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We consider elliptic and parabolic variational equations and inequalities governed by integro-differential operators of order $${2s \in (0,2]}$$. Our main motivation is the pricing of European or American options under Lévy processes, in particular pure jump processes or jump diffusion processes with tempered stable processes. The problem is discretized using piecewise linear finite elements in space and the implicit Euler method in time. We construct a residual-type a posteriori error estimator which gives a computable upper bound for the actual error in H-norm. The estimator is localized in the sense that the residuals are restricted to the discrete non-contact region. Numerical experiments illustrate the accuracy of the space and time estimators, and show that they can be used to measure local errors and drive adaptive algorithms. [ABSTRACT FROM AUTHOR]

Motivated by the pricing of American options for baskets we consider a parabolic variational inequality in a bounded polyhedral domain ??Rdwith a continuous piecewise smooth obstacle. We formulate a fully discrete method by using piecewise linear finite elements in space and the backward Euler method in time. We define an a posteriori error estimator and show that it gives an upper bound for the error in L2(0,T;H1(?)). The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate non-contact region, and the approximability of the obstacle is only relevant in the approximate contact region. We also obtain lower bound results for the space error indicators in the non-contact region, and for the time error estimator. Numerical results for d=1,2 show that the error estimator decays with the same rate as the actual error when the space meshsize h and the time step ? tend to zero. Also, the error indicators capture the correct behavior of the errors in both the contact and the non-contact regions. [ABSTRACT FROM AUTHOR]

Efficient methods for solving linear algebraic equations are crucial to creating fast and accurate numerical simulations in many applications. In this paper, an algebraic multigrid (AMG) method, which combines the classical coarsening scheme by [19] with an energy-minimizing interpolation algorithm by [26], is employed and tested for subsurface water simulations. Based on numerical tests using real field data, our results suggest that the energy-minimizing algebraic multigrid method is efficient and, more importantly, very robust. [ABSTRACT FROM AUTHOR]

In metazoans, cells depend on extracellular growth factors for energy homeostasis. We found that glycogen synthase kinase-3 (GSK3), when deinhibited by default in cells deprived of growth factors, activates acetyltransferase TIP60 through phosphorylating TIP60-Ser86, which directly acetylates and stimulates the protein kinase ULK1, which is required for autophagy. Cells engineered to express TIP60S86A that cannot be phosphorylated by GSK3 could not undergo serum deprivation-induced autophagy. An acetylation-defective mutant of ULK1 failed to rescue autophagy in ULK1-/- mouse embryonic fibroblasts. Cells used signaling from GSK3 to TIP60 and ULK1 to regulate autophagy when deprived of serum but not glucose. These findings uncover an activating pathway that integrates protein phosphorylation and acetylation to connect growth factor deprivation to autophagy. [ABSTRACT FROM AUTHOR]