We present improved worldline numerical algorithms for high-precision calculations of Casimir interaction energies induced by scalar-field fluctuations with Dirichlet boundary conditions for various Casimir geometries. Significant reduction of numerical cost is gained by exploiting the symmetries of the worldline ensemble in combination with those of the configurations. This facilitates high-precision calculations on standard PCs or small clusters. We illustrate our strategies using the

more »

... ntally most relevant sphere-plate and cylinder-plate configuration. We compute Casimir curvature effects for a wide parameter range, revealing the tight validity bounds of the commonly used proximity force approximation (PFA). We conclude that data analysis of future experiments aiming at a precision of 0.1% must no longer be based on the PFA. Revisiting the parallel-plate configuration, we find a mapping between the D-dimensional Casimir energy and properties of a random-chain polymer ensemble.