833 research outputs found
Pinching of the first eigenvalue for second order operators on hypersurfaces of the Euclidean space
We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces
Neural network guided adjoint computations in dual weighted residual error estimation
In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main objective of our approach is to explore alternatives for solving the adjoint problem with greater potential of a numerical cost reduction. The proposed algorithm is based on the general goal-oriented error estimation theorem including both linear and nonlinear stationary partial differential equations and goal functionals. Our developments are substantiated with some numerical experiments that include comparisons of neural network computed adjoints and classical finite element solutions of the adjoints. In the programming software, the open-source library deal.II is successfully coupled with LibTorch, the PyTorch C++ application programming interface
Numerical Methods for Algorithmic Systems and Neural Networks
These lecture notes are devoted to numerical concepts and solution of algorithmic systems and neural networks. The course is divided into four parts: traditional AI (artificial intelligence), deep learning in neural networks, applications to (and with) differential equations, and project work. Throughout this course an emphasis is on mathematical ingredients from which several are rigorously proven. In the project work, the participants usually form groups and work together on a given problem to train themselves on mathematical modeling, design of algorithms, implementation, and analysis and intepretation of the simulation results
A monolithic space-time temporal multirate finite element framework for interface and volume coupled problems
In this work, we propose and computationally investigate a monolithic
space-time multirate scheme for coupled problems. The novelty lies in the
monolithic formulation of the multirate approach as this requires a careful
design of the functional framework, corresponding discretization, and
implementation. Our method of choice is a tensor-product Galerkin space-time
discretization. The developments are carried out for both prototype interface-
and volume coupled problems such as coupled wave-heat-problems and a
displacement equation coupled to Darcy flow in a poro-elastic medium. The
latter is applied to the well-known Mandel's benchmark. Detailed computational
investigations and convergence analyses give evidence that our monolithic
multirate framework performs well.Comment: 34 pages, 14 figures, 7 table
MORe DWR: Space-time goal-oriented error control for incremental POD-based ROM
In this work, the dual-weighted residual (DWR) method is applied to obtain a
certified incremental proper orthogonal decomposition (POD) based reduced order
model. A novel approach called MORe DWR (Model Order Rduction with
Dual-Weighted Residual error estimates) is being introduced. It marries
tensor-product space-time reduced-order modeling with time slabbing and an
incremental POD basis generation with goal-oriented error control based on
dual-weighted residual estimates. The error in the goal functional is being
estimated during the simulation and the POD basis is being updated if the
estimate exceeds a given threshold. This allows an adaptive enrichment of the
POD basis in case of unforeseen changes in the solution behavior which is of
high interest in many real-world applications. Consequently, the offline phase
can be skipped, the reduced-order model is being solved directly with the POD
basis extracted from the solution on the first time slab and -- if necessary --
the POD basis is being enriched on-the-fly during the simulation with
high-fidelity finite element solutions. Therefore, the full-order model solves
can be reduced to a minimum, which is demonstrated on numerical tests for the
heat equation and elastodynamics.Comment: 42 pages, 13 figure
Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity
In this work, the space-time MORe DWR (Model Order Reduction with
Dual-Weighted Residual error estimates) framework is extended and further
developed for single-phase flow problems in porous media. Specifically, our
problem statement is the Biot system which consists of vector-valued
displacements (geomechanics) coupled to a Darcy flow pressure equation. The
MORe DWR method introduces a goal-oriented adaptive incremental proper
orthogonal decomposition (POD) based-reduced-order model (ROM). The error in
the reduced goal functional is estimated during the simulation, and the POD
basis is enriched on-the-fly if the estimate exceeds a given threshold. This
results in a reduction of the total number of full-order-model solves for the
simulation of the porous medium, a robust estimation of the quantity of
interest and well-suited reduced bases for the problem at hand. We apply a
space-time Galerkin discretization with Taylor-Hood elements in space and a
discontinuous Galerkin method with piecewise constant functions in time. The
latter is well-known to be similar to the backward Euler scheme. We demonstrate
the efficiency of our method on the well-known two-dimensional Mandel benchmark
and a three-dimensional footing problem.Comment: 33 pages, 9 figures, 3 table
Streaming SPHINCS+ for Embedded Devices using the Example of TPMs
We present an implementation of the hash-based post-quantum signature scheme SPHINCS+ that enables heavily memory-restricted devices to sign messages by streaming-out a signature during its computation and to verify messages by streaming-in a signature. We demonstrate our implementation in the context of Trusted Platform Modules (TPMs) by proposing a SPHINCS+ integration and a streaming extension for the TPM specification. We evaluate the overhead of our signature-streaming approach for a stand-alone SPHINCS+ implementation and for its integration in a proof-of-concept TPM with the proposed streaming extension running on an ARM Cortex-M4 platform. Our streaming interface greatly reduces the memory requirements without introducing a significant performance penalty. This is achieved not only by removing the need to store an entire signature but also by reducing the stack requirements of the key generation, sign, and verify operations. Therefore, our streaming interface enables small embedded devices that do not have sufficient memory to store an entire SPHINCS+ signature or that previously were only able to use a parameter set that results in smaller signatures to sign and verify messages using all SPHINCS+ variants
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