78 research outputs found
Numerical homogenization of elliptic PDEs with similar coefficients
We consider a sequence of elliptic partial differential equations (PDEs) with
different but similar rapidly varying coefficients. Such sequences appear, for
example, in splitting schemes for time-dependent problems (with one coefficient
per time step) and in sample based stochastic integration of outputs from an
elliptic PDE (with one coefficient per sample member). We propose a
parallelizable algorithm based on Petrov-Galerkin localized orthogonal
decomposition (PG-LOD) that adaptively (using computable and theoretically
derived error indicators) recomputes the local corrector problems only where it
improves accuracy. The method is illustrated in detail by an example of a
time-dependent two-pase Darcy flow problem in three dimensions
Numerical upscaling for heterogeneous materials in fractured domains
We consider numerical solution of elliptic problems with heterogeneous
diffusion coefficients containing thin highly conductive structures. Such
problems arise e.g. in fractured porous media, reinforced materials, and
electric circuits. The main computational challenge is the high resolution
needed to resolve the data variation. We propose a multiscale method that
models the thin structures as interfaces and incorporate heterogeneities in
corrected shape functions. The construction results in an accurate upscaled
representation of the system that can be used to solve for several forcing
functions or to simulate evolution problems in an efficient way. By introducing
a novel interpolation operator, defining the fine scale of the problem, we
prove exponential decay of the shape functions which allows for a sparse
approximation of the upscaled representation. An a priori error bound is also
derived for the proposed method together with numerical examples that verify
the theoretical findings. Finally we present a numerical example to show how
the technique can be applied to evolution problems
Numerical upscaling of perturbed diffusion problems
In this paper we study elliptic partial differential equations with rapidly
varying diffusion coefficient that can be represented as a perturbation of a
reference coefficient. We develop a numerical method for efficiently solving
multiple perturbed problems by reusing local computations performed with the
reference coefficient. The proposed method is based on the Petrov--Galerkin
Localized Orthogonal Decomposition (PG-LOD) which allows for straightforward
parallelization with low communcation overhead and memory consumption. We focus
on two types of perturbations: local defects which we treat by recomputation of
multiscale shape functions and global mappings of a reference coefficient for
which we apply the domain mapping method. We analyze the proposed method for
these problem classes and present several numerical examples
Iterative solution of spatial network models by subspace decomposition
We present and analyze a preconditioned conjugate gradient method (PCG) for
solving spatial network problems. Primarily, we consider diffusion and
structural mechanics simulations for fiber based materials, but the methodology
can be applied to a wide range of models, fulfilling a set of abstract
assumptions. The proposed method builds on a classical subspace decomposition
into a coarse subspace, realized as the restriction of a finite element space
to the nodes of the spatial network, and localized subspaces with support on
mesh stars. The main contribution of this work is the convergence analysis of
the proposed method. The analysis translates results from finite element
theory, including interpolation bounds, to the spatial network setting. A
convergence rate of the PCG algorithm, only depending on global bounds of the
operator and homogeneity, connectivity and locality constants of the network,
is established. The theoretical results are confirmed by several numerical
experiments.Comment: Journal article draft, not peer-reviewe
Study and Comparsion of Data Lakehouse Systems
This thesis presents a comprehensive study and comparative analysis of three distinct data lakehouse systems: Delta Lake, Apache Iceberg, and Apache Hudi. Data lakehouse systems are an emergent concept that combines the capabilities of data warehouses and data lakes to provide a unified platform for large-scale data management and analysis.
Three experimental scenarios were conducted focusing on data ingestion, query performance, and scaling, each assessing a different aspect of the system’s capabilities. The results show that each data lakehouse system possesses unique strengths and weaknesses: Apache Iceberg demonstrated the best data ingestion speed, Delta Lake exhibited consistent performance across all testing scenarios, while Apache Hudi excelled with smaller datasets.
Furthermore, the study also considered the ease of implementation and use for each system. Apache Iceberg emerged as the most user-friendly, with comprehensive documentation. Delta Lake provided a slightly steeper learning curve, while Apache Hudi was the most challenging to implement.
This study underscores the promising potential of data lakehouses as alternatives to traditional database architectures. However, further research is necessary to solidify the positioning of data lakehouses as the new generation of database architectures
Nanofluidic Trapping of Faceted Colloidal Nanocrystals for Parallel Single-Particle Catalysis
Catalyst activity can depend distinctly on nano -particle size and shape. Therefore, understanding the structure sensitivity of catalytic reactions is of fundamental and technical importance. Experiments with single-particle resolution, where ensemble-averaging is eliminated, are required to study it. Here, we implement the selective trapping of individual spherical, cubic, and octahedral colloidal Au nanocrystals in 100 parallel nanofluidic channels to determine their activity for fluorescein reduction by sodium borohydride using fluorescence microscopy. As the main result, we identify distinct structure sensitivity of the rate-limiting borohydride oxidation step originating from different edge site abundance on the three particle types, as confirmed by first -principles calculations. This advertises nanofluidic reactors for the study of structure-function correlations in catalysis and identifies nanoparticle shape as a key factor in borohydride-mediated catalytic reactions
Multiscale and multilevel methods for porous media flow problems
We consider two problems encountered in simulation of fluid flow through porous media. In macroscopic models based on Darcy's law, the permeability field appears as data. The first problem is that the permeability field generally is not entirely known. We consider forward propagation of uncertainty from the permeability field to a quantity of interest. We focus on computing p-quantiles and failure probabilities of the quantity of interest. We propose and analyze improved standard and multilevel Monte Carlo methods that use computable error bounds for the quantity of interest. We show that substantial reductions in computational costs are possible by the proposed approaches. The second problem is fine scale variations of the permeability field. The permeability often varies on a scale much smaller than that of the computational domain. For standard discretization methods, these fine scale variations need to be resolved by the mesh for the methods to yield accurate solutions. We analyze and prove convergence of a multiscale method based on the Raviart–Thomas finite element. In this approach, a low-dimensional multiscale space based on a coarse mesh is constructed from a set of independent fine scale patch problems. The low-dimensional space can be used to yield accurate solutions without resolving the fine scale
Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data
We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model. The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element. The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem
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