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arXiv 2021年

作者： Fornasier, Massimo Klock, Timo Riedl, Konstantin Technical University of Munich School of Computation Information and Technology Department of Mathematics Munich Germany Munich Center for Machine Learning Munich Germany Munich Data Science Institute Munich Germany Simula Research Laboratory Department of Numerical Analysis and Scientific Computing Oslo Norway University of San Diego Department of Mathematics San DiegoCA United States

In this paper, we study consensus-based optimization (CBO), which is a multi-agent metaheuristic derivative-free optimization method that can globally minimize nonconvex nonsmooth functions and is amenable to theoretical analysis. Based on an experimentally supported intuition that, on average, CBO performs a gradient descent of the squared Euclidean distance to the global minimizer, we devise a novel technique for proving the convergence to the global minimizer in mean-field law for a rich class of objective functions. The result unveils internal mechanisms of CBO that are responsible for the success of the method. In particular, we prove that CBO performs a convexification of a large class of optimization problems as the number of optimizing agents goes to infinity. Furthermore, we improve prior analyses by requiring mild assumptions about the initialization of the method and by covering objectives that are merely locally Lipschitz continuous. As a core component of this analysis, we establish a quantitative nonasymptotic Laplace principle, which may be of independent interest. From the result of CBO convergence in mean-field law, it becomes apparent that the hardness of any global optimization problem is necessarily encoded in the rate of the mean-field approximation, for which we provide a novel probabilistic quantitative estimate. The combination of these results allows to obtain probabilistic global convergence guarantees of the numerical CBO *** Codes 65K10, 90C26, 90C56, 35Q90, 35Q84 Copyright © 2021, The Authors. All rights reserved.

关键词： Fokker Planck equation

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Spatial modeling algorithms for reactions and transport (SMART) in biological cells

arXiv

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arXiv 2024年

作者： Francis, Emmet A. Laughlin, Justin G. Dokken, Jørgen S. Finsberg, Henrik N.T. Lee, Christopher T. Rognes, Marie E. Rangamani, Padmini Department of Mechanical and Aerospace Engineering University of California La Jolla San DiegoCA United States Computational Engineering Division Lawrence Livermore National Laboratory LivermoreCA United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Department of Computational Physiology Simula Research Laboratory Oslo Norway

Biological cells rely on precise spatiotemporal coordination of biochemical reactions to control their many functions. Such cell signaling networks have been a common focus for mathematical models, but they remain challenging to simulate, particularly in realistic cell geometries. Herein, we present our software, Spatial Modeling Algorithms for Reaction and Transport (SMART), a package that takes in high-level user specifications about cell signaling networks and molecular transport, and then assembles and solves the associated mathematical and computational systems. SMART uses state-of-the-art finite element analysis, via the FEniCS Project software, to efficiently and accurately resolve cell signaling events over discretized cellular and subcellular geometries. We demonstrate its application to several different biological systems, including YAP/TAZ mechanotransduction, calcium signaling in neurons and cardiomyocytes, and ATP generation in mitochondria. Throughout, we utilize experimentally-derived realistic cellular geometries represented by well-conditioned tetrahedral meshes. These scenarios demonstrate the applicability, flexibility, accuracy and efficiency of SMART across a range of temporal and spatial scales. © 2024, CC BY.

关键词： Cell signaling

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OPTIMIZATION OF HOPF BIFURCATION POINTS

arXiv

引用

arXiv 2022年

作者： Boullé, Nicolas Farrell, Patrick E. Rognes, Marie E. Isaac Newton Institute for Mathematical Sciences University of Cambridge CB3 0EH United Kingdom Mathematical Institute University of Oxford OxfordOX2 6GG United Kingdom Department of Scientific Computing and Numerical Analysis Simula Research Laboratory Oslo Norway Department of Mathematics University of Bergen Bergen Norway

We introduce a numerical technique for controlling the location and stability properties of Hopf bifurcations in dynamical systems. The algorithm consists of solving an optimization problem constrained by an extended system of nonlinear partial differential equations that characterizes Hopf bifurcation points. The flexibility and robustness of the method allows us to advance or delay a Hopf bifurcation to a target value of the bifurcation parameter, as well as controlling the oscillation frequency with respect to a parameter of the system or the shape of the domain on which solutions are defined. numerical applications are presented in systems arising from biology and fluid dynamics, such as the FitzHugh-Nagumo model, Ginzburg-Landau equation, Rayleigh-Bénard convection problem, and Navier-Stokes equations, where the control of the location and oscillation frequency of periodic solutions is of high *** Codes 65P30, 65P40, 37M20, 65K10, 49M41 Copyright © 2022, The Authors. All rights reserved.

关键词： Hopf bifurcation

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Laplacian preconditioning of elliptic pdes: Localization of the eigenvalues of the discretized operator

arXiv

引用

arXiv 2018年

作者： Gergelits, Tomáš Mardal, Kent-André Nielsen, Bjørn Fredrik Strakoš, Zdeněk Institute of Computer Sciences Czech Academy of Science Prague Czech Republic Faculty of Mathematics and Physics Charles University Prague Czech Republic Department of Mathematics University of Oslo Oslo0316 Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Lysaker1325 Norway Faculty of Science and Technology Norwegian University of Life Sciences ÅsNO-1432 Norway

In the paper Preconditioning by inverting the Laplacian;an analysis of the eigenvalues. IMA Journal of numerical analysis 29, 1 (2009), 24-42, Nielsen, Hackbusch and Tveito study the operator generated by using the inverse of the Laplacian as preconditioner for second order elliptic PDEs ∇ · (k(x)∇u) = f. They prove that the range of k(x) is contained in the spectrum of the preconditioned operator, provided that k is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. Motivated by this investigation, we analyze the eigenvalues of the matrix L−1A, where L and A are the stiffness matrices associated with the Laplace operator and general second order elliptic operators, respectively. Without any assumption about the continuity of k(x), we prove the existence of a one-to-one pairing between the eigenvalues of L−1A and the intervals determined by the images under k(x) of the supports of the FE nodal basis functions. As a consequence, we can show that the nodal values of k(x) yield accurate approximations of the eigenvalues of L−1A. Our theoretical results are illuminated by several numerical experiments.65F08, 65F15, 65N12, 35J99 Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions

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Application of a dispersive wave hydro-sediment-morphodynamic model in the Ria Formosa lagoon

arXiv

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arXiv 2022年

作者： Kazhyken, Kazbek Valseth, Eirik Videman, Juha Dawson, Clint Oden Institute for Computational Engineering and Sciences The University of Texas at Austin AustinTX78712 United States CAMGSD Departamento de Matemática Universidade de Lisboa Lisbon1049-001 Portugal Department of Data Science The Norwegian University of Life Sciences Ås1433 Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

Results of an application of a dispersive wave hydro-sediment-morphodynamic model in the western circulation cell of the Ria Formosa lagoon located in the Algarve region of the southern Portugal are presented. This area of interest has a couple of features that complicate the application of the dispersive wave model: (1) the area has a complex irregular geometry with a number of barrier islands that separate the lagoon from the Atlantic Ocean, artificial and naturally occurring tidal inlets, and a number of curling channels inside the lagoon that interconnect the inlets and serve as waterways between the lagoon settlements;(2) the tidal range in the area can reach up to 3.5 m;therefore, the terrain inside the lagoon is characterized by vast salt marshes and tidal flats, and the wetting-drying process is a key component of any hydrodynamic simulation in this area. A model representation of the area has been developed by generating an unstructured finite element mesh of the circulation cell, and collecting data on parameters that characterize the tidal waves in the area, and bottom friction and sediment transport models used in the simulations. The results of the simulations indicate that the dispersive wave model can be applied in coastal areas with nontrivial underlying physical processes, and complex irregular geometries. Moreover, the dispersive term of the model is capable of capturing additional flow characteristics that are otherwise not present in hydrodynamic simulations that involve the nonlinear shallow water equations;and these additional flow features can, in their turn, affect the resulting sediment transport and bed morphodynamic process simulations. Copyright © 2022, The Authors. All rights reserved.

关键词： Equations of motion

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Investigating molecular transport in the human brain from MRI with physics-informed neural networks

arXiv

引用

arXiv 2022年

作者： Zapf, Bastian Haubner, Johannes Kuchta, Miroslav Ringstad, Geir Eide, Per Kristian Mardal, Kent-Andre University of Oslo Faculty of Mathematics and Natural Sciences Oslo0851 Norway Simula Research Laboratory Department of Numerical Analysis and Scientific Computing Oslo0164 Norway Oslo University Hospital Department of Radiology Oslo0372 Norway Sorlandet Hospital Department of Geriatrics and Internal Medicine Arendal4838 Norway Oslo University Hospital Department of Neurosurgery Oslo0372 Norway University of Oslo Institute of Clinical Medicine Oslo0372 Norway

In recent years, a plethora of methods combining neural networks and partial differential equations have been developed. A widely known example are physics-informed neural networks, which solve problems involving partial differential equations by training a neural network. We apply physics-informed neural networks and the finite element method to estimate the diffusion coefficient governing the long term spread of molecules in the human brain from magnetic resonance images. Synthetic testcases are created to demonstrate that the standard formulation of the physics-informed neural network faces challenges with noisy measurements in our application. Our numerical results demonstrate that the residual of the partial differential equation after training needs to be small for accurate parameter recovery. To achieve this, we tune the weights and the norms used in the loss function and use residual based adaptive refinement of training points. We find that the diffusion coefficient estimated from magnetic resonance images with PINNs becomes consistent with results from a finite element based approach when the residuum after training becomes small. The observations presented here are an important first step towards solving inverse problems on cohorts of patients in a semi-automated fashion with physics-informed neural *** Codes 49M41, 65L09 © 2022, CC BY-NC-SA.

关键词： Partial differential equations

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POLYLLA: Polygonal meshing algorithm based on terminal-edge regions

arXiv

引用

arXiv 2022年

作者： Salinas, Sergio Hitschfeld-Kahler, Nancy Ortiz-Bernardin, Alejandro Si, Hang Department of Computer Sciences Universidad de Chile Av. Beauchef 851 RM Santiago8370456 Chile Computational and Applied Mechanics Laboratory Department of Mechanical Engineering Universidad de Chile Av. Beauchef 851 RM Santiago8370456 Chile Numerical Mathematics and Scientific Computing Weierstrass Institute for Applied Analysis and Stochastics Mohrenstr. 39 Berlin Berlin10117 Germany

This paper presents an algorithm to generate a new kind of polygonal mesh obtained from triangulations. Each polygon is built from a terminal-edge region surrounded by edges that are not the longest-edge of any of the two triangles that share them. The algorithm is termed Polylla and is divided into three phases. The first phase consists of labeling each edge of the input triangulation according to its size;the second phase builds polygons (simple or not) from terminal-edges regions using the label system;and the third phase transforms each non simple polygon into simple ones. The final mesh contains polygons with convex and non convex shape. Since Voronoi based meshes are currently the most used polygonal meshes, we compare some geometric properties of our meshes against constrained Voronoi meshes. Several experiments were run to compare the shape and size of polygons, the number of final mesh points and polygons. For the same input, Polylla meshes contain less polygons than Voronoi meshes, and the algorithm is simpler and faster than the algorithm to generate constrained Voronoi meshes. Finally, we have validated Polylla meshes by solving the Laplace equation on an L-shaped domain using the Virtual Element Method (VEM). We show that the numerical performance of the VEM using Polylla meshes and Voronoi meshes is similar. Copyright © 2022, The Authors. All rights reserved.

关键词： Mesh generation

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Magnetic Optical Rotation from Real-Time Simulations in Finite Magnetic Fields

arXiv

引用

arXiv 2023年

作者： Ofstad, Benedicte Sverdrup Wibowo-Teale, Meilani Kristiansen, Håkon Emil Aurbakken, Einar Kitsaras, Marios Petros Schøyen, Øyvind Sigmundson Hauge, Eirill Irons, Tom J.P. Kvaal, Simen Stopkowicz, Stella Wibowo-Teale, Andrew M. Pedersen, Thomas Bondo Hylleraas Centre for Quantum Molecular Sciences Department of Chemistry University of Oslo Norway School of Chemistry University of Nottingham University Park NottinghamNG7 2RD United Kingdom University of Saarland Physical and Theoretical Chemistry Campus B2.2 Saarbruecken66123 Germany Department of Physics University of Oslo Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

We present a numerical approach to magnetic optical rotation based on real-time time-dependent electronic-structure theory. Not relying on perturbation expansions in the magnetic-field strength, the formulation allows us to test the range of validity of the linear relation between the rotation angle per unit path length and the magnetic-field strength that was established empirically by Verdet 160 years ago. Results obtained from time-dependent coupled-cluster and time-dependent current density-functional theory are presented for the closed-shell molecules H2, HF, and CO in magnetic fields up to 55 kT at standard temperature and pressure conditions. We find that Verdet’s linearity remains valid up to roughly 10–20 kT, above which significant deviations from linearity are observed. Among the three current density-functional approximations tested in this work, the current-dependent Tao-Perdew-Staroverov–Scuseria hybrid functional performs the best in comparison with time-dependent coupled-cluster singles and doubles results for the magnetic optical rotation. Copyright © 2023, The Authors. All rights reserved.

关键词： numerical methods

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Parametric sequential method for MRI-based wall shear stress quantification

arXiv

引用

arXiv 2020年

作者： Shokina, Nina Teschner, Gabriel Bauer, Andreas Tropea, Cameron Egger, Herbert Hennig, Jürgen Krafft, Axel J. Department of Radiology Medical Physics Medical Center University of Freiburg Faculty of Medicine University of Freiburg Freiburg79106 Germany Institute for Numerical Analysis and Scientific Computing Department of Mathematics Technische Universität Darmstadt Darmstadt64293 Germany Institute for Fluid Mechanics and Aerodynamics Department of Mechanical Engineering Technische Universität Darmstadt Darmstadt-Griesheim64347 Germany Department of Radiology Medical Physics Medical Center University of Freiburg Faculty of Medicine University of Freiburg Freiburg79106 Germany

Wall shear stress (WSS) has been suggested as a potential biomarker in various cardiovascular diseases and it can be estimated from phase-contrast Magnetic Resonance Imaging (PC-MRI) velocity measurements. We present a new parametric sequential method for MRI-based WSS quantification consisting of a geometry identification and a subsequent approximation of the velocity field. The new method was validated on well controlled, high resolution in vitro measurements of turbulent stationary flows and physiological pulsatile flows in phantoms. Basic applicability to in vivo was demonstrated using 2D PC-MRI data of the ascending aorta of three volunteers. Copyright © 2020, The Authors. All rights reserved.

关键词： Magnetic resonance imaging

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Exploration of core concepts required for mid- and domain-level ontology development to facilitate explainable-AI-readiness of data and models 4

Exploration of core concepts required for mid- and domain-le...

引用

4th International Workshop on Data meets Ontologies in Explainable AI, DAO-XAI 2024

作者： Horsch, Martin T. Chiacchiera, Silvia Todorov, Ilian T. Correia, Ana Teresa Dey, Aditya Konchakova, Natalia A. Scholze, Sebastian Stephan, Simon Tøndel, Kristin Sarkar, Arkopaul Karray, M. Hedi Al Machot, Fadi Schembera, Björn Norwegian University of Life Sciences Department of Data Science Postboks 5003 Ås1432 Norway UKRI Science and Technology Facilities Council Scientific Computing Department DaresburyWA4 4AD United Kingdom ATB Institut für Angewandte Systemtechnik Bremen GmbH Wiener Str. 1 Bremen28359 Germany Helmholtz-Zentrum Hereon Institute of Surface Science Max-Planck-Str. 1 Geesthacht21502 Germany RPTU Kaiserslautern Laboratory of Engineering Thermodynamics Kaiserslautern67663 Germany University of Technology Tarbes Occitanie Pyrénées Production Engineering Lab. 47 av. d’Azereix Tarbes France University of Stuttgart Institute of Applied Analysis and Numerical Simulation Stuttgart70569 Germany

This position paper reports on the initial discussions within the Knowledge Graph Alliance’s working group on explainable-AI-ready data and metadata principles, which was created in March 2024. At present, we are taking initial steps toward capturing core concepts related to explanation, grounding, reliance, and trust;the scope also extends to potential dual notions such as explainability, verifiability/reproducibility, reliability, and trustworthiness. These initial steps consist in reviewing core concepts as they are discussed in the literature and exploring what could be practically useful definitions of these most central concepts. One of the conclusions is that the metadata standards will need to be suitable for documenting three kinds of grounding: Grounding of knowledge, grounding of reliance, and grounding of trust. Pre-existing metadata standards at the mid and domain level are presently undergoing a redesign in order to become more modular, computationally tractable, intelligible to humans, and adjustable, which will be needed as we continue our work toward actionable recommendations. The development of this system of lite (OWL 2 EL) ontologies, called MSO-EM: Ontologies for modelling, simulation, optimization (MSO) and epistemic metadata (EM), is carried out on a public repository. © 2024 Copyright for this paper by its authors.

关键词： Applied ontology epistemic metadata explainable-AI-ready (XAIR) ontology redesign reliability reproducibility

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