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Strong convergence rates for numerical approximations of stochastic partial differential equations
(2018)
In this thesis and in the research articles which this thesis consists of, respectively, we focus on strong convergence rates for numerical approximations of stochastic partial differential equations (SPDEs). In Part I of this thesis, i.e., Chapter 2 and Chapter 3, we study higher order numerical schemes for SPDEs with multiplicative trace class noise based on suitable Taylor expansions of the Lipschitz continuous coefficients of the SPDEs under consideration. More precisely, Chapter 2 proves strong convergence rates for a linear implicit Euler-Milstein scheme for SPDEs and is based on an unpublished manuscript written by the author of this thesis. This chapter extends an earlier result1 by slightly lowering the assumptions posed on the diffusion coefficient and a different approximation of the semigroup. In Chapter 3 we introduce an exponential Wagner-Platen type numerical scheme for SPDEs and prove that this numerical approximation method converges in the strong sense with oder up to 3/2−. Moreover, we illustrate how the (mixed) iterated stochastic-deterministic integrals, that are part of our numerical scheme, can be simulated exactly under suitable assumptions.
The second part of this thesis, i.e. Chapter 4 and Chapter 5, is devoted to strong convergence rates for numerical approximations of SPDEs with superlinearly growing nonlinearities driven by additive space-time white noise. More specifically, in Chapter 4, we prove strong convergence with rate in the time variable for a class of nonlinearity-truncated numerical approximation schemes for SPDEs and provide examples that fit into our abstract setting like stochastic Allen-Cahn equations. Finally, in Chapter 5, we extend this result with spatial approximations and establish strong convergence rates for a class of full-discrete nonlinearity truncated numerical approximation schemes for SPDEs. Moreover, we apply our strong convergence result to stochastic Allen-Cahn equations and provide lower and upper bounds which show that our strong convergence result can, in general, not essentially be improved.
Purpose: Anastomotic leakage is a major surgical complication following esophagectomy and gastric pull-up. Specific risk factors such as celiac trunk (TC) stenosis and high calcification score of the aorta have been identified, but no data are available on their relative prognostic values. This retrospective study aimed to compare and evaluate calcification score versus stenosis quantification with regards to prognostic impact on anastomotic leakage.
Patients and methods: Preoperative contrast-enhanced computed tomography scans of 164 consecutive patients with primary esophageal cancer were evaluated by two radiologists to apply a calcification score (0–3 scale) assessing the aorta, the celiac axis and the right and left postceliac arteries. Concurrently, the presence and degree of stenosis of TC and superior mesenteric artery were recorded for stenosis quantification.
Results: Anastomotic leakage was noted in 14/164 patients and 12/14 showed stenosis of TC (n=11). The presence of TC stenosis was found to have a significant impact on anastomotic healing (p=0.004). The odds ratio for the prediction of anastomotic leakage by the degree of stenosis was 1.04 (95% CI, 1.02–1.07). Ten of 14 patients had aortic calcification scores of 1 or 2, but calcification scores of the aorta, the celiac axis and the right and left postceliac arteries did not correlate with the corresponding TC stenosis values and showed no influence on patient outcome as defined by the occurrence of anastomotic insufficiency (p=0.565, 0.855, 0.518 and 1.000, respectively). Inter-reader reliability of computed tomography analysis and absolute agreement on calcium scoring was mostly over 90%. No significant differences in preoperative comorbidities and patient characteristics were found between those with and without anastomotic leakage.
Conclusion: Measurement of TC stenosis in preoperative contrast-enhanced computed tomography scans proved to be more reliable than calcification scores in predicting anastomotic leakage and should, therefore, be used in the risk assessment of patients undergoing esophagectomy and gastric pull-up.