Learning Elliptic Partial Differential Equations with Randomized Linear Algebra

On the Solutions of Quasi-linear Elliptic Partial Differential Equations*

The literature concerning these equations being very extensive, we shall not attempt to give a complete list of references. The starting point for many more modern researches has been the work of S. Bernstein,f who was the first to prove the analyticity of the solutions of the general equation with analytic and who was able to obtain a priori bounds for the second and higher derivatives of ...

Numerical Linear Algebra and Solvability of Partial Differential Equations

Differential Equations and Linear Algebra

where D = d/dt is the basic differentiation operator on functions. (The j-th derivative operator, for j ≥ 1, is Dj , and we consider the identity operator I as D0, since it is standard to regard the zero-th derivative of a function as the function itself: D0(y) = y for all functions y.) We call (1.2) an n-th order linear differential operator since the highest derivative appearing in it is the ...

Linear Partial Differential Equations with Random Forcing

where (0 l ) is the ensemble-averaging operation. The higher order moments can be determined by forming the ensemble average of different combinations of u(x, t) with the help of (1.4). Unfortunately, the Green’s function, G(x, t ; x’, t’), of (l.l), (1.2) and (1.3) cannot be obtained in terms of elementary or special functions except for the simplest cases. In general, we will have to obtain G...

Lecture Notes on Elliptic Partial Differential Equations