Mathematical models and numerical methods for water management in soils
Duration: 02/2022 – 02/2024
Funding scheme: Funding for Mobility Cooperation with Germany
Funder: Academy of Finland
Budget: 18,000 €
My role: Finnish principal investigator
Sibling project: Discontinuous Galerkin methods and parameter estimation for microstructure models in porous media
German principal investigator: Nadja Ray
Public description
Nowadays, soils should meet more and more requirements. For example, they should be stable to build houses on them safely. In flood-prone areas, they should be able to absorb a lot of water, and in arid regions, they should hold water as well as possible. Soils also play an essential role in the greenhouse effect, as they (and not just the plants growing on their surface) can sequester enormous amounts of carbon and release it if not treated properly.
However, if we want to treat soils to meet all or at least some of these requirements, we need to understand the basic building blocks (i.e., microaggregates; see the picture) of soils and how their development affects soil properties. To this end, we develop two models and numerical methods to simulate them. First, we describe the formation of soil microaggregates, and second, we consider the flow of fluids through consolidated soil. Our goal is to apply the results to support water management.
The image is taken from Microaggregates in soils by Kai Uwe Totsche et al., licensed under CC BY 4.0.
Scientific abstract
This exchange program develops new mathematical tools for porous media applications. The main goal is to support water management by providing models for the consolidation of soils and the flow and transport through them as well as numerical methods to simulate them on the computer. We use these models and techniques to optimize soil parameters and ultimately serve the goal of treating the soil in such a way that, for example, plants find optimal growing conditions or icroorganisms in the ground can sequester as much CO2 as possible.
The first research focuses on parameter estimation for cellular automaton (CA) based models. Such models have been successfully developed and applied in soil science research by Andreas Rupp (Lappeenranta-Lahti University of Technology, abbreviated as LUT) and the group at the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU). However, physically meaningful parameters that enter the CA model are not experimentally accessible. Consequently, the parameterization of the underlying models remains a challenge. Heikki Haario’s group (LUT) has published research articles on determining relevant parameters in partial differential equations. We use adapted versions of these methods to find appropriate parameters for the CA models.
In the second part of the exchange program, we address discontinuous Galerkin methods for models used to simulate soil applications with two porosities. Dual porosity models accurately describe flow through a porous medium whose solid matrix is composed of a porous medium. Here, we characterize the flow within the large pores by a Stokes equation while formulating the flow through the (micropores of the) porous matrix itself with Darcy’s law. Coupling strategies between the different flow models are constructed and analyzed using techniques developed in the context of petroleum applications or the interactions of lakes with their bottoms. Ultimately, our research may facilitate the design of more efficient batteries and accumulators. Thus, this research topic can be a starting point for collaborating with Prof. Bernardo Barbiellini (LUT) to optimize electric batteries further.
Keywords
cellular automaton model, coupled Stokes-Darcy flow, coupling condition, discontinuous Galerkin, dual-porosity flow, inverse problem, mathematical modeling, microaggregate, numerical methods, parameter estimation, soil science, water management
Related publications
Liu, Zhi-Song; Büttner, Markus; Aizinger, Vadym; Rupp, Andreas
Super-resolution works for coastal simulations Online Forthcoming
Forthcoming, visited: 30.08.2024.
@online{LiuBAR24,
title = {Super-resolution works for coastal simulations},
author = {Zhi-Song Liu and Markus Büttner and Vadym Aizinger and Andreas Rupp},
url = {https://arxiv.org/abs/2408.16553},
year = {2024},
date = {2024-08-30},
urldate = {2024-08-30},
abstract = {Learning fine-scale details of a coastal ocean simulation from a coarse representation is a challenging task. For real-world applications, high-resolution simulations are necessary to advance understanding of many coastal processes, specifically, to predict flooding resulting from tsunamis and storm surges. We propose a Deep Network for Coastal Super-Resolution (DNCSR) for spatiotemporal enhancement to efficiently learn the high-resolution numerical solution. Given images of coastal simulations produced on low-resolution computational meshes using low polynomial order discontinuous Galerkin discretizations and a coarse temporal resolution, the proposed DNCSR learns to produce high-resolution free surface elevation and velocity visualizations in both time and space. To efficiently model the dynamic changes over time and space, we propose grid-aware spatiotemporal attention to project the temporal features to the spatial domain for non-local feature matching. The coordinate information is also utilized via positional encoding. For the final reconstruction, we use the spatiotemporal bilinear operation to interpolate the missing frames and then expand the feature maps to the frequency domain for residual mapping. Besides data-driven losses, the proposed physics-informed loss guarantees gradient consistency and momentum changes. Their combination contributes to the overall 24% improvements in RMSE. To train the proposed model, we propose a large-scale coastal simulation dataset and use it for model optimization and evaluation. Our method shows superior super-resolution quality and fast computation compared to the state-of-the-art methods.},
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Learning fine-scale details of a coastal ocean simulation from a coarse representation is a challenging task. For real-world applications, high-resolution simulations are necessary to advance understanding of many coastal processes, specifically, to predict flooding resulting from tsunamis and storm surges. We propose a Deep Network for Coastal Super-Resolution (DNCSR) for spatiotemporal enhancement to efficiently learn the high-resolution numerical solution. Given images of coastal simulations produced on low-resolution computational meshes using low polynomial order discontinuous Galerkin discretizations and a coarse temporal resolution, the proposed DNCSR learns to produce high-resolution free surface elevation and velocity visualizations in both time and space. To efficiently model the dynamic changes over time and space, we propose grid-aware spatiotemporal attention to project the temporal features to the spatial domain for non-local feature matching. The coordinate information is also utilized via positional encoding. For the final reconstruction, we use the spatiotemporal bilinear operation to interpolate the missing frames and then expand the feature maps to the frequency domain for residual mapping. Besides data-driven losses, the proposed physics-informed loss guarantees gradient consistency and momentum changes. Their combination contributes to the overall 24% improvements in RMSE. To train the proposed model, we propose a large-scale coastal simulation dataset and use it for model optimization and evaluation. Our method shows superior super-resolution quality and fast computation compared to the state-of-the-art methods.
Hauck, Moritz; Målqvist, Axel; Rupp, Andreas
Arbitrary order approximations at constant cost for Timoshenko beam network models Online Forthcoming
Forthcoming.
@online{HauckMR24,
title = {Arbitrary order approximations at constant cost for Timoshenko beam network models},
author = {Moritz Hauck and Axel Målqvist and Andreas Rupp},
url = {https://arxiv.org/abs/2407.14388},
year = {2024},
date = {2024-07-19},
abstract = {This paper considers the numerical solution of Timoshenko beam network models, comprised of Timoshenko beam equations on each edge of the network, which are coupled at the nodes of the network using rigid joint conditions. Through hybridization, we can equivalently reformulate the problem as a symmetric positive definite system of linear equations posed on the network nodes. This is possible since the nodes, where the beam equations are coupled, are zero-dimensional objects. To discretize the beam network model, we propose a hybridizable discontinuous Galerkin method that can achieve arbitrary orders of convergence under mesh refinement without increasing the size of the global system matrix. As a preconditioner for the typically very poorly conditioned global system matrix, we employ a two-level overlapping additive Schwarz method. We prove uniform convergence of the corresponding preconditioned conjugate gradient method under appropriate connectivity assumptions on the network. Numerical experiments support the theoretical findings of this work.},
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This paper considers the numerical solution of Timoshenko beam network models, comprised of Timoshenko beam equations on each edge of the network, which are coupled at the nodes of the network using rigid joint conditions. Through hybridization, we can equivalently reformulate the problem as a symmetric positive definite system of linear equations posed on the network nodes. This is possible since the nodes, where the beam equations are coupled, are zero-dimensional objects. To discretize the beam network model, we propose a hybridizable discontinuous Galerkin method that can achieve arbitrary orders of convergence under mesh refinement without increasing the size of the global system matrix. As a preconditioner for the typically very poorly conditioned global system matrix, we employ a two-level overlapping additive Schwarz method. We prove uniform convergence of the corresponding preconditioned conjugate gradient method under appropriate connectivity assumptions on the network. Numerical experiments support the theoretical findings of this work.
Vedral, Joshua; Rupp, Andreas; Kuzmin, Dmitri
Strongly consistent low-dissipation WENO schemes for finite elements Online Forthcoming
Forthcoming, visited: 08.07.2024.
@online{VedralRK24,
title = {Strongly consistent low-dissipation WENO schemes for finite elements},
author = {Joshua Vedral and Andreas Rupp and Dmitri Kuzmin},
url = {https://arxiv.org/abs/2407.04646},
year = {2024},
date = {2024-07-08},
urldate = {2024-07-08},
abstract = {We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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We propose a way to maintain strong consistency and facilitate error analysis in the context of dissipation-based WENO stabilization for continuous and discontinuous Galerkin discretizations of conservation laws. Following Kuzmin and Vedral (J. Comput. Phys. 487:112153, 2023) and Vedral (arXiv preprint arXiv:2309.12019), we use WENO shock detectors to determine appropriate amounts of low-order artificial viscosity. In contrast to existing WENO methods, our approach blends candidate polynomials using residual-based nonlinear weights. The shock-capturing terms of our stabilized Galerkin methods vanish if residuals do. This enables us to achieve improved accuracy compared to weakly consistent alternatives. As we show in the context of steady convection-diffusion-reaction (CDR) equations, nonlinear local projection stabilization terms can be included in a way that preserves the coercivity of local bilinear forms. For the corresponding Galerkin-WENO discretization of a CDR problem, we rigorously derive a priori error estimates. Additionally, we demonstrate the stability and accuracy of the proposed method through one- and two-dimensional numerical experiments for hyperbolic conservation laws and systems thereof. The numerical results for representative test problems are superior to those obtained with traditional WENO schemes, particularly in scenarios involving shocks and steep gradients.
Cheng, Hanz Martin; Helin, Tapio; Manninen, Ville-Petteri; Holopainen, Timo; Jokinen, Juha; Sorvari, Samu; Rupp, Andreas
Recovery of transversely-isotropic elastic material parameters in induction motor rotors Online Forthcoming
Forthcoming, visited: 10.05.2024.
@online{ChengHMHJSR24,
title = {Recovery of transversely-isotropic elastic material parameters in induction motor rotors},
author = {Hanz Martin Cheng and Tapio Helin and Ville-Petteri Manninen and Timo Holopainen and Juha Jokinen and Samu Sorvari and Andreas Rupp},
url = {https://arxiv.org/abs/2405.06388},
year = {2024},
date = {2024-05-10},
urldate = {2024-05-10},
abstract = {We propose numerical algorithms for recovering parameters in eigenvalue problems for linear elasticity of transversely isotropic materials. Specifically, the algorithms are used to recover the elastic constants of a rotor core. Numerical tests show that in the noiseless setup, two pairs of bending modes are sufficient for recovering one to four parameters accurately. To recover all five parameters that govern the elastic properties of electric engines accurately, we require three pairs of bending modes and one torsional mode. Moreover, we study the stability of the inversion method against multiplicative noise; for tests in which the data contained multiplicative noise of at most 1%, we find that all parameters can be recovered with an error less than 10%. },
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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We propose numerical algorithms for recovering parameters in eigenvalue problems for linear elasticity of transversely isotropic materials. Specifically, the algorithms are used to recover the elastic constants of a rotor core. Numerical tests show that in the noiseless setup, two pairs of bending modes are sufficient for recovering one to four parameters accurately. To recover all five parameters that govern the elastic properties of electric engines accurately, we require three pairs of bending modes and one torsional mode. Moreover, we study the stability of the inversion method against multiplicative noise; for tests in which the data contained multiplicative noise of at most 1%, we find that all parameters can be recovered with an error less than 10%.
Di Pietro, Daniele Antonio; Dong, Zhaonan; Kanschat, Guido; Matalon, Pierre; Rupp, Andreas
Homogeneous multigrid for hybrid discretizations: application to HHO methods Online Forthcoming
Forthcoming, visited: 25.03.2024.
@online{DiPietroDKMR24,
title = {Homogeneous multigrid for hybrid discretizations: application to HHO methods},
author = {Di Pietro, Daniele Antonio and Zhaonan Dong and Guido Kanschat and Pierre Matalon and Andreas Rupp},
url = {https://arxiv.org/abs/2403.15858
https://inria.hal.science/hal-04518103},
year = {2024},
date = {2024-03-25},
urldate = {2024-03-25},
abstract = {We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
We prove the uniform convergence of the geometric multigrid V-cycle for hybrid high-order (HHO) and other discontinuous skeletal methods. Our results generalize previously established results for HDG methods, and our multigrid method uses standard smoothers and local solvers that are bounded, convergent, and consistent. We use a weak version of elliptic regularity in our proofs. Numerical experiments confirm our theoretical results.
Lu, Peipei; Wang, Wei; Kanschat, Guido; Rupp, Andreas
Homogeneous multigrid for HDG applied to the Stokes equation Journal Article
In: IMA Journal of Numerical Analysis, vol. 44, no. 5, pp. 3124–3152, 2023, ISBN: 0272-4979.
@article{LuWKR23,
title = {Homogeneous multigrid for HDG applied to the Stokes equation},
author = {Peipei Lu and Wei Wang and Guido Kanschat and Andreas Rupp},
url = {https://academic.oup.com/imajna/advance-article-pdf/doi/10.1093/imanum/drad079/52422702/drad079.pdf},
doi = {10.1093/imanum/drad079},
isbn = {0272-4979},
year = {2023},
date = {2023-10-23},
urldate = {2023-10-23},
journal = {IMA Journal of Numerical Analysis},
volume = {44},
number = {5},
pages = {3124--3152},
abstract = {We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn and Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput., 74, 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings.},
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We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn and Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput., 74, 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings.
Altunay, Rabia; Vesterinen, Kalevi; Alander, Pasi; Immonen, Eero; Rupp, Andreas; Roininen, Lassi
Denture reinforcement via topology optimization Online Forthcoming
Forthcoming, visited: 01.09.2023.
@online{AltunayVAIRR23,
title = {Denture reinforcement via topology optimization},
author = {Rabia Altunay and Kalevi Vesterinen and Pasi Alander and Eero Immonen and Andreas Rupp and Lassi Roininen},
url = {https://arxiv.org/abs/2309.00396},
year = {2023},
date = {2023-09-01},
urldate = {2023-09-01},
abstract = {We present a computational design method that optimizes the reinforcement of dental prostheses and increases the durability and fracture resistance of dentures. Our approach optimally places reinforcement, which could be implemented by modern multi-material, three-dimensional printers. The study focuses on reducing deformation by identifying regions within the structure that require reinforcement (E-glass material). Our method is applied to a three-dimensional removable lower jaw dental prosthesis and aims to improve the living quality of denture patients and pretend fracture of dental reinforcement in clinical studies. To do this, we compare the deformation results of a non-reinforced denture and a reinforced denture that has two materials. The results indicate the maximum deformation is lower and node-based displacement distribution demonstrates that the average displacement distribution is much better in the reinforced denture.},
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
}
We present a computational design method that optimizes the reinforcement of dental prostheses and increases the durability and fracture resistance of dentures. Our approach optimally places reinforcement, which could be implemented by modern multi-material, three-dimensional printers. The study focuses on reducing deformation by identifying regions within the structure that require reinforcement (E-glass material). Our method is applied to a three-dimensional removable lower jaw dental prosthesis and aims to improve the living quality of denture patients and pretend fracture of dental reinforcement in clinical studies. To do this, we compare the deformation results of a non-reinforced denture and a reinforced denture that has two materials. The results indicate the maximum deformation is lower and node-based displacement distribution demonstrates that the average displacement distribution is much better in the reinforced denture.
Lu, Peipei; Maier, Roland; Rupp, Andreas
A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods Online Forthcoming
Forthcoming, visited: 28.07.2023.
@online{LuMR23,
title = {A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods},
author = {Peipei Lu and Roland Maier and Andreas Rupp},
url = {https://arxiv.org/abs/2307.14961},
year = {2023},
date = {2023-07-28},
urldate = {2023-07-28},
abstract = {We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is a first step to reliably merge hybrid skeletal formulations and localized orthogonal decomposition and unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings. },
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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We formulate and analyze a multiscale method for an elliptic problem with an oscillatory coefficient based on a skeletal (hybrid) formulation. More precisely, we employ hybrid discontinuous Galerkin approaches and combine them with the localized orthogonal decomposition methodology to obtain a coarse-scale skeletal method that effectively includes fine-scale information. This work is a first step to reliably merge hybrid skeletal formulations and localized orthogonal decomposition and unite the advantages of both strategies. Numerical experiments are presented to illustrate the theoretical findings.
Musch, Markus; Rupp, Andreas; Aizinger, Vadym; Knabner, Peter
Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps Journal Article
In: GEM - International Journal on Geomathematics, vol. 14, no. 18, pp. 25, 2023, ISSN: 1869-2680.
@article{MuschRAK23,
title = {Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps},
author = {Markus Musch and Andreas Rupp and Vadym Aizinger and Peter Knabner},
url = {https://link.springer.com/article/10.1007/s13137-023-00228-7},
doi = {10.1007/s13137-023-00228-7},
issn = {1869-2680},
year = {2023},
date = {2023-07-13},
urldate = {2023-07-13},
journal = {GEM - International Journal on Geomathematics},
volume = {14},
number = {18},
pages = {25},
abstract = {We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns. },
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pubstate = {published},
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We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.
Kazarnikov, Alexey; Ray, Nadja; Haario, Heikki; Lappalainen, Joona; Rupp, Andreas
Parameter estimation for cellular automata Online Forthcoming
Forthcoming, visited: 01.02.2023.
@online{KazarnikovRHLR23,
title = {Parameter estimation for cellular automata},
author = {Alexey Kazarnikov and Nadja Ray and Heikki Haario and Joona Lappalainen and Andreas Rupp},
url = {https://arxiv.org/abs/2301.13320},
year = {2023},
date = {2023-02-01},
urldate = {2023-02-01},
abstract = {Self organizing complex systems can be modeled using cellular automaton models. However, the parametrization of these models is crucial and significantly determines the resulting structural pattern. In this research, we introduce and successfully apply a sound statistical method to estimate these parameters. The method is based on constructing Gaussian likelihoods using characteristics of the structures such as the mean particle size. We show that our approach is robust with respect to the method parameters, domain size of patterns, or CA iterations. },
keywords = {},
pubstate = {forthcoming},
tppubtype = {online}
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Self organizing complex systems can be modeled using cellular automaton models. However, the parametrization of these models is crucial and significantly determines the resulting structural pattern. In this research, we introduce and successfully apply a sound statistical method to estimate these parameters. The method is based on constructing Gaussian likelihoods using characteristics of the structures such as the mean particle size. We show that our approach is robust with respect to the method parameters, domain size of patterns, or CA iterations.
Kaarnioja, Vesa; Rupp, Andreas
Quasi-Monte Carlo and discontinuous Galerkin Online Forthcoming
Forthcoming, visited: 15.07.2022.
@online{KaarniojaR22,
title = {Quasi-Monte Carlo and discontinuous Galerkin},
author = {Vesa Kaarnioja and Andreas Rupp},
url = {https://arxiv.org/abs/2207.07698},
year = {2022},
date = {2022-07-15},
urldate = {2022-07-15},
abstract = {In this study, we design and develop Quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficient. That is, we consider the affine and uniform, and the lognormal models for the input random field, and investigate the QMC cubatures to compute the response statistics (expectation and variance) of the discretized PDE. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Numerical results underline our analytical findings. Moreover, we present a novel analysis for the parametric regularity in the lognormal setting.},
keywords = {},
pubstate = {forthcoming},
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In this study, we design and develop Quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficient. That is, we consider the affine and uniform, and the lognormal models for the input random field, and investigate the QMC cubatures to compute the response statistics (expectation and variance) of the discretized PDE. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Numerical results underline our analytical findings. Moreover, we present a novel analysis for the parametric regularity in the lognormal setting.
