SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1372-1393, June 2024. Abstract. This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a [math]-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources.
{"title":"Inverse Wave-Number-Dependent Source Problems for the Helmholtz Equation","authors":"Hongxia Guo, Guanghui Hu","doi":"10.1137/23m1572696","DOIUrl":"https://doi.org/10.1137/23m1572696","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1372-1393, June 2024. <br/> Abstract. This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the inverse Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a [math]-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141287149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francisco Bersetche, Francisco Fuica, Enrique Otárola, Daniel Quero
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1344-1371, June 2024. Abstract. We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first- and necessary and sufficient second-order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.
{"title":"Bilinear Optimal Control for the Fractional Laplacian: Analysis and Discretization","authors":"Francisco Bersetche, Francisco Fuica, Enrique Otárola, Daniel Quero","doi":"10.1137/23m154947x","DOIUrl":"https://doi.org/10.1137/23m154947x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1344-1371, June 2024. <br/> Abstract. We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic PDE as the state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first- and necessary and sufficient second-order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141320015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1313-1343, June 2024. Abstract. In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter [math]. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of [math] into a mesh resolution condition, which can be done through error estimates that are explicit with respect to [math] and the spatial mesh width [math]. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted [math]-weighted norm. Afterward we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived [math]- and [math]-error estimates are indeed optimal in [math] and [math].
{"title":"Error Bounds for Discrete Minimizers of the Ginzburg–Landau Energy in the High-[math] Regime","authors":"Benjamin Dörich, Patrick Henning","doi":"10.1137/23m1560938","DOIUrl":"https://doi.org/10.1137/23m1560938","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1313-1343, June 2024. <br/> Abstract. In this work, we study discrete minimizers of the Ginzburg–Landau energy in finite element spaces. Special focus is given to the influence of the Ginzburg–Landau parameter [math]. This parameter is of physical interest as large values can trigger the appearance of vortex lattices. Since the vortices have to be resolved on sufficiently fine computational meshes, it is important to translate the size of [math] into a mesh resolution condition, which can be done through error estimates that are explicit with respect to [math] and the spatial mesh width [math]. For that, we first work in an abstract framework for a general class of discrete spaces, where we present convergence results in a problem-adapted [math]-weighted norm. Afterward we apply our findings to Lagrangian finite elements and a particular generalized finite element construction. In numerical experiments we confirm that our derived [math]- and [math]-error estimates are indeed optimal in [math] and [math].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peter Binev, Andrea Bonito, Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1278-1312, June 2024. Abstract. We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution. We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.
{"title":"Solving PDEs with Incomplete Information","authors":"Peter Binev, Andrea Bonito, Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova","doi":"10.1137/23m1546671","DOIUrl":"https://doi.org/10.1137/23m1546671","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1278-1312, June 2024. <br/> Abstract. We consider the problem of numerically approximating the solutions to a partial differential equation (PDE) when there is insufficient information to determine a unique solution. Our main example is the Poisson boundary value problem, when the boundary data is unknown and instead one observes finitely many linear measurements of the solution. We view this setting as an optimal recovery problem and develop theory and numerical algorithms for its solution. The main vehicle employed is the derivation and approximation of the Riesz representers of these functionals with respect to relevant Hilbert spaces of harmonic functions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141182385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1259-1277, June 2024. Abstract. We generalize Bernoulli’s classical method for finding poles of rational functions using the rational orthogonal Malmquist–Takenaka system. We show that our approach overcomes the limitations of previous methods, especially their dependence on the existence of a so-called dominant pole, while significantly simplifying the required calculations. A description of the identifiable poles is provided, as well as an iterative algorithm that can be applied to find every pole of a rational function. We discuss automatic parameter choice for the proposed algorithm and demonstrate its effectiveness through numerical examples.
{"title":"On Bernoulli’s Method","authors":"Tamás Dózsa, Ferenc Schipp, Alexandros Soumelidis","doi":"10.1137/22m1528501","DOIUrl":"https://doi.org/10.1137/22m1528501","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1259-1277, June 2024. <br/> Abstract. We generalize Bernoulli’s classical method for finding poles of rational functions using the rational orthogonal Malmquist–Takenaka system. We show that our approach overcomes the limitations of previous methods, especially their dependence on the existence of a so-called dominant pole, while significantly simplifying the required calculations. A description of the identifiable poles is provided, as well as an iterative algorithm that can be applied to find every pole of a rational function. We discuss automatic parameter choice for the proposed algorithm and demonstrate its effectiveness through numerical examples.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Benedict J. Leimkuhler, Daniel Paulin, Peter A. Whalley
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1226-1258, June 2024. Abstract. We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for [math]-[math]Lipschitz, [math]-convex potentials. Our approach gives convergence rates of [math], with explicit step size restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Further, we introduce the property “[math]-limit convergent” to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have step size restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. Finally, we provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure.
{"title":"Contraction and Convergence Rates for Discretized Kinetic Langevin Dynamics","authors":"Benedict J. Leimkuhler, Daniel Paulin, Peter A. Whalley","doi":"10.1137/23m1556289","DOIUrl":"https://doi.org/10.1137/23m1556289","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1226-1258, June 2024. <br/>Abstract. We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for [math]-[math]Lipschitz, [math]-convex potentials. Our approach gives convergence rates of [math], with explicit step size restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Further, we introduce the property “[math]-limit convergent” to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have step size restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. Finally, we provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141085304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1212-1225, June 2024. Abstract. Let [math] be a convex polytope ([math]). The Ritz projection is the best approximation, in the [math]-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, the gradient at any point in [math] is controlled by the Hardy–Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of interest in the analysis of PDEs immediately follows. Among those are weighted spaces, Orlicz spaces, and Lorentz spaces.
{"title":"Pointwise Gradient Estimate of the Ritz Projection","authors":"Lars Diening, Julian Rolfes, Abner J. Salgado","doi":"10.1137/23m1571800","DOIUrl":"https://doi.org/10.1137/23m1571800","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1212-1225, June 2024. <br/> Abstract. Let [math] be a convex polytope ([math]). The Ritz projection is the best approximation, in the [math]-norm, to a given function in a finite element space. When such finite element spaces are constructed on the basis of quasiuniform triangulations, we show a pointwise estimate on the Ritz projection. Namely, the gradient at any point in [math] is controlled by the Hardy–Littlewood maximal function of the gradient of the original function at the same point. From this estimate, the stability of the Ritz projection on a wide range of spaces that are of interest in the analysis of PDEs immediately follows. Among those are weighted spaces, Orlicz spaces, and Lorentz spaces.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141079251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1191-1211, June 2024. Abstract. We show that generalized multiquadric radial basis functions (RBFs) on [math] have a mean dimension that is [math] as [math] with an explicit bound for the implied constant, under moment conditions on their inputs. Under weaker moment conditions the mean dimension still approaches 1. As a consequence, these RBFs become essentially additive as their dimension increases. Gaussian RBFs by contrast can attain any mean dimension between 1 and [math]. We also find that a test integrand due to Keister that has been influential in quasi-Monte Carlo theory has a mean dimension that oscillates between approximately 1 and approximately 2 as the nominal dimension [math] increases.
{"title":"Mean Dimension of Radial Basis Functions","authors":"Christopher Hoyt, Art B. Owen","doi":"10.1137/23m1614833","DOIUrl":"https://doi.org/10.1137/23m1614833","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1191-1211, June 2024. <br/> Abstract. We show that generalized multiquadric radial basis functions (RBFs) on [math] have a mean dimension that is [math] as [math] with an explicit bound for the implied constant, under moment conditions on their inputs. Under weaker moment conditions the mean dimension still approaches 1. As a consequence, these RBFs become essentially additive as their dimension increases. Gaussian RBFs by contrast can attain any mean dimension between 1 and [math]. We also find that a test integrand due to Keister that has been influential in quasi-Monte Carlo theory has a mean dimension that oscillates between approximately 1 and approximately 2 as the nominal dimension [math] increases.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141074216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1171-1190, June 2024. Abstract. We prove a new numerical approximation result for the solutions of semilinear parabolic stochastic partial differential equations, driven by additive space-time white noise in dimension 1. The temporal discretization is performed using an accelerated exponential Euler scheme, and we show that, under appropriate regularity conditions on the nonlinearity, the total variation distance between the distributions of the numerical approximation and of the exact solution at a given time converges to 0 when the time-step size vanishes, with order of convergence [math]. Equivalently, weak error estimates with order [math] are thus obtained for bounded measurable test functions. This is an original and major improvement compared with the performance of the standard linear implicit Euler scheme or exponential Euler methods, which do not converge in the sense of total variation when the time-step size vanishes. Equivalently weak error estimates for the standard schemes require twice differentiable test functions. The proof of the total variation error bounds for the accelerated exponential Euler scheme exploits some regularization property of the associated infinite-dimensional Kolmogorov equations.
{"title":"Total Variation Error Bounds for the Accelerated Exponential Euler Scheme Approximation of Parabolic Semilinear SPDEs","authors":"Charles-Edouard Bréhier","doi":"10.1137/22m152596x","DOIUrl":"https://doi.org/10.1137/22m152596x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1171-1190, June 2024. <br/> Abstract. We prove a new numerical approximation result for the solutions of semilinear parabolic stochastic partial differential equations, driven by additive space-time white noise in dimension 1. The temporal discretization is performed using an accelerated exponential Euler scheme, and we show that, under appropriate regularity conditions on the nonlinearity, the total variation distance between the distributions of the numerical approximation and of the exact solution at a given time converges to 0 when the time-step size vanishes, with order of convergence [math]. Equivalently, weak error estimates with order [math] are thus obtained for bounded measurable test functions. This is an original and major improvement compared with the performance of the standard linear implicit Euler scheme or exponential Euler methods, which do not converge in the sense of total variation when the time-step size vanishes. Equivalently weak error estimates for the standard schemes require twice differentiable test functions. The proof of the total variation error bounds for the accelerated exponential Euler scheme exploits some regularization property of the associated infinite-dimensional Kolmogorov equations.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141304419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christa Cuchiero, Christoph Reisinger, Stefan Rigger
SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1145-1170, June 2024. Abstract.We consider two approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme studied by Kaushansky et al. [Ann. Appl. Probab., 33 (2023), pp. 274–298], but here the flux over the free boundary and its velocity are coupled implicitly. Moreover, we extend the analysis to more general driving processes than Brownian motion. The second scheme is a Donsker-type approximation, also interpretable as an implicit finite difference scheme, for which global convergence is shown under minor technical conditions. With stronger assumptions, which apply in cases without blow-ups, we obtain additionally a convergence rate arbitrarily close to 1/2. Our numerical results suggest that this rate also holds for less regular solutions, in contrast to explicit schemes, and allow a sharper resolution of the discontinuous free boundary in the blow-up regime.
{"title":"Implicit and Fully Discrete Approximation of the Supercooled Stefan Problem in the Presence of Blow-Ups","authors":"Christa Cuchiero, Christoph Reisinger, Stefan Rigger","doi":"10.1137/22m1509722","DOIUrl":"https://doi.org/10.1137/22m1509722","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 3, Page 1145-1170, June 2024. <br/> Abstract.We consider two approximation schemes of the one-dimensional supercooled Stefan problem and prove their convergence, even in the presence of finite time blow-ups. All proofs are based on a probabilistic reformulation recently considered in the literature. The first scheme is a version of the time-stepping scheme studied by Kaushansky et al. [Ann. Appl. Probab., 33 (2023), pp. 274–298], but here the flux over the free boundary and its velocity are coupled implicitly. Moreover, we extend the analysis to more general driving processes than Brownian motion. The second scheme is a Donsker-type approximation, also interpretable as an implicit finite difference scheme, for which global convergence is shown under minor technical conditions. With stronger assumptions, which apply in cases without blow-ups, we obtain additionally a convergence rate arbitrarily close to 1/2. Our numerical results suggest that this rate also holds for less regular solutions, in contrast to explicit schemes, and allow a sharper resolution of the discontinuous free boundary in the blow-up regime.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140902996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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