| 1. |
- Glaubitz, Jan, et al.
(författare)
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Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form
- 2024
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Ingår i: Journal of Scientific Computing. - : SPRINGER/PLENUM PUBLISHERS. - 0885-7474 .- 1573-7691. ; 98:1
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Tidskriftsartikel (refereegranskat)abstract
- Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop a stability theory for global radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities. Although we address their practical construction, we restrict the discussion to basic numerical simulations and focus on providing a proof of concept.
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| 2. |
- Glaubitz, Jan, et al.
(författare)
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Multi-dimensional summation-by-parts operators for general function spaces: Theory and construction
- 2023
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 491
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Tidskriftsartikel (refereegranskat)abstract
- Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods. © 2023 The Author(s)
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| 3. |
- Glaubitz, Jan, et al.
(författare)
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Summation-by-Parts Operators for General Function Spaces
- 2023
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 61:2, s. 733-754
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Tidskriftsartikel (refereegranskat)abstract
- Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.
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| 4. |
- Glaubitz, Jan, et al.
(författare)
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Summation-by-parts operators for general function spaces: The second derivative
- 2024
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Ingår i: Journal of Computational Physics. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0021-9991 .- 1090-2716. ; 504
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Tidskriftsartikel (refereegranskat)abstract
- Many applications rely on solving time-dependent partial differential equations (PDEs) that include second derivatives. Summation-by-parts (SBP) operators are crucial for developing stable, high-order accurate numerical methodologies for such problems. Conventionally, SBP operators are tailored to the assumption that polynomials accurately approximate the solution, and SBP operators should thus be exact for them. However, this assumption falls short for a range of problems for which other approximation spaces are better suited. We recently addressed this issue and developed a theory for first-derivative SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the innovation of FSBP operators to accommodate second derivatives. The developed second-derivative FSBP operators maintain the desired mimetic properties of existing polynomial SBP operators while allowing for greater flexibility by being applicable to a broader range of function spaces. We establish the existence of these operators and detail a straightforward methodology for constructing them. By exploring various function spaces, including trigonometric, exponential, and radial basis functions, we illustrate the versatility of our approach. The work presented here opens up possibilities for using second-derivative SBP operators based on suitable function spaces, paving the way for a wide range of applications in the future.
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| 5. |
- Vinyals, Marc, et al.
(författare)
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Simplified and Improved Separations Between Regular and General Resolution by Lifting
- 2020
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Ingår i: Theory and Applications of Satisfiability Testing – SAT 2020 : 23rd International Conference, Proceedings - 23rd International Conference, Proceedings. - Cham : Springer International Publishing. - 1611-3349 .- 0302-9743. - 9783030518240 - 9783030518257 ; 12178, s. 182-200
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Konferensbidrag (refereegranskat)abstract
- We give a significantly simplified proof of the exponential separation between regular and general resolution of Alekhnovich et al. (2007) as a consequence of a general theorem lifting proof depth to regular proof length in resolution. This simpler proof then allows us to strengthen the separation further, and to construct families of theoretically very easy benchmarks that are surprisingly hard for SAT solvers in practice.
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| 6. |
- Abel, Edvard, 1970, et al.
(författare)
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Fatigue in Patients With Head and Neck Cancer Treated With Radiation Therapy: A Prospective Study of Patient-Reported Outcomes and Their Association With Radiation Dose to the Cerebellum
- 2022
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Ingår i: Advances in Radiation Oncology. - : Elsevier BV. - 2452-1094. ; 7:5
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Tidskriftsartikel (refereegranskat)abstract
- Purpose: Although fatigue is a known side effect in patients with head and neck cancer (HNC) receiving radiation therapy, knowledge regarding long-term fatigue and dose-response relationships to organs at risk is scarce. The aim of this prospective study was to analyze patient-reported fatigue in patients with HNC receiving radiation therapy and to explore any possible association with organ-at-risk doses. Methods and Materials: Patients with HNC referred for curative radiation therapy were eligible for inclusion in the study. To assess patient-reported fatigue, quality of life questionnaires (European Organization for Research and Treatment of Cancer QLQ-C30 and QLQ-FA12) were distributed before treatment and 1, 3, 6, 12, 24, and 60 months after the start of treatment. Mean dose (Dmean) and near maximum dose (D2%) of the cerebellum and brain stem were evaluated in relation to baseline-adjusted fatigue scores at 3 months. Results: One hundred twenty-six patients treated with intensity modulated radiation therapy between 2008 and 2010 were available for final analysis. Female sex and age <60 years were associated with higher fatigue at baseline, whereas patients also treated with chemotherapy had reduced physical and emotional fatigue at 6 months. Physical fatigue (QLQ-FA12 scale) increased from baseline up to 3 months (29 vs 59; P < .0001) but showed no difference compared with baseline from 1 to 5 years. Emotional fatigue was significantly lower at 5 years compared with baseline (14 vs 28; P < .0001). Patients with cerebellum Dmean > 3.5 Gy had higher mean physical fatigue scores at 3 months (38 vs 27; P = .036). Conclusions: Although there is a significant increase in fatigue scores for patients with HNC up to 1 year after radiation therapy, this study showed a return to baseline levels at 5 years. A possible association was found between physical fatigue and a higher mean dose to the cerebellum.
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| 7. |
- Abgrall, Remi, et al.
(författare)
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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems
- 2020
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 85:2
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Tidskriftsartikel (refereegranskat)abstract
- In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.
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| 8. |
- Abgrall, Remi, et al.
(författare)
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Analysis of the SBP-SAT Stabilization for Finite Element Methods Part II: Entropy Stability
- 2023
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Ingår i: Communications on Applied Mathematics and Computation. - : Springer. - 2096-6385 .- 2661-8893. ; 5:2, s. 573-595
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Tidskriftsartikel (refereegranskat)abstract
- In the hyperbolic research community, there exists the strong belief that a continuous Galerkin scheme is notoriously unstable and additional stabilization terms have to be added to guarantee stability. In the first part of the series [6], the application of simultaneous approximation terms for linear problems is investigated where the boundary conditions are imposed weakly. By applying this technique, the authors demonstrate that a pure continuous Galerkin scheme is indeed linearly stable if the boundary conditions are imposed in the correct way. In this work, we extend this investigation to the nonlinear case and focus on entropy conservation. By switching to entropy variables, we provide an estimation of the boundary operators also for nonlinear problems, that guarantee conservation. In numerical simulations, we verify our theoretical analysis.
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| 9. |
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| 10. |
- Conti, David, V, et al.
(författare)
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Trans-ancestry genome-wide association meta-analysis of prostate cancer identifies new susceptibility loci and informs genetic risk prediction
- 2021
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Ingår i: Nature Genetics. - : Springer Nature. - 1061-4036 .- 1546-1718. ; 53:1, s. 65-75
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Tidskriftsartikel (refereegranskat)abstract
- Prostate cancer is a highly heritable disease with large disparities in incidence rates across ancestry populations. We conducted a multiancestry meta-analysis of prostate cancer genome-wide association studies (107,247 cases and 127,006 controls) and identified 86 new genetic risk variants independently associated with prostate cancer risk, bringing the total to 269 known risk variants. The top genetic risk score (GRS) decile was associated with odds ratios that ranged from 5.06 (95% confidence interval (CI), 4.84-5.29) for men of European ancestry to 3.74 (95% CI, 3.36-4.17) for men of African ancestry. Men of African ancestry were estimated to have a mean GRS that was 2.18-times higher (95% CI, 2.14-2.22), and men of East Asian ancestry 0.73-times lower (95% CI, 0.71-0.76), than men of European ancestry. These findings support the role of germline variation contributing to population differences in prostate cancer risk, with the GRS offering an approach for personalized risk prediction. A meta-analysis of genome-wide association studies across different populations highlights new risk loci and provides a genetic risk score that can stratify prostate cancer risk across ancestries.
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