this provides a reasonable surrogate in cases where a direct reconstruction is unfeasible due to a mismatch between the coarse data scale and the microscopic quantities to be reconstructed. On the other hand, the approach allows us to investigate the requirement for a certain non-locality in the context of numerical homogenization. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments. © 2020, The Author(s).