We introduce a stable numerical space marching scheme based on discrete mollification-implemented as an automatic adaptive filter--for the approximate identification of temperature, temperature gradient, and source terms in the two-dimensional inverse heat conduction problem (IHCP).The stability and error analysis of the algorithm, together with some numerical examples, are provided. (~)

In this paper, by designing a normalized nonmonotone search strategy with the Barzilai-Borwein-type step-size, a novel local minimax method (LMM), which is a globally convergent iterative method, is proposed and analyzed to find multiple (unstable) saddle points of nonconvex functionals in Hilbert spaces. Compared to traditional LMMs with monotone search strategies, this approach, which does not require strict decrease of the objective functional value at each iterative step, is observed to converge faster with less computations. Firstly, based on a normalized iterative scheme coupled with a local peak selection that pulls the iterative point back onto the solution submanifold, by generalizing the Zhang-Hager (ZH) search strategy in the optimization theory to the LMM framework, a kind of normalized ZH-type nonmonotone step-size search strategy is introduced, and then a novel nonmonotone LMM is constructed. Its feasibility and global convergence results are rigorously carried out under the relaxation of the monotonicity for the functional at the iterative sequences. Secondly, in order to speed up the convergence of the nonmonotone LMM, a globally convergent Barzilai-Borwein-type LMM (GBBLMM) is presented by explicitly constructing the Barzilai-Borwein-type step-size as a trial step-size of the normalized ZH-type nonmonotone step-size search strategy in each iteration. Finally, the GBBLMM algorithm is implemented to find multiple unstable solutions of two classes of semilinear elliptic boundary value problems with variational structures: one is the semilinear elliptic equations with the homogeneous Dirichlet boundary condition and another is the linear elliptic equations with semilinear Neumann boundary conditions. Extensive numerical results indicate that our approach is very effective and speeds up the LMMs significantly.

The study describes the changing epidemiology of tick-borne encephalitis (TBE) based on a literature review. TBE case numbers were obtained from Austria, Germany, and Estonia for incidence calculations and for trend analyses at the county level. Currently, the TBEV is circulating in an area from the United Kingdom and France in the West to Japan in the East and from the arctic circle in Norway and Siberia down to Northern Italy, Kazakhstan and China. Over the last two decades, the TBEV was detected for the first time in Denmark, The Netherlands, the United Kingdom, France, Norway, Japan and also in higher altitudes of previously known endemic regions. TBE case numbers have been fluctuating with huge annual variations in central Europe (Germany, Austria), reaching an all-time high in 2020. Case numbers have been continuously increasing over recent decades in Norway, Sweden and Finland, whereas the TBE-epidemic curve was bell-shaped in 2 Northern-most Baltic States (Latvia, Estonia) with a huge peak in 1995 and 1997. However, the opposite (decreasing) trend was noted in some countries and TBE even disappeared from some previously highly endemic areas. Vaccination has a clear effect on TBE case numbers, which for example dropped from 677 in 1979 to 41 in 1999 (vaccine uptake at that time >80%) in Austria. Incidence rates are an inappropriate tool to predict the risk for TBE in a given region due to a lack of valid surveillance and the unpredictability of the main driver for exposure to the TBEV: human outdoor activities and the risk definition by the European Center for Disease Prevention and Control for arbovirus infections should be used instead.