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Critical slowing down is a major problem in Monte Carlo calculations. It is especially becoming an urgent problem in lattice QCD because the required precision is getting high for the test of the standard model. In this talk, after a review of the problem, we consider developing the idea of the trivializing map proposed by Lüscher using a Schwinger-Dyson equation. In this method, the basis for the flow kernel can be chosen arbitrarily by hand, and the coefficients in the kernel are determined by lattice estimates of the observables, which does not require analytic calculations beforehand. We perform the HMC with the exact effective action obtained by the Schwinger-Dyson method, and show that we can have better control of the effective action. However, the algorithmic overhead is still large and overwhelming the gain though faster decorrelation (in the Monte Carlo steps) is observed for long-range observables in some cases. We report the preliminary results of this attempt.