This chapter discusses the challenges and methods for posterior exploration in computationally intensive inverse problems, particularly focusing on electrical impedance tomography (EIT). The authors introduce the problem setup, where the goal is to infer an unknown spatial field \( x \) from indirect observations \( y \) through a physical system \( \zeta(x) \). The likelihood \( L(y|x) \) accounts for both observation errors and systematic differences between the physical system and the simulator \( \eta(x) \). The posterior distribution \( \pi(x|y) \) is then derived, which is often high-dimensional and computationally demanding. The chapter explores various Markov Chain Monte Carlo (MCMC) methods to sample from the posterior distribution. Single-site Metropolis updates are straightforward but computationally expensive, requiring a large number of simulator evaluations. Multivariate random walk Metropolis (RWM) schemes are proposed to reduce computational burden by updating multiple components of \( x \) simultaneously, though they still require significant computational effort. To address the high computational cost, the chapter introduces two approaches using fast, approximate simulators: Metropolis-coupled MCMC (MC-MCMC) and delayed acceptance (DA) schemes. MC-MCMC uses a joint posterior with an auxiliary distribution to balance the computational load between the fine-scale and approximate posterior. DA schemes use the approximate simulator to "filter" proposals, reducing the number of expensive exact simulator evaluations. The adaptive, multi-level delayed acceptance (AMSDA) method is highlighted for its effectiveness in handling the coarse-scale simulator, which is much faster but less accurate than the fine-scale simulator. The chapter concludes with a discussion on the trade-offs between computational efficiency and posterior exploration, emphasizing the importance of choosing appropriate prior distributions and approximate models to balance the complexity of the problem. It also reviews other methods for accelerating MCMC in inverse problems, such as reduced-order models and parallelization techniques.This chapter discusses the challenges and methods for posterior exploration in computationally intensive inverse problems, particularly focusing on electrical impedance tomography (EIT). The authors introduce the problem setup, where the goal is to infer an unknown spatial field \( x \) from indirect observations \( y \) through a physical system \( \zeta(x) \). The likelihood \( L(y|x) \) accounts for both observation errors and systematic differences between the physical system and the simulator \( \eta(x) \). The posterior distribution \( \pi(x|y) \) is then derived, which is often high-dimensional and computationally demanding. The chapter explores various Markov Chain Monte Carlo (MCMC) methods to sample from the posterior distribution. Single-site Metropolis updates are straightforward but computationally expensive, requiring a large number of simulator evaluations. Multivariate random walk Metropolis (RWM) schemes are proposed to reduce computational burden by updating multiple components of \( x \) simultaneously, though they still require significant computational effort. To address the high computational cost, the chapter introduces two approaches using fast, approximate simulators: Metropolis-coupled MCMC (MC-MCMC) and delayed acceptance (DA) schemes. MC-MCMC uses a joint posterior with an auxiliary distribution to balance the computational load between the fine-scale and approximate posterior. DA schemes use the approximate simulator to "filter" proposals, reducing the number of expensive exact simulator evaluations. The adaptive, multi-level delayed acceptance (AMSDA) method is highlighted for its effectiveness in handling the coarse-scale simulator, which is much faster but less accurate than the fine-scale simulator. The chapter concludes with a discussion on the trade-offs between computational efficiency and posterior exploration, emphasizing the importance of choosing appropriate prior distributions and approximate models to balance the complexity of the problem. It also reviews other methods for accelerating MCMC in inverse problems, such as reduced-order models and parallelization techniques.