@article{askham_adaptive_2016, title = {An adaptive fast multipole accelerated {Poisson} solver for complex geometries}, url = {https://arxiv.org/abs/1610.00823}, urldate = {2016-10-12}, journal = {arXiv preprint arXiv:1610.00823}, author = {Askham, Travis and Cerfon, Antoine J.}, year = {2016}, file = {[PDF] from arxiv.org:/home/andreas/data/zotero/storage/HXFF2IV5/Askham und Cerfon - 2016 - An adaptive fast multipole accelerated Poisson sol.pdf:application/pdf;Snapshot:/home/andreas/data/zotero/storage/4QK3WVTH/1610.html:text/html} } @article{beylkin_multiresolution_2007, title = {Multiresolution separated representations of singular and weakly singular operators}, volume = {23}, issn = {1063-5203}, url = {http://www.sciencedirect.com/science/article/pii/S1063520307000048}, doi = {10.1016/j.acha.2007.01.001}, abstract = {For a finite but arbitrary precision, we construct efficient low separation rank representations for the Poisson kernel and for the projector on the divergence free functions in the dimension d=3. Our construction requires computing only one-dimensional integrals. We use scaling functions of multiwavelet bases, thus making these representations available for a variety of multiresolution algorithms. Besides having many applications, these two operators serve as examples of weakly singular and singular operators for which our approach is applicable. Our approach provides a practical implementation of separated representations of a class of weakly singular and singular operators in dimensions d⩾2.}, number = {2}, urldate = {2016-10-12}, journal = {Applied and Computational Harmonic Analysis}, author = {Beylkin, Gregory and Cramer, Robert and Fann, George and Harrison, Robert J.}, month = sep, year = {2007}, keywords = {Integral operators, Multiwavelet bases, Poisson kernel, Projector on the divergence free functions, Separated representation}, pages = {235--253}, file = {ScienceDirect Snapshot:/home/andreas/data/zotero/storage/TBRFV2PT/S1063520307000048.html:text/html} } @article{hackbusch_efficient_2008, title = {Efficient convolution with the {Newton} potential in d dimensions}, volume = {110}, issn = {0029-599X, 0945-3245}, url = {http://link.springer.com/article/10.1007/s00211-008-0171-9}, doi = {10.1007/s00211-008-0171-9}, abstract = {The paper is concerned with the evaluation of the convolution integral ∫ℝd1∥x−y∥f(y)dy∫Rd1∥x−y∥f(y)dy\{{\textbackslash}int\_\{{\textbackslash}mathbb\{R\}{\textasciicircum}d\}{\textbackslash}frac\{1\}\{{\textbackslash}left{\textbackslash}Vert x-y{\textbackslash}right{\textbackslash}Vert\} f(y)\{{\textbackslash}rm d\}y\} in d dimensions (usually d = 3), when f is given as piecewise polynomial of possibly large degree, i.e., f may be considered as an hp-finite element function. The underlying grid is locally refined using various levels of dyadically organised grids. The result of the convolution is approximated in the same kind of mesh. If f is given in tensor product form, the d-dimensional convolution can be reduced to one-dimensional convolutions. Although the details are given for the kernel 1/∥x∥,1/∥x∥,\{\{1 / {\textbackslash}left {\textbackslash}Vert x {\textbackslash}right{\textbackslash}Vert,\}\} the basis techniques can be generalised to homogeneous kernels, e.g., the fundamental solution const⋅∥x∥2−dconst⋅∥x∥2−d\{\{const{\textbackslash}cdot{\textbackslash}left{\textbackslash}Vert x{\textbackslash}right{\textbackslash}Vert {\textasciicircum}\{2-d\}\}\} of the d-dimensional Poisson equation.}, language = {en}, number = {4}, urldate = {2016-10-12}, journal = {Numerische Mathematik}, author = {Hackbusch, W.}, month = sep, year = {2008}, pages = {449--489}, file = {Full Text PDF:/home/andreas/data/zotero/storage/MP2B9UB9/Hackbusch - 2008 - Efficient convolution with the Newton potential in.pdf:application/pdf;Snapshot:/home/andreas/data/zotero/storage/GNWFA5VI/s00211-008-0171-9.html:text/html} } @article{lanzara_fast_2011, title = {On the fast computation of high dimensional volume potentials}, volume = {80}, issn = {0025-5718, 1088-6842}, url = {http://www.ams.org/mcom/2011-80-274/S0025-5718-2010-02425-1/}, doi = {10.1090/S0025-5718-2010-02425-1}, abstract = {A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of numerical experiments, which show approximation order for the Newton potential in high dimensions, for example, for , are provided. The computation time scales linearly in the space dimension. New one-dimensional integral representations with separable integrands of the potentials of advection-diffusion and heat equations are obtained.}, number = {274}, urldate = {2016-10-12}, journal = {Mathematics of Computation}, author = {Lanzara, Flavia and Maz’ya, Vladimir and Schmidt, Gunther}, year = {2011}, pages = {887--904}, file = {Full Text PDF:/home/andreas/data/zotero/storage/NSB74EDC/Lanzara et al. - 2011 - On the fast computation of high dimensional volume.pdf:application/pdf;Snapshot:/home/andreas/data/zotero/storage/I49XF5VK/S0025-5718-2010-02425-1.html:text/html} } @article{lee_fast_1999, title = {A fast {Poisson} solver on disks}, volume = {6}, url = {http://link.springer.com/article/10.1007/BF02941907}, number = {1}, urldate = {2016-10-12}, journal = {Korean Journal of Computational and Applied Mathematics}, author = {Lee, Daeshik}, year = {1999}, pages = {65--78}, file = {[PDF] springer.com:/home/andreas/data/zotero/storage/ZHRNJU29/Lee - 1999 - A fast Poisson solver on disks.pdf:application/pdf;Snapshot:/home/andreas/data/zotero/storage/296P9BNQ/BF02941907.html:text/html} } @article{of_fast_2010, title = {Fast {Evaluation} of {Volume} {Potentials} in {Boundary} {Element} {Methods}}, volume = {32}, issn = {1064-8275}, url = {http://epubs.siam.org/doi/abs/10.1137/080744359}, doi = {10.1137/080744359}, abstract = {The solution of inhomogeneous partial differential equations by boundary element methods requires the evaluation of volume potentials. A direct standard computation of the classical Newton potentials is possible but expensive. Here, a fast evaluation of the Newton potentials by using the fast multipole method is described and analyzed. In particular, an approximation by the fast multipole method is investigated and related error estimates are given. Furthermore, an indirect evaluation of the normal derivative of the Newton potential is presented. A numerical analysis is presented for all approaches mentioned above. Numerical results are presented for the Poisson equation and for the system of linear elastostatics.}, number = {2}, urldate = {2016-10-12}, journal = {SIAM Journal on Scientific Computing}, author = {Of, G. and Steinbach, O. and Urthaler, P.}, month = jan, year = {2010}, pages = {585--602}, file = {Full Text PDF:/home/andreas/data/zotero/storage/EMQ4UQAR/Of et al. - 2010 - Fast Evaluation of Volume Potentials in Boundary E.pdf:application/pdf;Snapshot:/home/andreas/data/zotero/storage/AM42UAG7/080744359.html:text/html} }