Accurate estimates of flow induced surface forces over a body are typically difficult to achieve in an experimental setting. However, such information would provide considerable insight into fluid-structure interactions. Here, we consider distributed load estimation over structures described by linear elliptic partial differential equations (PDEs) from an array of noisy structural measurements. For this, we propose a new algorithm using Tikhonov regularization. Our approach differs from existing distributed load estimation procedures in that we pose and solve the problem at the PDE level. Although this approach requires up-front mathematical work, it also offers many advantages including the ability to: obtain an exact form of the load estimate, obtain guarantees in accuracy and convergence to the true load estimate, and utilize existing numerical methods and codes intended to solve PDEs (e.g., finite element, finite difference, or finite volume codes). We investigate the proposed algorithm with a two-dimensional membrane test problem with respect to various forms of distributed loads, measurement patterns, and measurement noise. We find that by posing the load estimation problem in a suitable Hilbert space, highly accurate distributed load and measurement noise magnitude estimates may be obtained.