DC Field | Value | Language |
---|---|---|
dc.contributor.author | Koswara,Ivan | ko |
dc.contributor.author | Pogudin, Gleb | ko |
dc.contributor.author | 스벳라나 셀리바노바 | ko |
dc.contributor.author | Ziegler, Martin | ko |
dc.date.accessioned | 2021-11-03T06:47:21Z | - |
dc.date.available | 2021-11-03T06:47:21Z | - |
dc.date.created | 2021-10-26 | - |
dc.date.created | 2021-10-26 | - |
dc.date.issued | 2021-07 | - |
dc.identifier.citation | 16th International Computer Science Symposium in Russia, CSR 2021, pp.223 - 241 | - |
dc.identifier.issn | 0302-9743 | - |
dc.identifier.uri | http://hdl.handle.net/10203/288647 | - |
dc.description.abstract | Finite Elements are a common method for solving differential equations via discretization. Under suitable hypotheses, the solution u= u(t, 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 An to the vector from the previous time step, starting with the initial condition u(0 ), approximated by the spatial grid vector u(0 ) n. The dimension of the matrix An 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 AnK·u(0)n, K∼ 2 O(n), of matrices and vectors of exponential dimension for various classes of such difference schemes An. 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 ≤ t≤ 1 and space x→ ∈ [ 0 ; 1 ] d, the computational cost of evaluating the solution u(t, 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. © 2021, Springer Nature Switzerland AG. | - |
dc.language | English | - |
dc.publisher | Springer Science and Business Media Deutschland GmbH | - |
dc.title | Bit-Complexity of Solving Systems of Linear Evolutionary Partial Differential Equations | - |
dc.type | Conference | - |
dc.identifier.scopusid | 2-s2.0-85111867502 | - |
dc.type.rims | CONF | - |
dc.citation.beginningpage | 223 | - |
dc.citation.endingpage | 241 | - |
dc.citation.publicationname | 16th International Computer Science Symposium in Russia, CSR 2021 | - |
dc.identifier.conferencecountry | RU | - |
dc.identifier.conferencelocation | Sochi | - |
dc.identifier.doi | 10.1007/978-3-030-79416-3_13 | - |
dc.contributor.localauthor | Ziegler, Martin | - |
dc.contributor.nonIdAuthor | Koswara,Ivan | - |
dc.contributor.nonIdAuthor | Pogudin, Gleb | - |
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