Finding maximum using DFT I'm trying to find an efficient way to compute maximum of a signal using its DFT. More formally:

$$\max\left\{ \mathcal F^{-1}\left(X_k\right)\right\}, X_k\text{ is the DFT of the signal and } \mathcal F^{-1} \text{ is the IDFT}$$

The original signal $x(n)$ is real, and it has some noise when sometimes there are wide peaks.

I'm looking for a solution that is quicker than the actual IFFT since the signal is very long (but we have its DFT).
This sounds like a good candidate for the Sparse Fast Fourier Transform (sFFT). Have a look at H. Hassanieh, P. Indyk, D. Katabi, and E. Price, sFFT: Sparse Fast Fourier Transform, 2012. I don't know details of the algorithm. You have the domains swapped, which should be fine as the discrete Fourier transform (DFT) and its inverse (IDFT) are nearly identical.

sFFT has been discussed also here on dsp.stackexchange.com.