Rapid computation of far-field statistics for random obstacle scattering

In this article, we consider the numerical approximation of far-field statistics for acoustic scattering problems in the case of random obstacles. In particular, we consider the computation of the expected far-field pattern and the expected scattered wave away from the scatterer as well as the computation of the corresponding variances. To that end, we introduce an artificial interface, which almost surely contains all realizations of the random scatterer. At this interface, we directly approximate the second order statistics, i.e., the expectation and the variance, of the Cauchy data by means of boundary integral equations. From these quantities, we are able to rapidly evaluate statistics of the scattered wave everywhere in the exterior domain, including the expectation and the variance of the far-field. By employing a low-rank approximation of the Cauchy data's two-point correlation function, we drastically reduce the cost of the computation of the scattered wave's variance. Numerical results are provided in order to demonstrate the feasibility of the proposed approach

Samplets: Construction and scattered data compression

We introduce the concept of samplets by transferring the construction of Tausch-White wavelets to scattered data. This way, we obtain a multiresolution analysis tailored to discrete data which directly enables data compression, feature detection and adaptivity. The cost for constructing the samplet basis and for the fast samplet transform, respectively, is $O(N)$, where $N$ is the number of data points. Samplets with vanishing moments can be used to compress kernel matrices, arising, for instance, kernel based learning and scattered data approximation. The result are sparse matrices with only $O(N \log N )$ remaining entries. We provide estimates for the compression error and present an algorithm that computes the compressed kernel matrix with computational cost $O(N \log N )$. The accuracy of the approximation is controlled by the number of vanishing moments. Besides the cost efficient storage of kernel matrices, the sparse representation enables the application of sparse direct solvers for the numerical solution of corresponding linear systems. In addition to a comprehensive introduction to samplets and their properties, we present numerical studies to benchmark the approach. Our results demonstrate that samplets mark a considerable step in the direction of making large scattered data sets accessible for multiresolution analysis

Multilevel Quadrature for Elliptic Parametric Partial Differential Equations in Case of Polygonal Approximations of Curved Domains

Multilevel quadrature methods for parametric operator equations such as the multilevel (quasi-) Monte Carlo method resemble a sparse tensor product approximation between the spatial variable and the parameter. We employ this fact to reverse the multilevel quadrature method by applying differences of quadrature rules to finite element discretizations of increasing resolution. Besides being algorithmically more efficient if the underlying quadrature rules are nested, this way of performing the sparse tensor product approximation enables the easy use of nonnested and even adaptively refined finite element meshes. We moreover provide a rigorous error and regularity analysis addressing the variational crimes of using polygonal approximations of curved domains and numerical quadrature of the bilinear form. Our results facilitate the construction of efficient multilevel quadrature methods based on deterministic high order quadrature rules for the stochastic parameter. Numerical results in three spatial dimensions are provided to illustrate the approach

A fast direct solver for nonlocal operators in wavelet coordinates

In this article, we consider fast direct solvers for nonlocal operators. The pivotal idea is to combine a wavelet representation of the system matrix, yielding a quasi-sparse matrix, with the nested dissection ordering scheme. The latter drastically reduces the fill-in during the factorization of the system matrix by means of a Cholesky decomposition or an LU decomposition, respectively. This way, we end up with the exact inverse of the compressed system matrix with only a moderate increase of the number of nonzero entries in the matrix. To illustrate the efficacy of the approach, we conduct numerical experiments for different highly relevant applications of nonlocal operators: We consider (i) the direct solution of boundary integral equations in three spatial dimensions, issuing from the polarizable continuum model, (ii) a parabolic problem for the fractional Laplacian in integral form and (iii) the fast simulation of Gaussian random fields

Untersuchungen an Elektrolyten für Lithium-Ionen-Zellen sowie Entwicklung und Test eines computergesteuerten, modular aufgebauten, elektrochemischen Meßsystems mit Quarzmikrowaage

Lithiumbis(oxalato)borat (LiBOB) ist ein vielversprechendes Leitsalz für Lithium-Ionen-Zellen. Im Gegensatz zu Lithiumhexafluorophosphat (LiPF6), dem derzeit gebräuchlichen Leitsalz ist es fluoridfrei. Damit ist die Entstehung von HF als aggressivem Zersetzungsprodukt ausgeschlossen, was den Einsatz von kostengünstigem Manganspinell als Elektrodenmaterial ermöglicht. Über die Hydrolyse von LiBOB, speziell in organischen Lösungsmitteln ist bisher aber nur wenig bekannt. Deshalb beschäftigt sich der erste Teil der Arbeit mit der Hydrolyse von LiBOB in Acetonitril. Da sich die durchgeführten Leitfähigkeitsmessungen ohne weitere Informationen über die Zersetzungsprodukte nicht ausreichend interpretieren ließen, kamen die B11-, C13- und Protonen-NMR-Spektroskopie zum Einsatz. Durch Zusatz von Standardsubstanzen konnten die Spektren quantitativ ausgewertet werden. Dabei zeigte LiBOB eine sehr geringe Hydrolyseneigung, selbst bei Wassergehalten von 5 %. Die Reaktion läuft in ein Gleichgewicht weit auf der Eduktseite, das sich selbst bei 60°C erst nach vielen Stunden einstellt. Der zweite Teil der Arbeit behandelt die komplette Eigenentwicklung eines elektrochemischen Meßplatzes mit Quarzmikrowaage (QCM). Die einzelnen Komponenten werden über USB an einen PC angebunden und mit einer einheitlichen Steuersoftware in LabVIEW sowie in LabWindows/CVI von National Instruments bedient. Kernkomponente ist ein Potentiostat und Galvanostat, der in zwei Varianten aufgebaut wurde. Aus Transistoren diskret aufgebaute Gegentaktendstufen erlauben einen maximalen Zellstrom von 1 A bzw. 3 A. Der Spannungsbereich liegt bei +-10 V bei einer Auflösung von 305 µV. Die Schaltungstechnik der von Atmel AVR-Mikrocontrollern gesteuerten Geräte ist in der Arbeit detailliert dargestellt. Die Steuersoftware erlaubt die üblichen elektrochemischen Methoden Zyklovoltammetrie, Chronocoulometrie, Chronopotentiometrie und die stromlose Messung des Zellpotentials. Messungen an Hexacyanoferrat(II)/(III) und Hydrochinon/Chinon weisen die einwandfreie Funktion des Potentiostaten nach. Die Quarzmikrowaage, ebenfalls eine komplette Eigenentwicklung, arbeitet nach dem impedanz-scannenden Verfahren. Diese Technik wird häufig als zu kostspielig und langsam angesehen, was in der Arbeit eindrucksvoll widerlegt werden kann. Das in Frequenz und Amplitude einstellbare Anregungssignal erzeugt ein integrierter DDS-Generator. Die Quarzantwort wird mit einem TRMS-Wandler und einem A/D-Wandler erfaßt. Die Bestimmung der Serien- und Parallelresonanzfrequenz erfolgt in der PC Software. Durch die lineare Anpassung der frequenzabhängigen Impedanz an eine gebrochen rationale Funktion sind Datenraten von mehr als 1 Hz möglich. Ein Präzisionsthermometer mit einer Auflösung von bis zu 1,2 mK ergänzt den Meßplatz bei Untersuchungen der Temperaturabhängigkeit. Als Temperatursensoren dienen NTC-Thermistoren von BetaTHERM mit geringer Toleranz und hoher Langzeitstabilität. Für Messungen unter Schutzgas wurde eine Zelle konstruiert, die für kostengünstige 0,55�-Quarze ausgelegt ist. Durch Integration des Quarzes in den Zellboden konnte das notwendige Lösungsvolumen auf 13 ml reduziert werden. Zur Temperierung der Meßzelle wurden verschiedene Aufbauten erprobt. Neben der Temperaturabhängigkeit konnte der Einfluß von Longitudinalwellen auf die Resonanzfrequenz beobachtet werden. Messungen der Elektroplattierung von Kupfer in Gegenwart von Chlorid demonstrieren die Leistungsfähigkeit des elektrochemischen Meßplatzes

An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains

A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding Karhunen-Loeve expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined high-dimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problem-dependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems

An adaptive stochastic Galerkin tensor train discretization for randomly perturbed domains

A linear PDE problem for randomly perturbed domains is considered in an adaptive Galerkin framework. The perturbation of the domain's boundary is described by a vector valued random field depending on a countable number of random variables in an affine way. The corresponding Karhunen-Lo\`eve expansion is approximated by the pivoted Cholesky decomposition based on a prescribed covariance function. The examined high-dimensional Galerkin system follows from the domain mapping approach, transferring the randomness from the domain to the diffusion coefficient and the forcing. In order to make this computationally feasible, the representation makes use of the modern tensor train format for the implicit compression of the problem. Moreover, an a posteriori error estimator is presented, which allows for the problem-dependent iterative refinement of all discretization parameters and the assessment of the achieved error reduction. The proposed approach is demonstrated in numerical benchmark problems

Anisotropic multiresolution analyses for deepfake detection

Generative Adversarial Networks (GANs) have paved the path towards entirely new media generation capabilities at the forefront of image, video, and audio synthesis. However, they can also be misused and abused to fabricate elaborate lies, capable of stirring up the public debate. The threat posed by GANs has sparked the need to discern between genuine content and fabricated one. Previous studies have tackled this task by using classical machine learning techniques, such as k-nearest neighbours and eigenfaces, which unfortunately did not prove very effective. Subsequent methods have focused on leveraging on frequency decompositions, i.e., discrete cosine transform, wavelets, and wavelet packets, to preprocess the input features for classifiers. However, existing approaches only rely on isotropic transformations. We argue that, since GANs primarily utilize isotropic convolutions to generate their output, they leave clear traces, their fingerprint, in the coefficient distribution on sub-bands extracted by anisotropic transformations. We employ the fully separable wavelet transform and multiwavelets to obtain the anisotropic features to feed to standard CNN classifiers. Lastly, we find the fully separable transform capable of improving the state-of-the-art