In this paper we consider different regularization methods for solving the heat equation u + Au = 0 (0 < i < T) backward in time, where A : H-, H is a linear (unbounded) operator in a Hubert space H with norm and z 6 are the available (noisy) data for u(T) with 11 z6-u(T)ii < 5. Assuming 11 u(0)11 < E we consider different regularized solutions q(t) for u(t) and discuss the question how to choose the regularization parameter = cs(5,E,t) in order to obtain optimal estimates sup q(t)-u(t)11 < E'+'&+ where the supremum is taken over z 6 E H, ll u (0)11 < E and 11 z6-u(T)II < 5.

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In this paper we consider a general class of regularization methods for ill-posed problems Ax = y where A X-V is a linear operator between Hubert spaces X and V. The regularization methods have the general form x, = x+g0((AA)')(A*A)A(yo-Al) where y 6 are the available noisy data with flyy fl :5 5. Assuming x E M,5 = {x E X X-I = (AA)" 2 v, 11v11 < E, p > 0) we consider different functions g, and discuss the question how to choose the order s and the regularization parameter a = a(5, E,p) in order to obtain optimal estimates sup Ii xx li < E p+15P/(P+1) where the supremum is taken over x,E M,E, y E V and li Ar-v'il

Several key properties of aluminium semi-finished products, including formability, softening behaviour, strength and ductility, are influenced by the constitution of alloying elements in solid solution or in precipitated form, i.e. by the materials microchemistry. In the present paper, we study microchemistry reactions in two Al–Mg–Si automotive sheet alloys, AA 6005C and AA 6016, with different Mg:Si ratios. The formation of constituent phases and dispersoids during solidification and subsequent homogenisation is analysed by scanning electron microscopy and measurements of the specific electrical resistivity. Type, composition and volume fraction of the constituent phases in the as-cast state are assessed with the Alstruc solidification model. The changes in solute level and precipitation during homogenisation are traced with a statistical microchemistry simulation tool termed ClaNG.

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