Many engineering materials exhibit heterogeneous microstructures, whose compositions and formations may to some extend be controlled during manufacturing processes. Homogenization methods predicting effective macroscopic properties of microheterogeneous materials are important tools for the development of high performance materials and for multiscale analyses of structures made thereof. Conventional computational homogenization techniques usually suffer from long computing times. For

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... ures represented by pixel or voxel images, a reduction of computational effort can be achieved using fast Fourier transform algorithms. A method combining Fourier-based acceleration with a finite element discretization is presented with numerical examples of thermal and mechanical homogenization. In addition, a reduced basis approach for materials with viscoplastic constituents and imperfect interfaces at the phase boundaries is developed. Systematic offline analyses of precomputed training data allow for efficient online algorithms, which yield good predictions for test scenarios that deviate moderately from the training cases.