Error analysis for function representation by the sparse-grid combination technique

Detailed error analyses are given for sparse-grid function representations through the combination technique. Two- and three-dimensional, and smooth and discontinuous functions are considered, as well as piecewise-constant and piecewise-linear interpolation techniques. Where appropriate, the results of the analyses are verified in numerical experiments. Instead of the common vertex-based function representation, cell-centered function representation is considered. Explicit, pointwise error expressions for the representation error are given, rather than order estimates. The paper contributes to the theory of sparse-grid techniques

The sparse-grid combination technique applied to time-dependent advection problems

In the numerical technique considered in this paper, time-stepping is performed on a set of semi-coarsened space grids. At given time levels the solutions on the different space grids are combined to obtain the asymptotic convergence of a single, fine uniform grid. We present error estimates for the two-dimensional spatially constant-coefficient model problem and discuss numerical examples. A spatially variable-coefficient problem (Molenkamp-Crowley test) is used to assess the practical merits of the technique. The combination technique is shown to be more efficient than the single-grid approach, yet for the Molenkamp-Crowley test, standard Richardson extrapolation is still more efficient than the combination technique. However, parallelization is expected to significantly improve the combination technique's performance

On the equivalence of the Langevin and auxiliary field quantization methods for absorbing dielectrics

Recently two methods have been developed for the quantization of the electromagnetic field in general dispersing and absorbing linear dielectrics. The first is based upon the introduction of a quantum Langevin current in Maxwell's equations [T. Gruner and D.-G. Welsch, Phys. Rev. A 53, 1818 (1996); Ho Trung Dung, L. Kn\"{o}ll, and D.-G. Welsch, Phys. Rev. A 57, 3931 (1998); S. Scheel, L. Kn\"{o}ll, and D.-G. Welsch, Phys. Rev. A 58, 700 (1998)], whereas the second makes use of a set of auxiliary fields, followed by a canonical quantization procedure [A. Tip, Phys. Rev. A 57, 4818 (1998)]. We show that both approaches are equivalent.Comment: 7 pages, RevTeX, no figure

70-Gene signature-guided adjuvant systemic treatment adjustments in early-stage ER+ breast cancer patients:7-year follow-up of a prospective multicenter cohort study

Background: A previous prospective multicenter study revealed the change of the oncologists’ chemotherapy advice due to the 70-Gene signature (GS) test result in half of the estrogen receptor-positive (ER+) invasive early-stage breast cancer patients with disputable chemotherapy indication. This resulted in less patients receiving chemotherapy. This study aims to complement these results by the 7-year oncological outcomes according to the 70-GS test result and the oncologists’ pre-test advice.Methods: Patients operated for early-stage ER+ breast cancer with disputable chemotherapy indication, had been prospectively included between 2013 and 2015. Oncologists were asked whether they intended to administer adjuvant chemotherapy before deployment of the 70-GS test. Information on adjuvant systemic treatment and oncological outcome was obtained through active follow-up by data managers of the Netherlands Cancer Registry. The primary endpoint of this study was distant metastasis-free survival (DMFS) according to the genomic risk. Exploratory analyses were done to evaluate DMFS in relation to the oncologists’ pre-test advice.Results: After a median follow-up of 7 years, distant metastases were diagnosed in 23 of the 606 patients (3.8%) and 36 (5.9%) patients had died. The DMFS rate for the 357 70-GS genomic low-risk patients was 94.2% (95% CI 91.2–96.2) and 89.1% for the 249 genomic high-risk patients (95% CI 84.3–92.4). Of the low-risk patients 3% had received chemotherapy compared to 80% of the high-risk patients. For the subgroups based on the pre-test oncologists’ advice (no chemotherapy/chemotherapy/unsure) there were no clinically relevant differences in DMFS (89.8, 93.2 and 92.0%, respectively), while comparable proportions of patients had received chemotherapy.Conclusions: In patients with early-stage ER+ breast cancer with a disputable chemotherapy indication it is sensible to deploy the 70-GS to better select patients for adjuvant chemotherapy

70-Gene signature-guided adjuvant systemic treatment adjustments in early-stage ER+ breast cancer patients: 7-year follow-up of a prospective multicenter cohort study

Background: A previous prospective multicenter study revealed the change of the oncologists’ chemotherapy advice due to the 70-Gene signature (GS) test result in half of the estrogen receptor-positive (ER+) invasive early-stage breast cancer patients with disputable chemotherapy indication. This resulted in less patients receiving chemotherapy. This study aims to complement these results by the 7-year oncological outcomes according to the 70-GS test result and the oncologists’ pre-test advice. Methods: Patients operated for early-stage ER+ breast cancer with disputable chemotherapy indication, had been prospectively included between 2013 and 2015. Oncologists were asked whether they intended to administer adjuvant chemotherapy before deployment of the 70-GS test. Information on adjuvant systemic treatment and oncological outcome was obtained through active follow-up by data managers of the Netherlands Cancer Registry. The primary endpoint of this study was distant metastasis-free survival (DMFS) according to the genomic risk. Exploratory analyses were done to evaluate DMFS in relation to the oncologists’ pre-test advice. Results: After a median follow-up of 7 years, distant metastases were diagnosed in 23 of the 606 patients (3.8%) and 36 (5.9%) patients had died. The DMFS rate for the 357 70-GS genomic low-risk patients was 94.2% (95% CI 91.2–96.2) and 89.1% for the 249 genomic high-risk patients (95% CI 84.3–92.4). Of the low-risk patients 3% had received chemotherapy compared to 80% of the high-risk patients. For the subgroups based on the pre-test oncologists’ advice (no chemotherapy/chemotherapy/unsure) there were no clinically relevant differences in DMFS (89.8, 93.2 and 92.0%, respectively), while comparable proportions of patients had received chemotherapy. Conclusions: In patients with early-stage ER+ breast cancer with a disputable chemotherapy indication it is sensible to deploy the 70-GS to better select patients for adjuvant chemotherapy