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We perform a convergence analysis of a discrete-in-time minimization schemeapproximating a finite dimensional singularly perturbed gradient flow. We allowfor different scalings between the viscosity parameter $\varepsilon$ and thetime scale $\tau$. When the ratio $\frac{\varepsilon}{\tau}$ diverges, werigorously prove the convergence of this scheme to a (discontinuous) BalancedViscosity solution of the quasistatic evolution problem obtained as formallimit, when $\varepsilon\to 0$, of the gradient flow. We also characterize thelimit evolution corresponding to an asymptotically finite ratio between thescales, which is of a different kind. In this case, a discrete interfacialenergy is optimized at jump times.

Multiscale analysis of singularly perturbed finite dimensional gradient flows: the minimizing movement approach