This work is motivated by the need to study the impact of data uncertainties and material imperfections on the solution to optimal control problems constrained by partial differential equations. We consider a pathwise optimal control problem constrained by a diffusion equation with random coefficient together with box constraints for the control. For each realization of the diffusion coefficient we solve an optimal control problem using the variational discretization [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45-61]. Our framework allows for lognormal coefficients whose realizations are not uniformly bounded away from zero and infinity. We establish finite element error bounds for the pathwise optimal controls. This analysis is nontrivial due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the diffusion operator. We apply the error bounds to prove convergence of a multilevel Monte Carlo estimator for the expected value of the pathwise optimal controls. In addition we analyze the computational complexity of the multilevel estimator. We perform numerical experiments in 2D space to confirm the convergence result and the complexity bound.

De Garengeot's hernia is quite rare and is a femoral hernia that contains a vermiform appendix and can present as a painful, tender swelling or an asymptomatic lump. We present the case of a 70-year-old patient who presented to our surgical unit after being referred for diagnostic imaging of an asymptomatic groin lump which was found to be a De Garengeot's hernia. She had an open repair of her femoral hernia and laparoscopic appendicectomy. Her post-operative stay was uneventful. De Garengeot's hernia is rare; however, imaging is usually required to make a diagnosis preoperatively. Management is usually surgical with simultaneous repair of the femoral hernia and appendicectomy.

Background: Oophorectomy is being increasingly carried out in the management of breast disease, as either adjuvant treatment for breast cancer or for prevention of ovarian and fallopian tube cancer in BRCA gene mutation carriers. The aims of this study were to determine the surgical outcome of laparoscopic oophorectomy when carried out by breast surgeons and whether laparoscopic oophorectomy can be safely carried out during the same anaesthesia as breast surgery. Methods: Patients who had laparoscopic oophorectomy carried out by two breast surgeons were reviewed with regard to the indication, surgical outcome and concurrent procedures. Salpingectomy was also carried out when the indication was prevention. Results: Seventy patients with breast disease had laparoscopic oophorectomy between January 2000 and June 2007. Forty-three patients had laparoscopic oophorectomy for adjuvant endocrine treatment of early breast cancer, 13 for prophylaxis, 7 for endocrine and prophylactic reasons and 7 for treatment of metastatic breast cancer. Sixteen patients had laparoscopic oophorectomy and breast surgery at the same time, without complication. Of note, four BRCA mutation carriers had prophylactic mastectomies, bilateral breast reconstruction and bilateral laparoscopic salpingo-oophorectomy. No patient required conversion to an open procedure, including 29 patients with previous abdominal surgery. There were no significant complications. Three patients had ovarian cancer or breast cancer detected in an ovary. Conclusion: Laparoscopic oophorectomy can be safely and efficiently carried out by breast surgeons with expertise in laparoscopic surgery. Previous abdominal surgery did not prevent a successful laparoscopic approach. Breast oncological and/or reconstructive surgery and laparoscopic oophorectomy can be reliably carried out as a combined procedure.

An optimal control problem subject to an elliptic obstacle problem is studied. We obtain a numerical approximation of this problem by discretising the PDE obtained via a Moreau--Yosida type penalisation. For the resulting discrete control problem we provide a condition that allows to decide whether a solution of the necessary first order conditions is a global minimum. In addition we show that the corresponding result can be transferred to the limit problem provided that the above condition holds uniformly in the penalisation and discretisation parameters. Numerical examples with unique global solutions are presented.

Abstract:We consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization. We provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. Numerical examples with unique global solutions are presented. (2000): 49J20, 35K20, 49M05, 49M25, 49M29, 65M12, 65M60 Mathematics Subject Classification