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arXiv 2024年

作者： Holzhausen, Konstantin Merlid, Mia Torvik, Håkon Olav Malthe-Sørenssen, Anders Lepperød, Mikkel Elle Department of Physics University of Oslo Sem Sælands vei 24 Fysikkbygningen Oslo0371 Norway Department of Numerical Analysis and Scientific Computing Simula Reserach Laboratory Kristian Augusts gate 23 Oslo0164 Norway

Backpropagation-optimized artificial neural networks, while precise, lack robustness, leading to unforeseen behaviors that affect their safety. Biological neural systems do solve some of these issues already. Unlike artificial models, biological neurons adjust connectivity based on neighboring cell activity. Understanding the biological mechanisms of robustness can pave the way towards building trustworthy and safe systems. Robustness in neural representations is hypothesized to correlate with the smoothness of the encoding manifold. Recent work suggests power law covariance spectra, which were observed studying the primary visual cortex of mice, to be indicative of a balanced trade-off between accuracy and robustness in representations. Here, we show that unsupervised local learning models with winner takes all dynamics learn such power law representations, providing upcoming studies a mechanistic model with that characteristic. Our research aims to understand the interplay between geometry, spectral properties, robustness, and expressivity in neural representations. Hence, we study the link between representation smoothness and spectrum by using weight, Jacobian and spectral regularization while assessing performance and adversarial robustness. Our work serves as a foundation for future research into the mechanisms underlying power law spectra and optimally smooth encodings in both biological and artificial systems. The insights gained may elucidate the mechanisms that realize robust neural networks in mammalian brains and inform the development of more stable and reliable artificial systems. © 2024, CC BY-NC-ND.

关键词： Adversarial machine learning

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MEDICAL IMAGE REGISTRATION USING OPTIMAL CONTROL OF A LINEAR HYPERBOLIC TRANSPORT EQUATION WITH A DG DISCRETIZATION

arXiv

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arXiv 2023年

作者： Zapf, Bastian Haubner, Johannes Baumgärtner, Lukas Schmidt, Stephan Expert Analytics AS Oslo Norway Department of Mathematics University of Oslo Oslo Norway Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo Norway Institute of Mathematics and Scientific Computing University of Graz Austria Institut of Mathematics Humboldt University of Berlin Berlin10099 Germany Department of Mathematics University of Trier Trier54296 Germany

Patient specific brain mesh generation from MRI can be a time consuming task and require manual corrections, e.g., for meshing the ventricular system or defining subdomains. To address this issue, we consider an image registration approach. The idea is to use the registration of an input magnetic resonance image (MRI) to a respective target in order to obtain a new mesh from a template mesh. To obtain the transformation, we solve an optimization problem that is constrained by a linear hyperbolic transport equation. We use a higher-order discontinuous Galerkin finite element method for discretization and motivate the numerical upwind scheme and its limitations from the continuous weak space–time formulation of the transport equation. We present a numerical implementation that builds on the finite element packages FEniCS and dolfin-adjoint. To demonstrate the efficacy of the proposed approach, numerical results for the registration of an input to a target MRI of two distinct individuals are presented. Moreover, it is shown that the registration transforms a manually crafted input mesh into a new mesh for the target subject whilst preserving mesh quality. Challenges of the algorithm are *** Codes 35D30, 35L02, 35Q49, 35Q93, 35R30, 49M05, 49M41, 49N45, 65D18, 65K10, 65M32, 65M50, 68T99 © 2023, CC BY.

关键词： numerical methods

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analysis OF THE SIMPL METHOD FOR DENSITY-BASED TOPOLOGY OPTIMIZATION

arXiv

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arXiv 2024年

作者： Keith, Brendan Kim, Dohyun Lazarov, Boyan S. Surowiec, Thomas M. Division of Applied Mathematics Brown University ProvidenceRI02912 United States Lawrence Livermore National Laboratory LivermoreCA94550 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [31]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. We refer to the new algorithm as the SiMPL method, pronounced like "simple", which stands for Sigmoidal Mirror descent with a Projected Latent *** Codes 68Q25, 68R10, 68U05 Copyright © 2024, The Authors. All rights reserved.

关键词： Lagrange multipliers

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A Simple Introduction to the SiMPL Method for Density-Based Topology Optimization

arXiv

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arXiv 2024年

作者： Kim, Dohyun Lazarov, Boyan S. Surowiec, Thomas M. Keith, Brendan Division of Applied Mathematics Brown University ProvidenceRI02912 United States Lawrence Livermore National Laboratory LivermoreCA94550 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as "the simple method") optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi–Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems. Copyright © 2024, The Authors. All rights reserved.

关键词： Mirrors

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A model reduction approach for inverse problems with operator valued data

arXiv

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arXiv 2020年

作者： Dölz, Jürgen Egger, Herbert Schlottbom, Matthias Institute for Numerical Simulation University of Bonn Friedrich-Hirzebruch-Allee 7 Bonn53115 Germany Numerical Analysis and Scientific Computing Department of Mathematics TU Darmstadt Dolivostr. 15 Darmstadt64293 Germany Department of Applied Mathematics University of Twente P.O. Box 217 Enschede7500 AE Netherlands

We study the efficient numerical solution of linear inverse problems with operator valued data which arise, e.g., in seismic exploration, inverse scattering, or tomographic imaging. The high-dimensionality of the data space implies extremely high computational cost already for the evaluation of the forward operator which makes a numerical solution of the inverse problem, e.g., by iterative regularization methods, practically infeasible. To overcome this obstacle, we take advantage of the underlying tensor product structure of the problem and propose a strategy for constructing low-dimensional certified reduced order models of quasi-optimal rank for the forward operator which can be computed much more efficiently than the truncated singular value decomposition. A complete analysis of the proposed model reduction approach is given in a functional analytic setting and the efficient numerical construction of the reduced order models as well as of their application for the numerical solution of the inverse problem is discussed. In summary, the setup of a low-rank approximation can be achieved in an offline stage at essentially the same cost as a single evaluation of the forward operator, while the actual solution of the inverse problem in the online phase can be done with extremely high efficiency. The theoretical results are illustrated by application to a typical model problem in fluorescence optical tomography. Copyright © 2020, The Authors. All rights reserved.

关键词： Inverse problems

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SMART: Spatial Modeling Algorithms for Reactions and Transport

arXiv

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arXiv 2023年

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

Recent advances in microscopy and 3D reconstruction methods have allowed for characterization of cellular morphology in unprecedented detail, including the irregular geometries of intracellular subcompartments such as membrane-bound organelles. These geometries are now compatible with predictive modeling of cellular function. Biological cells respond to stimuli through sequences of chemical reactions generally referred to as cell signaling pathways. The propagation and reaction of chemical substances in cell signaling pathways can be represented by coupled nonlinear systems of reaction-transport equations. These reaction pathways include numerous chemical species that react across boundaries or interfaces (e.g., the cell membrane and membranes of organelles within the cell) and domains (e.g., the bulk cell volume and the interior of organelles). Such systems of multi-dimensional partial differential equations (PDEs) are notoriously difficult to solve because of their high dimensionality, non-linearities, strong coupling, stiffness, and potential instabilities. In this work, we describe Spatial Modeling Algorithms for Reactions and Transport (SMART), a high-performance finite-element-based simulation package for model specification and numerical simulation of spatially-varying reaction-transport processes. SMART is based on the FEniCS finite element library, provides a symbolic representation framework for specifying reaction pathways, and supports geometries in 2D and 3D including large and irregular cell geometries obtained from modern ultra-structural characterization methods. © 2023, CC BY.

关键词： Cell signaling

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Finite element simulation of ionic electrodiffusion in cellular geometries

arXiv

引用

arXiv 2019年

作者： Ellingsrud, A.J. Solbrå, A. Einevoll, G.T. Halnes, G. Rognes, M.E. Department for Scientific Computing and Numerical Analysis Simula Research Laboratory Norway Centre for Integrative Neuroplasticity University of Oslo Oslo Norway Department of Physics University of Oslo Oslo Norway Faculty of Science and Technology Norwegian University of Life Sciences Ås Norway

Mathematical models for excitable cells are commonly based on cable theory, which considers a homogenized domain and spatially constant ionic concentrations. Although such models provide valuable insight, the effect of altered ion concentrations or detailed cell morphology on the electrical potentials cannot be captured. In this paper, we discuss an alternative approach to detailed modelling of electrodiffusion in neural tissue. The mathematical model describes the distribution and evolution of ion concentrations in a geometrically-explicit representation of the intra- and extracellular domains. As a combination of the electroneutral Kirchhoff-Nernst-Planck (KNP) model and the Extracellular-Membrane-Intracellular (EMI) framework, we refer to this model as the KNP-EMI model. Here, we introduce and numerically evaluate a new, finite element-based numerical scheme for the KNP-EMI model, capable of efficiently and flexibly handling geometries of arbitrary dimension and arbitrary polynomial degree. Moreover, we compare the electrical potentials predicted by the KNP-EMI and EMI models. Finally, we study ephaptic coupling induced in an unmyelinated axon bundle and demonstrate how the KNP-EMI framework can give new insights in this setting. Copyright © 2019, The Authors. All rights reserved.

关键词： Finite element method

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The Deep Ritz Method for Parametric p-Dirichlet Problems

arXiv

引用

arXiv 2022年

作者： Kaltenbach, Alex Zeinhofer, Marius Department of Applied Mathematics University of Freiburg Ernst–Zermelo–Straße 1 Freiburg i. Br.79104 Germany Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Kristian Augusts Gate 23 Oslo0164 Norway

We establish error estimates for the approximation of parametric p-Dirichlet problems deploying the Deep Ritz Method. Parametric dependencies include, e.g., varying geometries and exponents p ∈ (1, ∞). Combining the derived error estimates with quantitative approximation theorems yields error decay rates and establishes that the Deep Ritz Method retains the favorable approximation capabilities of neural networks in the approximation of high dimensional functions which makes the method attractive for parametric problems. Finally, we present numerical examples to illustrate potential *** Codes 68T07, 35A35, 65N15 Copyright © 2022, The Authors. All rights reserved.

关键词： Deep neural networks

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Preconditioned smoothers for the full approximation scheme for the RANS equations

arXiv

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arXiv 2017年

作者： Birken, Philipp Bull, Jonathan Jameson, Antony Centre for the Mathematical Sciences Numerical Analysis Lund University Lund Sweden Division of Scientific Computing Dept. of Information Technology Uppsala University Box 337 Uppsala75105 Sweden Stanford University Department of Aeronautics and Astronautics StanfordCA94305 United States

We consider multigrid methods for finite volume discretizations of the Reynolds Averaged Navier-Stokes (RANS) equations for both steady and unsteady flows. We analyze the effect of different smoothers based on pseudo time iterations, such as explicit and additive Runge-Kutta (ARK) methods. Furthermore, we derive the new class of additive W (AW) methods from Rosenbrock smoothers. This gives rise to two classes of preconditioned smoothers, preconditioned ARK and additive W (AW), which are implemented the exact same way, but have different parameters and properties. The new derivation allows to choose some of these based on results for time integration methods. As preconditioners, we consider SGS preconditioners based on flux vector splitting discretizations with a cutoff function for small eigenvalues. We compare these methods based on a discrete Fourier analysis. numerical results on pitching and plunging airfoils identify AW3 as the best smoother regarding overall efficiency. Specifically, for the NACA 64A010 airfoil steady-state convergence rates of as low as 0.85 were achieved, or a reduction of 6 orders of magnitude in approximately 25 pseudo-time iterations. Unsteady convergence rates of as low as 0.77 were achieved, or a reduction of 11 orders of magnitude in approximately 70 pseudo-time iterations. Copyright © 2017, The Authors. All rights reserved.

关键词： Runge Kutta methods

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A Risk Management Perspective on Statistical Estimation and Generalized Variational Inference

arXiv

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arXiv 2023年

作者： Javeed, Aurya S. Kouri, Drew P. Surowiec, Thomas M. Optimization and Uncertainty Quantification Sandia National Laboratories P.O. Box 5800 MS-1320 AlbuquerqueNM87125 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Kristian Augusts gate 23 Oslo0164 Norway

Generalized variational inference (GVI) provides an optimization-theoretic framework for statistical estimation that encapsulates many traditional estimation procedures. The typical GVI problem is to compute a distribution of parameters that maximizes the expected payoff minus the divergence of the distribution from a specified prior. In this way, GVI enables likelihood-free estimation with the ability to control the influence of the prior by tuning the so-called learning rate. Recently, GVI was shown to outperform traditional Bayesian inference when the model and prior distribution are misspecified. In this paper, we introduce and analyze a new GVI formulation based on utility theory and risk management. Our formulation is to maximize the expected payoff while enforcing constraints on the maximizing distribution. We recover the original GVI distribution by choosing the feasible set to include a constraint on the divergence of the distribution from the prior. In doing so, we automatically determine the learning rate as the Lagrange multiplier for the constraint. In this setting, we are able to transform the infinite-dimensional estimation problem into a two-dimensional convex program. This reformulation further provides an analytic expression for the optimal density of parameters. In addition, we prove asymptotic consistency results for empirical approximations of our optimal distributions. Throughout, we draw connections between our estimation procedure and risk management. In fact, we demonstrate that our estimation procedure is equivalent to evaluating a risk measure. We test our procedure on an estimation problem with a misspecified model and prior distribution, and conclude with some extensions of our *** Codes 62A10, 62A15, 62C12, 62G05, 62G07 © 2023, CC BY.

关键词： Risk management

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