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)).
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Notes
- 1.
Definitions of the real-valued counterparts of the complexity classes are given in Subsect. 1.3; for simplicity we use same notation as for the “discrete” case for them.
- 2.
The reader may forgive us for identifying decision and function complexity classes.
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Koswara, I., Pogudin, G., Selivanova, S., Ziegler, M. (2021). Bit-Complexity of Solving Systems of Linear Evolutionary Partial Differential Equations. In: Santhanam, R., Musatov, D. (eds) Computer Science – Theory and Applications. CSR 2021. Lecture Notes in Computer Science(), vol 12730. Springer, Cham. https://doi.org/10.1007/978-3-030-79416-3_13
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