The paper by Assyr Abdulle discusses the efficiency of numerical homogenization methods, particularly focusing on finite element (FE) discretizations of second-order elliptic problems with rapidly oscillating coefficients. The problem is formulated as: \[ -\nabla \cdot (a^{\varepsilon}(x) \nabla u^{\varepsilon}(x)) = f(x) \text{ in } \Omega, \qquad u^{\varepsilon}(x) = 0 \text{ on } \partial \Omega, \] where \(a^{\varepsilon}(x)\) is a tensor that varies on a small spatial scale \(\varepsilon\). Numerical homogenization methods aim to compute an effective solution \(u^0(x)\) by solving a homogenized problem with unknown data. The FE-HMM method involves a macro FE space and micro FEMs for solving cell problems, followed by averaging the micro solutions to define the effective bilinear form. The complexity of the FE-HMM method is superlinear with respect to the macroscopic degrees of freedom, requiring simultaneous refinement of both macro and micro meshes for optimal convergence rates. To reduce this complexity, three strategies are discussed: 1. **Very regular micro oscillations**: Coupling the macro solver with pseudo-spectral methods on sampling domains can significantly reduce computational cost if the oscillating tensor has sufficient regularity. 2. **Regular dependence on the macro variable**: Computing a low-dimensional subspace of micro solutions offline using a greedy algorithm can reduce the computational burden. 3. **Low regularity of \(u^0\)**: Adaptive mesh refinement can be used to refine the macro and micro meshes, saving computational costs compared to uniform refinement. The work is supported by the Swiss National Science Foundation.The paper by Assyr Abdulle discusses the efficiency of numerical homogenization methods, particularly focusing on finite element (FE) discretizations of second-order elliptic problems with rapidly oscillating coefficients. The problem is formulated as: \[ -\nabla \cdot (a^{\varepsilon}(x) \nabla u^{\varepsilon}(x)) = f(x) \text{ in } \Omega, \qquad u^{\varepsilon}(x) = 0 \text{ on } \partial \Omega, \] where \(a^{\varepsilon}(x)\) is a tensor that varies on a small spatial scale \(\varepsilon\). Numerical homogenization methods aim to compute an effective solution \(u^0(x)\) by solving a homogenized problem with unknown data. The FE-HMM method involves a macro FE space and micro FEMs for solving cell problems, followed by averaging the micro solutions to define the effective bilinear form. The complexity of the FE-HMM method is superlinear with respect to the macroscopic degrees of freedom, requiring simultaneous refinement of both macro and micro meshes for optimal convergence rates. To reduce this complexity, three strategies are discussed: 1. **Very regular micro oscillations**: Coupling the macro solver with pseudo-spectral methods on sampling domains can significantly reduce computational cost if the oscillating tensor has sufficient regularity. 2. **Regular dependence on the macro variable**: Computing a low-dimensional subspace of micro solutions offline using a greedy algorithm can reduce the computational burden. 3. **Low regularity of \(u^0\)**: Adaptive mesh refinement can be used to refine the macro and micro meshes, saving computational costs compared to uniform refinement. The work is supported by the Swiss National Science Foundation.