A Performance Study of Variational Quantum Algorithms for Solving the Poisson Equation on a Quantum Computer

Recent advances in quantum computing and their increased availability has led to a growing interest in possible applications. Among those is the solution of partial differential equations (PDEs) for, e.g., material or flow simulation. Currently, the most promising route to useful deployment of quantum processors in the short to near term are so-called hybrid variational quantum algorithms (VQAs). Thus, variational methods for PDEs have been proposed as a candidate for quantum advantage in the noisy intermediate scale quantum (NISQ) era. In this work, we conduct an extensive study of utilizing VQAs on real quantum devices to solve the simplest prototype of a PDE -- the Poisson equation. Although results on noiseless simulators for small problem sizes may seem deceivingly promising, the performance on quantum computers is very poor. We argue that direct resolution of PDEs via an amplitude encoding of the solution is not a good use case within reach of today's quantum devices -- especially when considering large system sizes and more complicated non-linear PDEs that are required in order to be competitive with classical high-end solvers.Comment: 19 pages, 18 figure

A multiscale damage model for composite materials using a FFT-Based method

Modeling failure and progressive damage of composite materials presents a challenging task and is currently subject of many research activities in the field of computational mechanics. Conventional methods which assume constant material coefficients or global failure criteria, are in many cases not sufficient to predict the appropriate mechanical material response. Composite failure occurs as a result of complex mesostructural damage mechanisms and therefore it is preferable to capture these nonlinear material effects directly on a finer scale. Hence, recent multiscale modeling and simulation techniques were developed to consider the mesoscopic material behavior. In this contribution we propose an alternative multiscale approach similar to FE2. Nonlinear material effects caused by progressive damage behavior are captured on a finer length scale. The constituents are modeled explicitly and simple isotropic damage laws are used to describe the constitutive behavior. Hence, the resulting material response is based on genuine physical effects and only a few material parameters are required which can be measured directly in physical experiments. The fine scale problem (material level) is reformulated into an integral equation of Lippmann-Schwinger type and solved efficiently using the fast Fourier transformation (FFT). The calculation is carried out on a regular voxel grid which can be obtained from 3D images like tomographies without using any complicated mesh generation. Furthermore, the fine scale problem is integrated in a standard Finite Element framework which is used to solve the macroscopic BVP (component level)

Using an independent geochronology based on palaeomagnetic secular variation (PSV) and atmospheric Pb deposition to date Baltic Sea sediments and infer 14C reservoir age

Dating of sediment cores from the Baltic Sea has proven to be difficult due to uncertainties surrounding the C-14 reservoir age and a scarcity of macrofossils suitable for dating. Here we present the results of multiple dating methods carried out on cores in the Gotland Deep area of the Baltic Sea. Particular emphasis is placed on the Littorina stage (8 ka ago to the present) of the Baltic Sea and possible changes in the C-14 reservoir age of our dated samples. Three geochronological methods are used. Firstly, palaeomagnetic secular variations (PSV) are reconstructed, whereby ages are transferred to PSV features through comparison with varved lake sediment based PSV records. Secondly, lead (Pb) content and stable isotope analysis are used to identify past peaks in anthropogenic atmospheric Pb pollution. Lastly, C-14 determinations were carried out on benthic foraminifera (Elphidium spec.) samples from the brackish Littorina stage of the Baltic Sea. Determinations carried out on smaller samples (as low as 4 mu g C) employed an experimental, state-of-the-art method involving the direct measurement of CO2 from samples by a gas ion source without the need for a graphitisation step - the first time this method has been performed on foraminifera in an applied study. The PSV chronology, based on the uppermost Littorina stage sediments, produced ten age constraints between 6.29 and 1.29 cal ka BP, and the Pb depositional analysis produced two age constraints associated with the Medieval pollution peak. Analysis of PSV data shows that adequate directional data can be derived from both the present Littorina saline phase muds and Baltic Ice Lake stage varved glacial sediments. Ferrimagnetic iron sulphides, most likely authigenic greigite (Fe3S4), present in the intermediate Ancylus Lake freshwater stage sediments acquire a gyroremanent magnetisation during static alternating field (AF) demagnetisation, preventing the identification of a primary natural remanent magnetisation for these sediments. An inferred marine reservoir age offset (Delta R) is calculated by comparing the foraminifera C-14 determinations to a PSV & Pb age model. This Delta R is found to trend towards younger values upwards in the core, possibly due to a gradual change in hydrographic conditions brought about by a reduction in marine water exchange from the open sea due to continued isostatic rebound. (C) 2012 Elsevier Ltd. All rights reserved

Mikrostruktursimulation der mechanischen Deformation von Fasermaterialien

Die Deformation von porösen Natur- und Kunstfasermaterialien unter Zug-, Druck- oder Biegebelastung hängt sehr stark von den geometrischen und mechanischen Eigenschaften der verwendeten Fasern und den Eigenschaften der Faser-Faser-Kontaktstellen ab. In den betrachteten Materialien besitzen die Fasern häufig eine Orientierung, die zu elastisch anisotropen Eigenschaften führt. Um das Materialverhalten beim Herstellungsprozess und im Einsatz vorherzusagen werden in dieser Arbeit Fasernetzwerkmodelle zur Beschreibung der Mikrostruktur verwendet. Im Vergleich zu ähnlichen Verfahren werden sehr komplizierte dreidimensionale Fasernetzwerke mit einem effizienten numerischen Verfahren gelöst. Das Lösungsverfahren basiert auf einer Formulierung der Elastizitätsgleichungen als Integralgleichung vom Lippmann-Schwinger-Typ. Diese Integralgleichungen werden iterativ mit Hilfe der schnellen Fourier-Transformation (FFT) gelöst. Die Anwendung dieser Lösungstechnik auf poröse Medien ist neu. Im Vortrag werden Simulationsergebnisse für verschiedene Fasermaterialien erläutert und diese mit entsprechenden Messungen verglichen. Dabei werden geometrisch und physikalisch nichtlineare Verformungen betrachtet. Mit Hilfe der entwickelten Mikrostruktursimulationstechnik (Softwarepaket FeelMath) lässt sich die Abhängigkeit der makroskopischen Deformationseigenschaften von den Eigenschaften der Einzelfasern und der Faserorientierung analysieren. Damit kann die Anzahl der notwendigen Messungen reduziert werden und die Eigenschaften der Materialien lassen sich für den speziellen Einsatzzweck optimieren. Das vorgestellte Lösungsverfahren ist ebenfalls für nichtporöse Verbundwerkstoffe und zur Lösung von Wärmeleitproblemen in Fasernetzwerken geeignet

Mass spectrometric fragmentation patterns discriminate C1- and C4-oxidised cello-oligosaccharides from their non-oxidised and reduced forms

Lytic polysaccharide monooxygenases (LPMOs) are powerful enzymes that degrade recalcitrant polysaccharides, such as cellulose. However, the identification of LPMO-generated C1- and/or C4-oxidised oligosaccharides is far from straightforward. In particular, their fragmentation patterns have not been well established when using mass spectrometry. Hence, we studied the fragmentation behaviours of non-, C1- and C4-oxidised cello-oligosaccharides, including their sodium borodeuteride-reduced forms, by using hydrophilic interaction chromatography and negative ion mode collision induced dissociation - mass spectrometry. Non-oxidised cello-oligosaccharides showed predominantly C- and A-type cleavages. In comparison, C4-oxidised ones underwent B-/Y- and X-cleavage close to the oxidised non-reducing end, while closer to the reducing end C-/Z- and A-fragmentation predominated. C1-oxidised cello-oligosaccharides showed extensively A-cleavage. Reduced oligosaccharides showed predominant glycosidic bond cleavage, both B-/Y- and C-/Z-, close to the non-reducing end. Our findings provide signature mass spectrometric fragmentation patterns to unambiguously elucidate the catalytic behaviour and classification of LPMOs.</p

Screening of novel fungal Carbohydrate Esterase family 1 enzymes identifies three novel dual feruloyl/acetyl xylan esterases

Feruloyl esterases (FAEs) and acetyl xylan esterases (AXEs) are important enzymes for plant biomass degradation and are both present in Carbohydrate Esterase family 1 (CE1) of the Carbohydrate-Active enZymes database. In this study, ten novel fungal CE1 enzymes from different subfamilies were heterologously produced and screened for their activity towards model and complex plant biomass substrates. CE1_1 enzymes possess AXE activity, while CE1_5 enzymes showed FAE activity. Two enzymes from CE1_2 and one from CE1_5 possess dual feruloyl/acetyl xylan esterase (FXE) activity, showing expansion of substrate specificity. The new FXEs from CE1 can efficiently release both feruloyl and acetyl residues from feruloylated xylan, making them particularly interesting novel components of industrial enzyme cocktails for plant biomass degradation

AutoMat -- Automatic Differentiation for Generalized Standard Materials on GPUs

We propose a universal method for the evaluation of generalized standard materials that greatly simplifies the material law implementation process. By means of automatic differentiation and a numerical integration scheme, AutoMat reduces the implementation effort to two potential functions. By moving AutoMat to the GPU, we close the performance gap to conventional evaluation routines and demonstrate in detail that the expression level reverse mode of automatic differentiation as well as its extension to second order derivatives can be applied inside CUDA kernels. We underline the effectiveness and the applicability of AutoMat by integrating it into the FFT-based homogenization scheme of Moulinec and Suquet and discuss the benefits of using AutoMat with respect to runtime and solution accuracy for an elasto-viscoplastic example.Comment: 28 pages, 15 figures, 7 tables; new layout, more detailed proof of Theorem

Numerische Lösung der stationären Diffusionsgleichung mit Homogenisierungsmethoden

For elliptic partial differential equations with periodically oscillating coefficients quadratic L2-convergence of a corrected asymptotic expansion, which is motivated by the theory of homogenization, is proven in the one-dimensional case. In the two-dimensional case the rate of convergence and its dependency on the symmetry of the diffusion coefficient is numerically analysed. The correction of the asymptotic expansion on a locally refined grid is then embedded inside a two-grid method and numerically compared with a classical PCG-method

The Lippmann-Schwinger equation in elasticity for porous media

The numerical computation of effective elastic properties of heterogeneous materials using the Lippmann-Schwinger equation in elasticity becomes problematic in the presence of pores. Under mild conditions on the interface of the pore the Lippmann-Schwinger equation is well-posed, but ill-conditioned. Introducing a fictitious soft material renders the problem well-conditioned again. We will discuss the exact rate of convergence of these regularized problems to the porous problem