GAMM Mitteilungen最新文献

{"title":"The mathematics of dots and pixels: On the theoretical foundations of image halftoning","authors":"Felix Krahmer,&nbsp;Anna Veselovska","doi":"10.1002/gamm.70000","DOIUrl":"https://doi.org/10.1002/gamm.70000","url":null,"abstract":"<p>The evolution of image halftoning, from its analog roots to contemporary digital methodologies, encapsulates a fascinating journey marked by technological advancements and creative innovations. Yet the theoretical understanding of halftoning is much more recent. In this article, we explore various approaches towards shedding light on the design of halftoning approaches and why they work. We discuss both halftoning in a continuous domain and on a pixel grid. We start by reviewing the mathematical foundation of the so-called electrostatic halftoning method, which departed from the heuristic of considering the back dots of the halftoned image as charged particles attracted by the grey values of the image in combination with mutual repulsion. Such an attraction-repulsion model can be mathematically represented via an energy functional in a reproducing kernel Hilbert space allowing for a rigorous analysis of the resulting optimization problem as well as a convergence analysis in a suitable topology. A second class of methods that we discuss in detail is the class of error diffusion schemes, arguably among the most popular halftoning techniques due to their ability to work directly on a pixel grid and their ease of application. The main idea of these schemes is to choose the locations of the black pixels via a recurrence relation designed to agree with the image in terms of the local averages. We discuss some recent mathematical understanding of these methods that is based on a connection to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∑</mo>\u0000 <mi>Δ</mi>\u0000 </mrow>\u0000 <annotation>$$ Sigma Delta $$</annotation>\u0000 </semantics></math> quantizers, a popular class of algorithms for analog-to-digital conversion.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.70000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability of N-front and N-back solutions in the Barkley model","authors":"Christian Kuehn,&nbsp;Pascal Sedlmeier","doi":"10.1002/gamm.70001","DOIUrl":"https://doi.org/10.1002/gamm.70001","url":null,"abstract":"<p>In this article, we establish for an intermediate Reynolds number domain the stability of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$$ N $$</annotation>\u0000 </semantics></math>-front and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$$ N $$</annotation>\u0000 </semantics></math>-back solutions for each <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$$ N&gt;1 $$</annotation>\u0000 </semantics></math> corresponding to traveling waves, in an experimentally validated model for the transition to turbulence in pipe flow proposed in <i>[Barkley et al., Nature 526(7574):550-553, 2015]</i>. We base our work on the existence analysis of a heteroclinic loop between a turbulent and a laminar equilibrium proved by Engel, Kuehn and de Rijk in <i>Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022</i>, as well as some results from this work. The stability proof follows the verification of a set of abstract stability hypotheses stated by Sandstede in <i>[SIAM Journal on Mathematical Analysis 29.1 (1998), pp. 183-207]</i> for traveling waves motivated by the FitzHugh–Nagumo equations. In particular, this completes the first detailed analysis of Engel, Kuehn and de Rijk in <i>[Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022]</i> leading to a complete existence and stability statement that nicely fits within the abstract framework of waves generated by twisted heteroclinic loops.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.70001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fast and interpretable support vector classification based on the truncated ANOVA decomposition","authors":"Kseniya Akhalaya,&nbsp;Franziska Nestler,&nbsp;Daniel Potts","doi":"10.1002/gamm.202470007","DOIUrl":"https://doi.org/10.1002/gamm.202470007","url":null,"abstract":"&lt;p&gt;Support vector machines (SVMs) are an important tool for performing classification on scattered data, where one usually has to deal with many data points in high-dimensional spaces. We propose solving SVMs in primal form using feature maps based on trigonometric functions or wavelets. In small dimensional settings the fast Fourier transform (FFT) and related methods are a powerful tool in order to deal with the considered basis functions. For growing dimensions the classical FFT-based methods become inefficient due to the curse of dimensionality. Therefore, we restrict ourselves to multivariate basis functions, each of which only depends on a small number of dimensions. This is motivated by the well-known sparsity of effects and recent results regarding the reconstruction of functions from scattered data in terms of truncated analysis of variance (ANOVA) decompositions, which makes the resulting model even interpretable in terms of importance of the features as well as their couplings. The usage of small superposition dimensions has the consequence that the computational effort no longer grows exponentially but only polynomially with respect to the dimension. In order to enforce sparsity regarding the basis coefficients, we use the frequently applied &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {ell}_2 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-norm and, in addition, &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {ell}_1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-norm regularization. The found classifying function, which is the linear combination of basis functions, and its variance can then be analyzed in terms of the classical ANOVA decomposition of functions. Based on numerical examples we show that we are able to recover the signum of a function that perfectly fits our model assumptions. Furthermore, we perform classification on different artificial and real-world data sets. We obtain better results with &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202470007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A continuum chemo-mechano-biological model for in-stent restenosis with consideration of hemodynamic effects","authors":"Kiran Manjunatha,&nbsp;Anna Ranno,&nbsp;Jianye Shi,&nbsp;Nicole Schaaps,&nbsp;Pakhwan Nilcham,&nbsp;Anne Cornelissen,&nbsp;Felix Vogt,&nbsp;Marek Behr,&nbsp;Stefanie Reese","doi":"10.1002/gamm.202370008","DOIUrl":"https://doi.org/10.1002/gamm.202370008","url":null,"abstract":"<p>The occurrence of in-stent restenosis following percutaneous coronary intervention highlights the need for the creation of computational tools that can extract pathophysiological insights and optimize interventional procedures on a patient-specific basis. In light of this, a modeling framework encompassing the chemo-mechano-biological interactions in the arterial wall and the effects of hemodynamic perturbations is introduced in this work.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202370008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143117512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Data-driven methods for quantitative imaging","authors":"Guozhi Dong,&nbsp;Moritz Flaschel,&nbsp;Michael Hintermüller,&nbsp;Kostas Papafitsoros,&nbsp;Clemens Sirotenko,&nbsp;Karsten Tabelow","doi":"10.1002/gamm.202470014","DOIUrl":"https://doi.org/10.1002/gamm.202470014","url":null,"abstract":"<p>In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative magnetic resonance imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202470014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143112676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularizations of forward-backward parabolic PDEs","authors":"Carina Geldhauser","doi":"10.1002/gamm.202470001","DOIUrl":"https://doi.org/10.1002/gamm.202470001","url":null,"abstract":"<p>Forward-backward parabolic equations have been studied since the 1980s, but a mathematically rigorous picture is still far from being established. As quite a number of new papers have appeared recently, we review in this work the current state of the art. We focus our analysis on the status quo regarding the three most common types of regularizations, namely semidiscretization, the viscous approximation, and regularization with higher order spatial derivatives.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"47 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202470001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parallel two-scale finite element implementation of a system with varying microstructure","authors":"Omar Lakkis,&nbsp;Adrian Muntean,&nbsp;Omar Richardson,&nbsp;Chandrasekhar Venkataraman","doi":"10.1002/gamm.202470005","DOIUrl":"https://doi.org/10.1002/gamm.202470005","url":null,"abstract":"<p>We propose a two-scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed pullback operator, we are able to model the different microscopic domains as macroscopically dependent deformations of a reference domain. This allows for a relatively simple finite element framework to approximate the underlying system of partial differential equations with a parallel computational structure. We apply this technique to a model problem where we focus on transport in plant tissues. We illustrate the accuracy of the implementation with convergence benchmarks and show satisfactory parallelization speed-ups. We further highlight the effect of the heterogeneous microscopic structure on the output of the two-scale systems. Our implementation (publicly available on GitHub) builds on the <span>deal.II</span> FEM library. Application of this technique allows for an increased capacity of microscopic detail in multiscale modeling, while keeping running costs manageable.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"47 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202470005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Low Mach number limit of a diffuse interface model for two-phase flows of compressible viscous fluids","authors":"Helmut Abels,&nbsp;Yadong Liu,&nbsp;Šárka Nečasová","doi":"10.1002/gamm.202470008","DOIUrl":"https://doi.org/10.1002/gamm.202470008","url":null,"abstract":"<p>In this paper, we consider a singular limit problem for a diffuse interface model for two immiscible compressible viscous fluids. Via a relative entropy method, we obtain a convergence result for the low Mach number limit to a corresponding system for incompressible fluids in the case of well-prepared initial data and same densities in the limit.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"47 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202470008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mathematical analysis of a mesoscale model for multiphase membranes","authors":"Jakob Fuchs,&nbsp;Matthias Röger","doi":"10.1002/gamm.202470009","DOIUrl":"https://doi.org/10.1002/gamm.202470009","url":null,"abstract":"<p>In this article, we introduce a mesoscale continuum model for membranes made of two different types of amphiphilic lipids. The model extends work by Peletier and the second author (<i>Arch. Ration. Mech. Anal. 193</i>, 2009) for the one-phase case. We present a mathematical analysis of the asymptotic reduction to the macroscale when a key length parameter becomes arbitrarily small. We identify two main contributions in the energy: one that can be connected to bending of the overall structure and a second that describes the cost of the internal phase separations. We prove the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 </mrow>\u0000 <annotation>$$ Gamma $$</annotation>\u0000 </semantics></math>-convergence towards a perimeter functional for the phase separation energy and construct, in two dimensions, recovery sequences for the convergence of the full energy towards a 2D reduction of the Jülicher–Lipowsky bending energy with a line tension contribution for phase separated hypersurfaces.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"47 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202470009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Robustness and exploration of variational and machine learning approaches to inverse problems: An overview","authors":"Alexander Auras,&nbsp;Kanchana Vaishnavi Gandikota,&nbsp;Hannah Droege,&nbsp;Michael Moeller","doi":"10.1002/gamm.202470003","DOIUrl":"https://doi.org/10.1002/gamm.202470003","url":null,"abstract":"<p>This paper provides an overview of current approaches for solving inverse problems in imaging using variational methods and machine learning. A special focus lies on point estimators and their robustness against adversarial perturbations. In this context results of numerical experiments for a one-dimensional toy problem are provided, showing the robustness of different approaches and empirically verifying theoretical guarantees. Another focus of this review is the exploration of the subspace of data-consistent solutions through explicit guidance to satisfy specific semantic or textural properties.</p>","PeriodicalId":53634,"journal":{"name":"GAMM Mitteilungen","volume":"47 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/gamm.202470003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}