Let us denote $\mathcal{C}$ the European call prices and consider the map $K\mapsto\mathcal{C}(t,T,S_t,K)$ the market price of calls maturing at $T$. We can obtain a link between the risk-neutral probability distribution of the stock price and current call prices via the Breeden-Litzenberger formula:
$$f^S_T(K)=f_{S_T|S_t}(S_T=K)=e^{r(T-t)}\frac{\partial^2\mathcal{C}_t}{\partial K^2}$$
Proof (sketch): Write the fair value of $\mathcal{C}_t$ as an integral over the risk-neutral probability density, differentiate twice wrt $K$.
This result tells us that we can infer all the risk-neutral distribution of the stock!
Unlike the implied volatility that is a function of $K$ and $T$, the local volatility is a function of $S$ and $T$. We need to chose a Lipschitz function $\sigma(S,t)$ to guarantee existence and uniqueness of solution of the stock price SDE.
Let us denote $h(S_T)$ the payoff of the European option with a deterministic risk-free rate $r$ and dividend yield $q$. Then, the fair value is:
$$V(S,t)=\mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^Tr_sds}h(S_T)|\mathcal{F}_t\right]$$
Note that this equation satisfies the Black-Scholes PDE:
$$\frac{\partial V}{\partial t}(S,t)+(r-q)S\frac{\partial V}{\partial S}(S,t)+\frac{1}{2}\sigma(S,t)^2S^2\frac{\partial^2 V}{\partial S^2}(S,t)=rV(S,t)$$
$$V(S,T)=h(S)$$
Proof (sketch): Write the fair value of the payoff $h$ at time $T$ in integral form, apply Ito's lemma and to compute the infinitesimal change in $h$ and integrate on both sides, thus obtaining the forward Kolmogorov equation.
Assuming that we have an arbitrage-free and smooth implied volatility surface (so that we can make use of the Breeden-Litzenberger formula), we can find a non-parametric function $\sigma(S, t)$ to uniquely determine the local volatility function.
We now introduce the Dupire formula:
$$\sigma^2(K,T)=\frac{\frac{\partial\mathcal{C}}{\partial T}+q_T\mathcal{C}+(r_T-q_T)\frac{\partial\mathcal{C}}{\partial K}}{\frac{1}{2}K^2\frac{\partial^2\mathcal{C}}{\partial K^2}}$$
This gives us the relation between the local volatility function and call option function using market prices.
Claim: The density $f_T^S$ of the SDE $dS_t=r_tS_tdt+\sigma(S_t,t)S_tdW_t$ is described by the Forward Kolmogorov equation:
$$\frac{\partial f}{\partial T}(S,T)=-\frac{\partial}{\partial S}((r_T-q_T)S f(S,T))+\frac{1}{2}\frac{\partial^2}{\partial S^2}(\sigma^2(S,T)S^2f(S,T))$$
This tells you that the initial density at time $t=0$ is a Dirca delta function centered at $S=S_0$, and this equation drives the dynamics of the density up to time $T$.
Proof: Look at Dupire (1994) paper.
Finally, we can re-write the Dupire formula as a function of implied volatilities
$$\sigma^2(K,T)=\frac{\sigma^2_{imp}+2\sigma_{imp}T\left(\frac{\partial\sigma_{imp}}{\partial T}+(r_t-q_T)K\frac{\partial\sigma_{imp}}{\partial K}\right)}{\left(1-\frac{K\ln\frac{k}{F_T}}{\sigma_{imp}}\frac{\partial\sigma_{imp}}{\partial K}\right)^2+K\sigma_{imp}T\left(\frac{\partial\sigma_{imp}}{\partial K}-\frac{1}{4}K\sigma_{imp}T(\frac{\partial\sigma_{imp}}{\partial K})^2+K\frac{\partial^2\sigma_{imp}}{\partial K^2}\right)}$$
Proof: Look at Gatheral(2006).
Finally, assuming $\sigma_{imp}$ constant in $K$ but time-varying, i.e. $\sigma_{imp}(K,T)=\sigma_{imp}(T)$, we can see by a Taylor expansion:
$$\sigma_{imp}(T)=\sigma^2_{imp}(T)+2T\sigma_{imp}(T)\sigma^\prime_{imp}(T)=\sigma^2_{imp}(T)+T(\sigma^2_{imp}(T))^\prime=(T\sigma^2_{imp}(T))^\prime(T)$$
integrating, we obtain:
$$\sigma^2_{imp}(T)=\frac{1}{T}\int_0^T\sigma^2(t)dt$$
Therefore, assuming that the implied volatility does not depend on the strike, we don't need numerical techniques to compute the option prices, since we can use the Black-Scholes formula plugging in the average local volatility from time $0$ to $T$.
For $\sigma_{imp}(K,T)$ one has to compare different shapes of mappings $(K,T)\mapsto\sigma_{imp}(K,T)$ and $(S,T)\mapsto\sigma_{imp}(K,T)$. One needs to pay attention at the interpolation to have a smooth surface.
