Bit-Complexity of Solving Systems of Linear Evolutionary Partial Differential Equations

Abstract

Finite Elements are a common method for solving differential equations via discretization. Under suitable hypotheses, the solution $\mathbf {u}=\mathbf {u}(t,\vec x)$ of a well-posed initial/boundary-value problem for a linear evolutionary system of PDEs is approximated up to absolute error $1/2^n$ by repeatedly (exponentially often in n) multiplying a matrix $\mathbf {A}_n$ to the vector from the previous time step, starting with the initial condition $\mathbf {u}(0)$, approximated by the spatial grid vector $\mathbf {u}(0)_n$. The dimension of the matrix $A_n$ is exponential in n, which is the number of the bits of the output.

We investigate the bit-cost of computing exponential powers and inner products $\mathbf {A}_n^K\cdot \mathbf {u}(0)_n$, $K\sim 2^{\mathcal {O}(n)}$, of matrices and vectors of exponential dimension for various classes of such difference schemes $\mathbf {A}_n$. Non-uniformly fixing any polynomial-time computable initial condition and focusing on single but arbitrary entries (instead of the entire vector/matrix) allows to improve naïve exponential sequential runtime EXP: Closer inspection shows that, given any time $0\le t\le 1$ and space $\vec x\in [0;1]^d$, the computational cost of evaluating the solution $\mathbf {u}(t,\vec x)$ corresponds to the discrete class PSPACE.

Many partial differential equations, including the Heat Equation, admit difference schemes that are (tensor products of constantly many) circulant matrices of constant bandwidth; and for these we show exponential matrix powering, and PDE solution computable in #P. This is achieved by calculating individual coefficients of the matrix’ multivariate companion polynomial’s powers using Cauchy’s Differentiation Theorem; and shown optimal for the Heat Equation. Exponentially powering twoband circulant matrices is established even feasible in P; and under additional conditions, also the solution to certain linear PDEs becomes computable in P.

Supported by the National Research Foundation of Korea (grant 2017R1E1A1A03071032) and by the International Research & Development Program of the Korean Ministry of Science and ICT (grant 2016K1A3A7A03950702) and by the NRF Brain Pool program (grant 2019H1D3A2A02102240). GP was supported by NSF grants CCF-1564132, CCF-1563942, DMS-1853482, DMS-1853650, and DMS-1760448, by PSC-CUNY grants #69827-0047 and #60098-0048. We thank Lina Bondar’ for a helpful discussion on different versions of the Sobolev Embedding Theorem (Example 2b)).