Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n1.a3
A. Sorrentino
{"title":"On John Mather’s seminal contributions in Hamiltonian dynamics","authors":"A. Sorrentino","doi":"10.4310/maa.2019.v26.n1.a3","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n1.a3","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n4.a2
Jian‐Guo Liu, Rong Yang
{"title":"Propagation of chaos for the Keller–Segel equation with a logarithmic cut-off","authors":"Jian‐Guo Liu, Rong Yang","doi":"10.4310/maa.2019.v26.n4.a2","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n4.a2","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n3.a5
O. Pironneau
. A simple fluid-structure problem is considered as a test to assess the feasibility of deep-learning algorithms for parameter identification. Tensorflow by Google is used and as it is a stochastic algorithm, provision must be made for the robustness of the large displacement fluid- structure simulator with respect to a wide range of values for the Lam´e coefficients and the density of the solid. Hence an Eulerian monolithic solver is introduced. The numerical tests validate the deep-learning approach.
{"title":"Parameter identification of a fluid-structure system by deep-learning with an Eulerian formulation","authors":"O. Pironneau","doi":"10.4310/maa.2019.v26.n3.a5","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n3.a5","url":null,"abstract":". A simple fluid-structure problem is considered as a test to assess the feasibility of deep-learning algorithms for parameter identification. Tensorflow by Google is used and as it is a stochastic algorithm, provision must be made for the robustness of the large displacement fluid- structure simulator with respect to a wide range of values for the Lam´e coefficients and the density of the solid. Hence an Eulerian monolithic solver is introduced. The numerical tests validate the deep-learning approach.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n3.a6
J. Rappaz, J. Rochat
{"title":"On von Karman modeling for turbulent flow near a wall","authors":"J. Rappaz, J. Rochat","doi":"10.4310/maa.2019.v26.n3.a6","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n3.a6","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-01DOI: 10.4310/maa.2019.v26.n4.a4
Shih-Wei Chou, John M. Hong, Ying-Chieh Lin
{"title":"Global existence and strong trace property of entropy solutions by the source-concentration Glimm scheme for nonlinear hyperbolic balance laws","authors":"Shih-Wei Chou, John M. Hong, Ying-Chieh Lin","doi":"10.4310/maa.2019.v26.n4.a4","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n4.a4","url":null,"abstract":"","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-19DOI: 10.4310/maa.2019.v26.n2.a3
Bjorn Engquist, Yunan Yang
Full waveform inversion (FWI) has recently become a favorite technique for the inverse problem of finding properties in the earth from measurements of vibrations of seismic waves on the surface. Mathematically, FWI is PDE constrained optimization where model parameters in a wave equation are adjusted such that the misfit between the computed and the measured dataset is minimized. In a sequence of papers, we have shown that the quadratic Wasserstein distance from optimal transport is to prefer as misfit functional over the standard $L^2$ norm. Datasets need however first to be normalized since seismic signals do not satisfy the requirements of optimal transport. There has been a puzzling contradiction in the results. Normalization methods that satisfy theorems pointing to ideal properties for FWI have not performed well in practical computations, and other scaling methods that do not satisfy these theorems have performed much better in practice. In this paper, we will shed light on this issue and resolve this contradiction.
{"title":"Seismic inversion and the data normalization for optimal transport","authors":"Bjorn Engquist, Yunan Yang","doi":"10.4310/maa.2019.v26.n2.a3","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n2.a3","url":null,"abstract":"Full waveform inversion (FWI) has recently become a favorite technique for the inverse problem of finding properties in the earth from measurements of vibrations of seismic waves on the surface. Mathematically, FWI is PDE constrained optimization where model parameters in a wave equation are adjusted such that the misfit between the computed and the measured dataset is minimized. In a sequence of papers, we have shown that the quadratic Wasserstein distance from optimal transport is to prefer as misfit functional over the standard $L^2$ norm. Datasets need however first to be normalized since seismic signals do not satisfy the requirements of optimal transport. There has been a puzzling contradiction in the results. Normalization methods that satisfy theorems pointing to ideal properties for FWI have not performed well in practical computations, and other scaling methods that do not satisfy these theorems have performed much better in practice. In this paper, we will shed light on this issue and resolve this contradiction.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49377775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-10-12DOI: 10.4310/maa.2020.v27.n1.a3
B. Temple, R. Young
The problem of applying Nash-Moser Newton methods to obtain periodic solutions of the compressible Euler equations has led authors to identify the main obstacle, namely, how to invert operators which impose periodicity when they are based on non-uniform shift operators. Here we begin a theory for finding the inverses of such operators by proving that a scalar non-uniform difference operator does in fact have a bounded inverse on its range. We argue that this is the simplest example which demonstrates the need to use direct rather than Fourier methods to analyze inverses of linear operators involving nonuniform shifts.
{"title":"Inversion of a non-uniform difference operator","authors":"B. Temple, R. Young","doi":"10.4310/maa.2020.v27.n1.a3","DOIUrl":"https://doi.org/10.4310/maa.2020.v27.n1.a3","url":null,"abstract":"The problem of applying Nash-Moser Newton methods to obtain periodic solutions of the compressible Euler equations has led authors to identify the main obstacle, namely, how to invert operators which impose periodicity when they are based on non-uniform shift operators. Here we begin a theory for finding the inverses of such operators by proving that a scalar non-uniform difference operator does in fact have a bounded inverse on its range. We argue that this is the simplest example which demonstrates the need to use direct rather than Fourier methods to analyze inverses of linear operators involving nonuniform shifts.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47036072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-06-12DOI: 10.4310/maa.2022.v29.n1.a3
Dehua Wang, Z. Ye
The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.
{"title":"Global existence and exponential decay of strong solutions for the inhomogeneous incompressible Navier–Stokes equations with vacuum","authors":"Dehua Wang, Z. Ye","doi":"10.4310/maa.2022.v29.n1.a3","DOIUrl":"https://doi.org/10.4310/maa.2022.v29.n1.a3","url":null,"abstract":"The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46629594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-01-17DOI: 10.4310/maa.2019.v26.n2.a5
Xiaobing H. Feng, T. Lewis, Stefan Schnake
This paper is concerned with continuous and discrete approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A $C^1$ finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the $H^2$ norm are shown. Lastly, numerical tests are given to show the validity of the method.
{"title":"Analysis of the vanishing moment method and its finite element approximations for second-order linear elliptic PDEs in non-divergence form","authors":"Xiaobing H. Feng, T. Lewis, Stefan Schnake","doi":"10.4310/maa.2019.v26.n2.a5","DOIUrl":"https://doi.org/10.4310/maa.2019.v26.n2.a5","url":null,"abstract":"This paper is concerned with continuous and discrete approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A $C^1$ finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the $H^2$ norm are shown. Lastly, numerical tests are given to show the validity of the method.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":" ","pages":""},"PeriodicalIF":0.3,"publicationDate":"2018-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49228685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-01-01DOI: 10.4310/maa.2018.v25.n3.a1
A. Aleksandrov
. We develop an original approach in computing analytic invariants of zero-dimensional singularities, which is based essentially on the study of properties of differential forms and the cotangent complex of multiple points. Among other things, we consider a series of specific tasks and problems for zero-dimensional complete intersections, graded and gradient singularities, including the computation of cotangent homology and cohomology for certain types of such singularities. We also examine the unimodular families of gradient zero-dimensional singularities, compile an adjacency diagram and compute the primitive ideals of these families. Finally, we briefly discuss the problem of nonexistence of negative weighted derivations, some relationships between the Milnor and Tjurina numbers and estimates of these invariants in the case of zero-dimensional complete intersections.
{"title":"Analytic invariants of multiple points","authors":"A. Aleksandrov","doi":"10.4310/maa.2018.v25.n3.a1","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n3.a1","url":null,"abstract":". We develop an original approach in computing analytic invariants of zero-dimensional singularities, which is based essentially on the study of properties of differential forms and the cotangent complex of multiple points. Among other things, we consider a series of specific tasks and problems for zero-dimensional complete intersections, graded and gradient singularities, including the computation of cotangent homology and cohomology for certain types of such singularities. We also examine the unimodular families of gradient zero-dimensional singularities, compile an adjacency diagram and compute the primitive ideals of these families. Finally, we briefly discuss the problem of nonexistence of negative weighted derivations, some relationships between the Milnor and Tjurina numbers and estimates of these invariants in the case of zero-dimensional complete intersections.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":"25 1","pages":"167-204"},"PeriodicalIF":0.3,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488845","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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