I'm currently studying the book Monte Carlo Methods in Financial Engineering by Glasserman. When covering the simulation of Brownian Motions, Glasserman explains the Principal Component Construction (PCC) and the Karhunen-Loeve (KL) expansion.
In the PCC, the Brownian path is approximated by $$\begin{pmatrix}W(t_1)\\W(t_2)\\\vdots\\W(t_n)\end{pmatrix} = \begin{pmatrix}a_{11}\\a_{21}\\\vdots\\ a_{n1}Z_1\end{pmatrix}Z_1 + \ldots + \begin{pmatrix}a_{1n} \\a_{2n}\\\vdots\\a_{nn}\end{pmatrix}Z_n$$ with $Z_i$ independent random $N(0,1)$ variables. This is a valid construction of the discrete Brownian path if $AA^T$ is the covariance matrix $C$ of $W = (W(t_1),\ldots,W(t_n))^T$. Specifically, $a_i = \sqrt{\lambda_i}v_i, i = 1,\ldots, n$, where $\lambda_1 >\lambda_2>\ldots >\lambda_n$ are the eigenvalues of $C$ and the $v_i$ are the corresponding eigenvectors.
Alternatively, in the KL expansion the Brownian path is equal to $$W(t) = \sum_{i = 0}^\infty \sqrt{\lambda_i}\psi(t)Z_i, 0\leq t \leq 1, $$ with $Z_i$ again independent random $N(0,1)$ variables. This infinite series an exact representation of the continuous Brownian path. By taking just the first $k$ terms in this series, we arrive at an approximation to the continuous path $\{W(t), 0\leq t\leq1\}$ that apparently is optimal in the sense of explained variability.
I'm trying to compare these methods, and basically the only difference I can think of is the coefficients. For PCC we have that $$\lambda_i = \dfrac{\Delta t}{4}\sin^{-2}\bigg(\dfrac{2i - 1}{2n + 1}\dfrac{\pi}{2}\bigg), v_i(j) = \dfrac{2}{\sqrt{2n + 1}}\sin\bigg(\dfrac{2i - 1}{2n + 1} j\pi\bigg), j = 1,\ldots, n$$ and for KL we have: $$\lambda_i = \bigg(\dfrac{2}{(2i +1)\pi}\bigg)^2, \psi_i = \sqrt{2}\sin\bigg(\dfrac{(2i + 1)\pi t}{2}\bigg)$$
Question: What are the advantages/disadvantages of these methods? When comparing the differences between these approximations at time $t$, what do we get? That is, what is $PCC_k(t) - KL_k(t)$ in general, where $k$ is the number of random normals used?
Edit: Perhaps, instead of answering my initial question, could somehow give me some hints on how I should analyse the coefficients of the different methods? I've tried looking at what they might be for different values for $i$ and $t$, but I didn't get very far.