Pub Date : 2024-01-01Epub Date: 2024-10-11DOI: 10.1007/s42985-024-00272-4
Christian Beck, Lukas Gonon, Arnulf Jentzen
Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.
{"title":"Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations.","authors":"Christian Beck, Lukas Gonon, Arnulf Jentzen","doi":"10.1007/s42985-024-00272-4","DOIUrl":"10.1007/s42985-024-00272-4","url":null,"abstract":"<p><p>Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear <i>parabolic</i> partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear <i>elliptic</i> PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.</p>","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"5 6","pages":"31"},"PeriodicalIF":0.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11469984/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142482611","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}
Pub Date : 2023-11-10DOI: 10.1007/s42985-023-00267-7
Jennifer Chepkorir, Fredrik Berntsson, Vladimir Kozlov
Abstract We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.
{"title":"Solving stationary inverse heat conduction in a thin plate","authors":"Jennifer Chepkorir, Fredrik Berntsson, Vladimir Kozlov","doi":"10.1007/s42985-023-00267-7","DOIUrl":"https://doi.org/10.1007/s42985-023-00267-7","url":null,"abstract":"Abstract We consider a steady state heat conduction problem in a thin plate. In the application, it is used to connect two cylindrical containers and fix their relative positions. At the same time it serves to measure the temperature on the inner cylinder. We derive a two dimensional mathematical model, and use it to approximate the heat conduction in the thin plate. Since the plate has sharp edges on the sides the resulting problem is described by a degenerate elliptic equation. To find the temperature in the interior part from the exterior measurements, we formulate the problem as a Cauchy problem for stationary heat equation. We also reformulate the Cauchy problem as an operator equation, with a compact operator, and apply the Landweber iteration method to solve the equation. The case of the degenerate elliptic equation has not been previously studied in this context. For numerical computation, we consider the case where noisy data is present and analyse the convergence.","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"93 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135092060","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 : 2023-11-06DOI: 10.1007/s42985-023-00268-6
Alexandra Neamţu, Tim Seitz
Abstract We investigate the pathwise well-posedness of stochastic partial differential equations perturbed by multiplicative Neumann boundary noise, such as fractional Brownian motion for $$Hin (1/3,1/2].$$ H∈(1/3,1/2]. Combining functional analytic tools with the controlled rough path approach, we establish global existence of solutions and flows for such equations. For Dirichlet boundary noise we obtain similar results for smoother noise, i.e. in the Young regime.
{"title":"Stochastic evolution equations with rough boundary noise","authors":"Alexandra Neamţu, Tim Seitz","doi":"10.1007/s42985-023-00268-6","DOIUrl":"https://doi.org/10.1007/s42985-023-00268-6","url":null,"abstract":"Abstract We investigate the pathwise well-posedness of stochastic partial differential equations perturbed by multiplicative Neumann boundary noise, such as fractional Brownian motion for $$Hin (1/3,1/2].$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>]</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Combining functional analytic tools with the controlled rough path approach, we establish global existence of solutions and flows for such equations. For Dirichlet boundary noise we obtain similar results for smoother noise, i.e. in the Young regime.","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135636442","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 : 2023-10-26DOI: 10.1007/s42985-023-00266-8
E. Compaan, N. Tzirakis
{"title":"Sharp well-posedness of the biharmonic Schrödinger equation in a quarter plane","authors":"E. Compaan, N. Tzirakis","doi":"10.1007/s42985-023-00266-8","DOIUrl":"https://doi.org/10.1007/s42985-023-00266-8","url":null,"abstract":"","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134906852","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 : 2023-10-25DOI: 10.1007/s42985-023-00265-9
Cheng, Hanz Martin, Boonkkamp, Jan ten thije, Janssen, Jesper, Mihailova, Diana, van Dijk, Jan
In this paper, we propose a numerical scheme for fluid models of magnetised plasmas. One important feature of the numerical scheme is that it should be able to handle the anisotropy induced by the magnetic field. In order to do so, we propose the use of the hybrid mimetic mixed (HMM) scheme for diffusion. This is combined with a hybridised variant of the Scharfetter-Gummel (SG) scheme for advection. The proposed hybrid scheme can be implemented very efficiently via static condensation. Numerical tests are then performed to show the applicability of the combined HMM-SG scheme, even for highly anisotropic magnetic fields.
{"title":"Combining the hybrid mimetic mixed method with the Scharfetter-Gummel scheme for magnetised transport in plasmas","authors":"Cheng, Hanz Martin, Boonkkamp, Jan ten thije, Janssen, Jesper, Mihailova, Diana, van Dijk, Jan","doi":"10.1007/s42985-023-00265-9","DOIUrl":"https://doi.org/10.1007/s42985-023-00265-9","url":null,"abstract":"In this paper, we propose a numerical scheme for fluid models of magnetised plasmas. One important feature of the numerical scheme is that it should be able to handle the anisotropy induced by the magnetic field. In order to do so, we propose the use of the hybrid mimetic mixed (HMM) scheme for diffusion. This is combined with a hybridised variant of the Scharfetter-Gummel (SG) scheme for advection. The proposed hybrid scheme can be implemented very efficiently via static condensation. Numerical tests are then performed to show the applicability of the combined HMM-SG scheme, even for highly anisotropic magnetic fields.","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"19 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972401","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 : 2023-10-11DOI: 10.1007/s42985-023-00264-w
Simone Farinelli
{"title":"Dirac cohomology on manifolds with boundary and spectral lower bounds","authors":"Simone Farinelli","doi":"10.1007/s42985-023-00264-w","DOIUrl":"https://doi.org/10.1007/s42985-023-00264-w","url":null,"abstract":"","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136057767","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 : 2023-09-26DOI: 10.1007/s42985-023-00258-8
Olivier Bokanowski, Averil Prost, Xavier Warin
We consider a deterministic optimal control problem, focusing on a finite horizon scenario. Our proposal involves employing deep neural network approximations to capture Bellman’s dynamic programming principle. This also corresponds to solving first-order Hamilton–Jacobi–Bellman (HJB) equations. Our work builds upon the research conducted by Huré et al. (SIAM J Numer Anal 59(1):525–557, 2021), which primarily focused on stochastic contexts. However, our objective is to develop a completely novel approach specifically designed to address error propagation in the absence of diffusion in the dynamics of the system. Our analysis provides precise error estimates in terms of an average norm. Furthermore, we provide several academic numerical examples that pertain to front propagation models incorporating obstacle constraints, demonstrating the effectiveness of our approach for systems with moderate dimensions (e.g., ranging from 2 to 8) and for nonsmooth value functions.
{"title":"Neural networks for first order HJB equations and application to front propagation with obstacle terms","authors":"Olivier Bokanowski, Averil Prost, Xavier Warin","doi":"10.1007/s42985-023-00258-8","DOIUrl":"https://doi.org/10.1007/s42985-023-00258-8","url":null,"abstract":"We consider a deterministic optimal control problem, focusing on a finite horizon scenario. Our proposal involves employing deep neural network approximations to capture Bellman’s dynamic programming principle. This also corresponds to solving first-order Hamilton–Jacobi–Bellman (HJB) equations. Our work builds upon the research conducted by Huré et al. (SIAM J Numer Anal 59(1):525–557, 2021), which primarily focused on stochastic contexts. However, our objective is to develop a completely novel approach specifically designed to address error propagation in the absence of diffusion in the dynamics of the system. Our analysis provides precise error estimates in terms of an average norm. Furthermore, we provide several academic numerical examples that pertain to front propagation models incorporating obstacle constraints, demonstrating the effectiveness of our approach for systems with moderate dimensions (e.g., ranging from 2 to 8) and for nonsmooth value functions.","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903236","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 : 2023-09-21DOI: 10.1007/s42985-023-00263-x
Michaël Ndjinga, Marcial Nguemfouo
{"title":"Well posedness for the Poisson problem on closed Lipschitz manifolds","authors":"Michaël Ndjinga, Marcial Nguemfouo","doi":"10.1007/s42985-023-00263-x","DOIUrl":"https://doi.org/10.1007/s42985-023-00263-x","url":null,"abstract":"","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136235221","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 : 2023-09-16DOI: 10.1007/s42985-023-00262-y
Cornelia Schneider, Flóra Orsolya Szemenyei
Abstract We study the regularity of solutions of parabolic partial differential equations with inhomogeneous boundary conditions on polyhedral domains $$Dsubset mathbb {R}^3$$ D⊂R3 in the specific scale $$ B^{alpha }_{tau ,tau }, frac{1}{tau }=frac{alpha }{3}+frac{1}{p} $$ Bτ,τα,1τ=α3+1p of Besov spaces. The regularity of the solution in this scale determines the order of approximation that can be achieved by adaptive numerical schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our results are in good agreement with the forerunner (Dahlke and Schneider in Anal Appl 17:235–291, 2019), where parabolic equations with homogeneous boundary conditions were investigated.
{"title":"Besov regularity of inhomogeneous parabolic PDEs","authors":"Cornelia Schneider, Flóra Orsolya Szemenyei","doi":"10.1007/s42985-023-00262-y","DOIUrl":"https://doi.org/10.1007/s42985-023-00262-y","url":null,"abstract":"Abstract We study the regularity of solutions of parabolic partial differential equations with inhomogeneous boundary conditions on polyhedral domains $$Dsubset mathbb {R}^3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> in the specific scale $$ B^{alpha }_{tau ,tau }, frac{1}{tau }=frac{alpha }{3}+frac{1}{p} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mspace /> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>τ</mml:mi> </mml:mrow> <mml:mi>α</mml:mi> </mml:msubsup> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>τ</mml:mi> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mi>α</mml:mi> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>p</mml:mi> </mml:mfrac> <mml:mspace /> </mml:mrow> </mml:math> of Besov spaces. The regularity of the solution in this scale determines the order of approximation that can be achieved by adaptive numerical schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our results are in good agreement with the forerunner (Dahlke and Schneider in Anal Appl 17:235–291, 2019), where parabolic equations with homogeneous boundary conditions were investigated.","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"198 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135306448","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 : 2023-09-09DOI: 10.1007/s42985-023-00261-z
Yanheng Ding, Qi Guo, Yuanyang Yu
{"title":"Semiclassical states of a type of Dirac–Klein–Gordon equations with nonlinear interacting terms","authors":"Yanheng Ding, Qi Guo, Yuanyang Yu","doi":"10.1007/s42985-023-00261-z","DOIUrl":"https://doi.org/10.1007/s42985-023-00261-z","url":null,"abstract":"","PeriodicalId":74818,"journal":{"name":"SN partial differential equations and applications","volume":"27 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86056597","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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