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I have the equation $\bf E = Y D^{-1} Y^\top$.

$\bf D$ is a potentially large sparse $m \times m$ matrix, and $\bf Y$ is a sparse $n \times m$ matrix, where $n \ll m$.

Is there a particularly efficient known method for calculating the answer $\bf E$?

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The inverse of a sparse matrix is not sparse in general. So it seems hard to exploit the potential sparseness of $D$. –  Marc van Leeuwen Jan 4 '12 at 8:07
    
But is there no way to exploit the fact that it is being pre and post multiplied by the sparse matrix Y, to produce a sparse E? –  Projectile Fish Jan 4 '12 at 23:02
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