This thesis investigates the computation of piecewise polynomial solutions to ill- conditioned linear systems of equations when noise on the linear measurements is observed. Specifically, we enhance our understanding of and provide qualifications on when such ill-conditioned systems of equations can be solved to a satisfactory accuracy. We show that the conventional condition number of the coefficient matrix is not sufficiently informative in this regard and propose a more relevant conditioning measure that takes into account the decay rate of singular values. We also discuss interactions of several factors affecting the solvability of such systems, including the number of discontinuities in solutions, as well as the distribution of nonzero entries in sparse matrices. In addition, we construct and test an approach for computing piecewise polynomial solutions of highly ill-conditioned linear systems using a randomized, SVD-based truncation, and L1-norm regularization. The randomized truncation is a stabilization technique that helps reduce the cost of the traditional SVD truncation for large and severely ill-conditioned matrices. For L1-minimization, we apply a solver based on the Alternating Direction Method. Numerical results are presented to compare our approach that is faster and can solve larger problems, called RTL1 (randomized truncation L1-minimization), with a well-known solver PP-TSVD.