Fast and Accurate Algorithms for Cosmic Microwave Background Radiation Data on HEALPix Points Kathryn P. Drake Grady B. Wright Supported by NASA Idaho Space Grant Consortium Boise State University Boise State University Motivation Current Method: HEALPix Numerical Test #1 Faintly glowing at the edge of the observable , the Cosmic Step 1: “Zeroth” Order Approximation

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first light to travel during the early stages of the universe's development 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 first using a low-order approximation to calculate the spherical harmonic coecients: and gives the strongest evidence for the theory. Since the Npix m 4⇡ m b` Y ` (i, ✓i)f(i, ✓i), 0 ` `max, `max m `max discovery of the CMBR in 1964, scientists have worked to measure it in ⇡ Npix    

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 i=1 full detail using a Hierarchical Equal Area isoLatitude Pixelization 3/2 Due to the isolatitudeX nature of the pixels, this computation is done with (N )

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 quasiuniform discretization of the sphere, they are not well suited for the In order to improve the estimate of the bms, HEALPix uses an iterative process. 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 ` fast algorithms necessary for mining the massive CMBR data sets. For `max ` m ¯ m this work, we apply the Double Fourier Sphere method and fast Fourier Analysis (function to coecients): b = Y (, ✓)f(, ✓) 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 b = Af 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 ` ` ()

transforms for uniform and non-uniform point distributions in order to 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 `=0 m=0 X `Xmax ` to efficiently compute the bivariate Fourier series coefficients. We then m m

Synthesis (coecients to function): 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 f(, ✓)= Y (, ✓)b f = Sb utilize a state-of-the-art method which computes spherical harmonic ` ` ()

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 operators are used in the iterativee refinement scheme, which convergese to the enables us to quickly and more accurately compute the angular power (0) (k) (k 1) (k 1)

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 squares solution to the coecients: b = Af, b = b + A f Sb

spectrum of CMBR data sampled at the HEALPix points. 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 Quadrature Weights ⇣ ⌘

Image credit: NASA AAACEHicZZDLSgMxFIYzXmu9VV26CRbRRSkzVVB3BTcuFWwr2FIymTNtMJMMyRmxDH0EN76KGxeKuHXpzrcxvSy0Hgh8+ZM/J+cPUyks+v63Nze/sLi0XFgprq6tb2yWtrabVmeGQ4Nrqc1NyCxIoaCBAiXcpAZYEkpohXfno/PWPRgrtLrGQQqdhPWUiAVn6KRu6aCN8IBhnLczFYEZPZNfZSwyDDMDtAWi10c7HHZLZb/qj4v+h2AKZTKty27pqx1pniWgkEtm7W3gp9jJmUHBJQyL7cxCyvgd68GtQ8USsJ18PNCQ7jslorE2bimkY/W3I2eJtYMkrFAHCcP+GLCfVOhoh1pLW5nYZjphfNrJhUozBMUnjeJMUtR0lA6NhAGOcuCAcSPcXynvM8M4ugyLLoRgduT/0KxVg6Nq7eq4XD+bxlEgu2SPHJKAnJA6uSCXpEE4eSTP5JW8eU/ei/fufUyuznlTzw75U97nD0Njnbk= 3

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 improve equal weight quadrature above, HEALPix o↵ers quadrature weights: 2 Npix TestAAACAHicZZDLSgMxFIYz3q23qgsXboJFcFHKTBUUcSG4cVmhF8GWkknP2GAmGZIzYhlm46u4caGIWx/DnW9j2s7Cy4HAd/7k5+T8YSKFRd//8mZm5+YXFpeWSyura+sb5c2tttWp4dDiWmpzHTILUihooUAJ14kBFocSOuHdxfi+cw/GCq2aOEqgF7NbJSLBGTqpX97pIjxgGGVNsEijVPGxnp/2yxW/5k+K/oeggAopqtEvf3YHmqcxKOSSWXsT+An2MmZQcAl5qZtaSBi/Y7dw41CxGGwvmyyQ032nDGikjTsK6UT96chYbO0oDqvUQcxwOAEcxlU67lBraatT259JGJ30MqGSFEHx6aAolRQ1HadBB8IARzlywLgR7q+UD5lhHF1mJRdC8Hfl/9Cu14LDWv3qqHJ+VsSxRHbJHjkgATkm5+SSNEiLcJKTJ/JCXr1H79l7896nT2e8wrNNfpX38Q3Mh5ZJ function :f(, ✓)=5g(, ✓,l ,t ) 3g(, ✓,l ,t )+8g(, ✓,l ,t ), AAACTXicbZHNS8MwGMbT+TXn19Sjl+AQHNbRboo7KAy8eJzgVHCjpGm6haVNSd4Ko/gPehG8+V948aCImG476PSFwC/Pk5c3eeIngmtwnBerMDe/sLhUXC6trK6tb5Q3t661TBVlHSqFVLc+0UzwmHWAg2C3iWIk8gW78YfnuX9zz5TmMr6CUcJ6EenHPOSUgJG8chDud4U5HhC7CwMGpHp23J+RbOG5Nnhu9bDxj1U3Vr160PzHahirUbVLXrni1Jxx4b/gTqGCptX2ys/dQNI0YjFQQbS+c50EehlRwKlgD6VuqllC6JD02Z3BmERM97JxGg94zygBDqUyKwY8Vn92ZCTSehT5NjYQERiMAQaRjfMdSCm0PWmbmQRhs5fxOEmBxXQyKEwFBonzaHHAFaMgRgYIVdzcFdMBUYSC+YA8BHf2yX/hul5zGzXn8qjSOp3GUUQ7aBftIxedoBa6QG3UQRQ9olf0jj6sJ+vN+rS+JkcL1rRnG/2qwtI313yvzQ== g(, ✓,l,t)=(2 2(x x(l, t)+y y(l, t)+z z(l, t)))

1 1 2 2 3 3 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sha1_base64="FEJVx4lNxgyhaG2PsHoPtV+Vzvg=">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sha1_base64="ellaVdlr4l/x4IlgUuW7/tjKSnQ=">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 m m · · · b` wiY ` (i, ✓i)f(i, ✓i), 0 ` `max, `max m `max x = cos()sin(✓),y=sin()sin(✓),z= cos(✓), 0 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sha1_base64="6O4cmLZG5HUKmNvIkP6hUqnlUPE=">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sha1_base64="vWi+I49F4FSUjWyzRSYH7WRp8M4=">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 1 2 3 1 2 3

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 i=1 X “Pixel weights” for each HEALPix point, computed using the algorithm from [5]

AAADZHicZVJNb9NAEHUSPkookFJxQkIjoqocoigJAnpBKkJIPYAUUNNWSqJkvR7Hq+6HtTtuGlnhT3Lixon/wI21E1Da7Ol5Zt7M85sJUykcdTo/K9Xanbv37u88qD/cffT4SWPv6ZkzmeU44EYaexEyh1JoHJAgiRepRaZCiefh5ccif36F1gmjT2mR4lixmRax4Ix8aLJXkSNthI5QE4wIrymM8+m0L65RwhzFLCF3eLiE2NhVWlCOjCdw8unDZ18FqSfTsgXcqDQjjCBzQs+AEgQmZ8YKShTE1igYvhlDvV4/ADK+CGE6tUXl/yEtmCfCdy5ykYhjtIWmdXpbQEFegonLUTfUgENybTj18ZWq8lf/lW4OBeEgMhrXop1QQjILIkLmCplFfbppRQvCjICByrwCp5iUaMEtHKGCZMUJEZyRVxh9H3FBmH9hloT+5vu7ZXvSaHbanfLBNuiuQTNYv/6k8WsUGZ4pbwWXzLlht5PSOC96conL+si7lTJ+yWY49FAzhW6cl2exhAMfiUrnYuNtKaObjJwp5xYqbIEHilFSAr+wFhRfZIx0rRXt1iSKj8a50MXGNV8NijNZWuZvzK/PIie58IBxfwKCA0+YZZz8JdbLNc5FRMn7t0J7S7q3DdgGZ71293W797XXPD5am7MTPA9eBq+CbvAuOA5Ogn4wCHjlR+VPtVqtVX/Xdmv7tWer0mplzdkPbrzai7/xlhS3

“RingAAACcHicZZHPbtNAEMbXhtISoA1wQeLAQoSKUBTZQYJekCpx4VgQaSs1UbrejO1R94+1Oya1rDwoJ3gGnoCNEyRK5/TbmZ2d/b7JKoWekuRHFN+5u3Nvd+9+78HDR/sH/cdPTr2tnYSJtMq680x4UGhgQkgKzisHQmcKzrKrT+v62XdwHq35Rk0FMy0KgzlKQSE17zdTY9EswBCfElxTlreXl1/RFHwJWJTkDw9XPLduU0VqQciSu3BhNeQeNSrhuFCFdUil5mR5hdeg/nYPeVYTXxKWXHCvhVLguG88gZ73B8ko6YLfhnQLA7aNk3n/13RhZa3DX6US3l+kSUWzVjhCqWDVm9YeKiGvRAEXAY3Q4GdtZ9GKvw6ZRSckt0Frl/23oxXa+0ZnQx5ACyo7CJKGfH0ia1UQs3ns5iTKj2YtmqomMHIzKK/V2om133yBDiSpJoCQwSSUXJbCCUlhK73O1SUuqPz4HsM6Bun/BtyG0/EofTcafxkPjo+25uyx5+wVe8NS9oEds8/shE2YZD+jnWg/Ooh+x8/iF/HLzdU42vY8ZTcifvsHBm29XA== weights” for each ring, similar algorithm to pixel weights, but wtih a smaller system Numerical Test #2 Numerical Test #3 Absolute error for the angular power spectrum of h for HP2SPH Angular power spectrum of CMBR temperature anisotropy map for New Method: HP2SPH (blue) and NASA’s HEALPix software (red) for 12582912 nodes.

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 HP2SPH (blue) and NASA HEALPix software (red), 50331648 nodes.

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 CMBR temperature anisotropy map (left) CMBR and the Big Bang (right) 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 Step 1: Upsample in

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 Spherical Harmonics In order to use fast Fourier analysis in , we need

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 upsample to an equal number of points on each ring

AAACEHicZZBNSwMxEIaz9avWr6pHL8Eieqhlt4oKIhS8eKxgP6AtJZumbWiyWZJZoZT+BC/+FS8eFPHq0Zv/xuxuD9oOhPfJZIbJvH4ouAHX/XEyS8srq2vZ9dzG5tb2Tn53r25UpCmrUSWUbvrEMMEDVgMOgjVDzYj0BWv4o9v4vfHItOEqeIBxyDqSDALe55SATXXzx/LmtIzb19iql6obS0IetppQuZsvuCU3CbwI3gwKaBbVbv673VM0kiwAKogxLc8NoTMhGjgVbJprR4aFhI7IgLUsBkQy05kkC03xkc30cF9pewLASfZvx4RIY8bSL2ILksAwARjKIo5voJQwxbRtbhL0rzoTHoQRsICmg/qRwKBw7A7ucc0oiLEFQjW3f8V0SDShYD3MWRO8+ZUXoV4ueWcl9/68ULmY2ZFFB+gQnSAPXaIKukNVVEMUPaEX9IbenWfn1flwPtPSjDPr2Uf/wvn6BatHlww= m = 2 m = 1 m =0 m =1 m =2

SphericalAAACDXicZZBNSwMxEIaz9bt+VT16CVbBQym79aDeBEE8KlpbaEvJprNtaDZZklmhlP4BL/4VLx4U8erdm//GdNuDtgOBZ2byZjJvmEhh0fd/vNzC4tLyyupafn1jc2u7sLP7YHVqOFS5ltrUQ2ZBCgVVFCihnhhgcSihFvYvx/3aIxgrtLrHQQKtmHWViARn6ErtwuFd0gPjUkmZYlJ3U6A6olfufQGGWtcD2y4U/bKfBZ2HYApFMo2bduG72dE8jUEhl8zaRuAn2Boyg4JLGOWbqYWE8T7rQsOhYjHY1jDbZkSPXKVDI23cUUiz6l/FkMXWDuKwRB3EDHsZYC8u0XGGWktbmshmJmF01hoKlaQIik8GRamkqOnYGtoRBjjKgQPGjXB/pbzHDOPoDMw7E4LZlefhoVIOTsr+baV4cT61Y5XskwNyTAJySi7INbkhVcLJE3khb+Tde/ZevQ/vc3I15001e+RfeF+/2iSbJg== analogue of Fourier series

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 using trigonometric interpolation and the DFT: Example: AAAB73icZZDLSgMxGIUz9VbrrerSTbAIFUqZUVA3YsGNywr2Ap2hZNJMG5qZjMk/Qil9BsGNC0Xc+g6ufAR3Poh705kutB4IfDnJ4b/4seAabPvLyi0sLi2v5FcLa+sbm1vF7Z2mlomirEGlkKrtE80Ej1gDOAjWjhUjoS9Yyx9eTt9bd0xpLqMbGMXMC0k/4gGnBIzVdpkQ+Bzb3WLJrtqp8H9wZlC6eC9/f9y7h/Vu8dPtSZqELAIqiNYdx47BGxMFnAo2KbiJZjGhQ9JnHYMRCZn2xmm/E3xgnB4OpDInApy6vxNjEmo9Cv0KNhASGKQAg7CCpzeQUuhKFpurBMGZN+ZRnACLaFYoSAQGiafD4x5XjIIYGSBUcdMrpgOiCAWzooJZgjM/8n9oHlWd46p9bZdqJyhTHu2hfVRGDjpFNXSF6qiBKBLoAT2hZ+vWerRerNfsa86aZXbRH1lvP31/kmk= ` =0 N 1 Let f(, ✓) be a smooth function 2⇡ikx,x[0, 1] f(x)= cke AAACRHicZZDLahRBFIar4y22t0ncxU3hILhohu4IKoIQ0IUbIYKTBLqaobr69EwxdelUndYZmsGVr+FWn8Kl7+A7uBNdijXTWWhyoOCr/9TPOfWXjZIe0/R7tHXp8pWr17avxzdu3rp9Z7Cze+Rt6wSMhVXWnZTcg5IGxihRwUnjgOtSwXE5f7HuH78D56U1b3HZQKH51MhaCo5Bmgx2E8pOT1te0QWTJk+TrJgMhuko3RS9CNkZDA/2Pg5/f3j99XCyE+2xyopWg0GhuPd5ljZYdNyhFApWMWs9NFzM+RTygIZr8EW3WX5FHwSlorV14RikG/VfR8e190tdJjSA5jjbAM50Qtc3tFb5pLedm4T106KTpmkRjOgH1a2iaOk6CVpJBwLVMgAXToZdqZhxxwWGvGKGsMD3ssIZfU4fSxOzCmr2smMNd8xYaarwW6b5HEq7yDPQRa6Kjq7YjJtp3w3iKg5pZuezuwhH+6Ps0Sh9E2J9RvraJvfIffKQZOQJOSCvyCEZE0EW5BP5TL5E36If0c/oV/90Kzrz3CX/VfTnLxJgsfA= 2

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 the surface of the sphere

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 k=0 ` X

AAAB73icZZDLSgMxGIUz9VbrrerSTbAIFUqZUVA3YsGNywr2Ap2hZNJMG5qZjMk/Qil9BsGNC0Xc+g6ufAR3Poh705kutB4IfDnJ4b/4seAabPvLyi0sLi2v5FcLa+sbm1vF7Z2mlomirEGlkKrtE80Ej1gDOAjWjhUjoS9Yyx9eTt9bd0xpLqMbGMXMC0k/4gGnBIzVdpkQ+Bw73WLJrtqp8H9wZlC6eC9/f9y7h/Vu8dPtSZqELAIqiNYdx47BGxMFnAo2KbiJZjGhQ9JnHYMRCZn2xmm/E3xgnB4OpDInApy6vxNjEmo9Cv0KNhASGKQAg7CCpzeQUuhKFpurBMGZN+ZRnACLaFYoSAQGiafD4x5XjIIYGSBUcdMrpgOiCAWzooJZgjM/8n9oHlWd46p9bZdqJyhTHu2hfVRGDjpFNXSF6qiBKBLoAT2hZ+vWerRerNfsa86aZXbRH1lvP38Dkmo= ` =1 1 f(, ✓)= bmY m(, ✓), Since the points are equispaced in latitude, we can solve ` ` for the upsampled function values using the fast Fourier

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 `=0 m= ` 2⇡ ⇡X X transform with a complexity of (N log N) 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 points upsampled in longitude m m AAAC9HicZVHbihNBEJ0Zb2u8ZfVNXwqzQoSwJCuoCMKCi/gUVzSbQDqEnp6apElP99hdkwvD/Ilv4qsf4V/4N3YmEXS3nk6frjpVdSrOlXTU7f4Oo2vXb9y8dXC7cefuvfsPmocPL5wprMCBMMrYUcwdKqlxQJIUjnKLPIsVDuPFu+3/cInWSaO/0CbHScZnWqZScPLUtPmLncFnqQUCzRFyIzU54BYBvxbS5VxgAlKD8ulUJNiBFYLgGpxRS2QMUmPryiJ3PMuVz04LLbbasOSqQAeFk3pW5zDCNcVpmXJH8N5vINFWXuMvT5Zr5wWzClaS5sBBmK3mWtIGTApHLOM0F1yVH6t2nykzg/7zI5g2W93jbh1wFfT2oBXs43x6GD5miRFFhpqE4s6Ne92cJiW3JIXCqsEKh371BZ/h2EPNM3STsja7gmeeSeq9U6MJavbfipJnzm2yuAMebAeuAc2zDmxfZIxynV3ZpU6Uvp6UUucFoRa7RmmhgAxsLweJtChIbTzgwko/K4g5t1yQv2+jNnElE+/bW3gpdYMlmLKzkuXcMu3vmvhtvYELjM163MNsMlaTEio253q2+/Vk1fBu9i57dxVcnBz3Xhx3P520Tt/sfT0IngRPg3bQC14Fp8GH4DwYBCJsh/1wGI6iZfQt+h792KVG4b7mUfBfRD//AJua7h8= AAAB73icZZDLSgMxGIUz9VbrrerSTbAIFUqZqaBuxIIblxXsBdpSMmnahiaTMflHKKXPILhxoYhb38GVj+DOB3FvOtOFtgcCX05y+C9+KLgB1/12UkvLK6tr6fXMxubW9k52d69mVKQpq1IllG74xDDBA1YFDoI1Qs2I9AWr+8Or6Xv9nmnDVXALo5C1JekHvMcpAWs1WkwIfIFLnWzOLbqx8CJ4M8hdfuR/Ph9ax5VO9qvVVTSSLAAqiDFNzw2hPSYaOBVskmlFhoWEDkmfNS0GRDLTHsf9TvCRdbq4p7Q9AeDY/ZsYE2nMSPoFbEESGMQAA1nA0xsoJUwhic1Vgt55e8yDMAIW0KRQLxIYFJ4Oj7tcMwpiZIFQzW2vmA6IJhTsijJ2Cd78yItQKxW9k6J74+bKpyhRGh2gQ5RHHjpDZXSNKqiKKBLoET2jF+fOeXJenbfka8qZZfbRPznvv4CHkms= ` =2 O b` = f(, ✓)Y ` (, ✓)sin✓d✓d Step 2: Double Fourier Sphere

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Spherical harmonics, Y , degree ` and order m 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 a function on the sphere to a rectangular grid while preserving periodicity 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 ` 0

30 The angular power spectrum measures 0 the amplitude of temperature fluctuations 60 30 90 75 100 75 25

AAACAHicZZDLSgMxFIYzXmu9VV24cBMsgotSZlRQXBXcuFSwrdAOJZOescFMMiRnxDLMxldx40IRtz6GO9/G9LLQeiDwnT/5OTl/lEph0fe/vbn5hcWl5dJKeXVtfWOzsrXdsjozHJpcS21uI2ZBCgVNFCjhNjXAkkhCO7q/GN23H8BYodUNDlMIE3anRCw4Qyf1KrtdhEeM4vwGLNI4U3ykF+e9StWv++Oi/yGYQpVM66pX+er2Nc8SUMgls7YT+CmGOTMouISi3M0spIzfszvoOFQsARvm4wUKeuCUPo21cUchHau/HTlLrB0mUY06SBgOxoCDpEZHHWotbW1im5mE8VmYC5VmCIpPBsWZpKjpKA3aFwY4yqEDxo1wf6V8wAzj6DIruxCC2ZX/Q+uoHhzXj65Pqo2zaRwlskf2ySEJyClpkEtyRZqEk4I8k1fy5j15L9679zF5OudNPTvkT3mfP8tTlkU= h(, ✓)=f(, ✓)+Y (, ✓)+Y (, ✓)+Y (, ✓)+Y (, ✓)+ Test function : 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 300 350 425 550 of the CMB and is calculated with the 120 60 50 75 25 25 50 50 50

⇡ 150 m ⇡ Y (, ✓)+Y (, ✓)+Y (, ✓)+Y (, ✓)+Y (, ✓)+Y (, ✓)+Y (, ✓)

AAADVXicfZLNa9swGMZlp+tH9tF0O+4iFjo6FoLkxkt3KBR62bGDpe2osyArci0iW0Z6PQgm/2QvY//JLoUpTg5bmuQFw6P3p0ePLN64UNICIb89v7HzbHdv/6D5/MXLV4eto9fXVpeGiwHXSpvbmFmhZC4GIEGJ28IIlsVK3MSTyzm/+SmMlTr/BtNCDDN2n8tEcgauNTryJsfpSaScYcw6EaQC2Ifz98lq6+P3UXVKyOxH1Q9na2E4h9RtWUd7QbjZGtbWYA2Moqbjn+rccP3J/W2XOiObTnbw88ZYBykJt4RSupUGm+io1SZdUhd+KuhStNGyrkath2iseZmJHLhi1t5RUsCwYgYkV2LWjEorCsYn7F7cOZmzTNhhVU/FDB+7zhgn2rgvB1x3/3VULLN2msUd7ETGIK0FpFkHz1egtbKdhW0lCZKzYSXzogSR80VQUioMGs9HDI+lERzU1AnGjXR3xTxlhnFwg9h0j0BXf/mpuA669LQbfO21L3rL59hHb9E7dIIo6qML9AVdoQHi3oP3x/d83//lPzZ2GruLrb639LxB/1Xj8C+LLwYV

/ 600 700 800 950 1050 1150 1250 /

spherical harmonic coecients, b : ✓

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 m 180 300

X The values of the angular power spectrum at higher degrees, AAACPHicZZBLaxRBEMd74iNxfCV6FEKTjeBhWWYUjAfFgDl4TMBNAttD6Omp2Wm2H0N3jWYZ9ujVq34UT0I+Ru65xVxzTu9ODpoUNPz6X/WnHnmtpMckOY2W7ty9d3955UH88NHjJ09X157te9s4AUNhlXWHOfegpIEhSlRwWDvgOldwkE8+zfMHX8F5ac0XnNaQaT42spSC41xioNTRai8ZJIugtyG9ht7Hk/Pv67/3/u4erUUbrLCi0WBQKO79KE1qzFruUAoFs5g1HmouJnwMo4CGa/BZuxh2Rl8GpaCldeEZpAv1X0fLtfdTnfdpAM2xWgBWuk/nP7RW+X5nu9EJy3dZK03dIBjRNSobRdHS+ea0kA4EqmkALpwMs1JRcccFhvvEDOEYv8kCK/qBvpUmZgWUbKdlNXfMWGmKsC3TfAK5PR6loLORytpNljdKAW7OWMXNuKsKyVkcrprevOFt2H89SN8Mkr2kt/2edLFCXpAN8oqkZItsk89klwyJIBX5QX6SX9Gf6Cw6jy660qXo2vOc/BfR5RXa+LE3 , are crucial to modern cosmology and are more difficult to calculate. The CMBR is strongest evidence for the Big 0 30 60 90 120 150 180 210 240 270 300 330 360 330 `

360 Bang theory. Refined analysis of the 180/⇡ 0 30 60 90 120 150 180 210 240 270 300 330 360 first two numerical tests demonstrate the accuracy of the new HP2SPH method relative to the HEALPix approach for problems involving

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(a) The surface of the Earth 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into dark matter (95% of the universe). AAACC3icZZC7SgNBFIZnvRtvUUubwSBYhLAbCy2jNjZCFBMFs4TZyWwyODdmziphSW/jq9hYKGLrC9j5Nk4uhcYfBr5zZn7OnD8xgjsIw+9gZnZufmFxabmwsrq2vlHc3Go6nVnKGlQLbW8S4pjgijWAg2A3xjIiE8Guk7vT4f31PbOOa3UFfcNiSbqKp5wS8K12cfdYdTNBLDb6gVnsDKNgM4l1ik/PTy6xJKZdLIWVcCT8H6IJlNBE9Xbxq9XRNJNMARXEudsoNBDnxAKngg0KrcwxQ+gd6bJbj4pI5uJ8tMsA7/lOB6fa+qMAj7q/HTmRzvVlUsYeJIHeCKAny3hYgdbClce2qUmQHsU5VyYDpuh4UJoJDBoPg8Edbv3uou+BUMv9XzHtEUso+PgKPoRoeuX/0KxWooNKeFEt1aqTOJbQDtpF+yhCh6iGzlAdNRBFj+gZvaK34Cl4Cd6Dj/HTmWDi2UZ/FHz+AFmrmak= Angular power spectrum of CMBR map 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 HP2SPH method can more accurately recover the angular power spectrum at high degrees when compared to the method used by the

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 Double Fourier Sphere technique allows for fast Fourier analysis in longitude HEALPix software. In the case of a real CMB map, which is inherently noisy, the power spectrums computed by the two methods show Step 3: Nonuniform FFT HEALPix Points 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 good agreement, as seen in the third numerical test. Advantages AAACCHicZZBLSwMxEMezPmt9VT16MFgED6Xs1oPiqeLFYwWrQltKNjvbBrPJkswWy9KjF7+KFw+KePUjePPbmD4OPgYCv5nkP5P5h6kUFn3/y5ubX1hcWi6sFFfX1jc2S1vb11ZnhkOTa6nNbcgsSKGgiQIl3KYGWBJKuAnvzsf3NwMwVmh1hcMUOgnrKRELztCVuqW9NsI9hnHezlQEZtwmP4sGTCHrgR2NuqWyX/UnQf9DMIMymUWjW/psR5pnCSjkklnbCvwUOzkzKLiEUbGdWUgZv3PtWw4VS8B28skiI3rgKhGNtXFHIZ1Ufypyllg7TMIKdZAw7E8A+0mFjjPUWtrKVPZnEsYnnVyoNENQfDooziRFTceu0EgY4CiHDhg3wv2V8j4zjKPzruhMCP6u/B+ua9XgqOpf1sr105kdBbJL9skhCcgxqZML0iBNwskDeSIv5NV79J69N+99+nTOm2l2yK/wPr4B8fiaSw== Our new method, HP2SPH, utilizes the FFT, NUFFT, and the FSHT to accurately compute the angular power spectrum of CMBR data

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 points are non-equispaced in longitude, so we use the Nonuniform FFT Hierarchically tessellated: maps, even at higher sphericalThis harmonic new method degrees, utilizes with near the optimal FFT, computational NUFFT, and complexity the FSHT ( (N log N), for N total nodes) to more

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 x be non-equispaced and 0 j, k N 1: the pixel structure lends itself to simple and   O N 1 accurately compute the angular power spectrum of CMBR data (even at higher spherical harmonic direct movements to more refined levels of 2⇡ixj k 2⇡ixj k degrees, `) with near optimal computational complexity. fj = cke f = F c, (F )jk = e 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 Future Work & References

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 resolution () AAACCXicZZDLSgMxFIYzXmu9VV26CRbBRSkzutCFi4IgLivYC7SlZNIzbWgmGZIzYhm6deOruHGhiFvfwJ1vY3pZaD0Q+PInf07OHyZSWPT9b29peWV1bT23kd/c2t7ZLezt161ODYca11KbZsgsSKGghgIlNBMDLA4lNMLh1eS8cQ/GCq3ucJRAJ2Z9JSLBGTqpW6BthAcMo6ydqh6YyTPZdYqpAdrQZjgedwtFv+xPi/6HYA5FMq9qt/DV7mmexqCQS2ZtK/AT7GTMoOASxvl2aiFhfMj60HKoWAy2k00nGdNjp/RopI1bCulU/e3IWGztKA5L1EHMcDAFHMQlOtmh1tKWZraFThhddDKhkhRB8VmjKJUUNZ3EQnvCAEc5csC4Ee6vlA+YYRxdeHkXQrA48n+on5aDs7J/e1qsXM7jyJFDckROSEDOSYXckCqpEU4eyTN5JW/ek/fivXsfs6tL3txzQP6U9/kDhVKalw== Future Work isoLatitude: 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 kX=0

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 Rigorously test our method against HEALPix’s method with more complicated deterministic and noisydata on the sphere. By relating F˜ to the standard DFT matrix F =e e 2⇡eijk/N with a low rank Pixels are distributed on lines of constant jk • (C or C++).

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 approximation (rank K), the NUFFT simplifies to K FFT computations [4] 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 Make the implementation of our method even faster by transcribing the code into a lower-level language latitude, which aids univariate Fourier 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 Use our method to compute the angular power spectrum of CMBR data maps and compare to the HEALPix points on the sphere (top) 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 given by HEALPix software.

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 and computations Step 4: Fast Spherical Harmonic Transform and mapped to a rectangle (bottom) 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 Disadvantages Modify our method to include the ability to perform Fourier synthesis on a CMBR map to 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 corresponding map values.

FinallyAAACPXicZVA9axtBEN1zPuwoX0pSplkiAimEuLOLhFSGgEjpgGQbLCHm9uakQXu7x+6cQQj/MTf5D+nSpUmRENKmzeikIrEHFt6+mTcfL68tRU7Tr8nenbv37u8fPOg8fPT4ydPus+en0TfB4Nh468N5DhEtORwzscXzOiBUucWzfPlhkz+7xBDJuxGvapxWMHdUkgEWatYdTZwnV6BjPSQH1q40wxI1L1APhyNNTlsp5aYQzus5cpvK6RICAUuRbEIYtPFYSluSTnHW7aWDtA19G2Q70FO7OJl1v0wKb5pKxMZCjBdZWvN0DYHJWLzqTJqINZglzPFCoIMK43TdXn+lXwtT6NIHeXJGy/6rWEMV46rK+1pABbxoAS+qvt782Hsb+1vZjUlcvpuuydUNozPbQWVjNz5srNQFBTQsjhUEJpDsqs0CAhgWwztiQnbz5Nvg9HCQHQ3ST4e94/c7Ow7US/VKvVGZequO1Ud1osbKqGv1Tf1QP5PPyffkV/J7W7qX7DQv1H+R/PkLGU6uUA== take the FFT in latitude to get the bivariate Fourier coecients AAACC3icZZBNSwMxEIaz9avWr6pHL8EieChltx4UT4IePFawrWBLyWZnbTCbLMmsWJbevfhXvHhQxKt/wJv/xmzbgx8DgWcmeWcyb5hKYdH3v7zS3PzC4lJ5ubKyura+Ud3c6lidGQ5trqU2VyGzIIWCNgqUcJUaYEkooRvenhb33TswVmh1iaMU+gm7USIWnKErDaq7PYR7DOO8l6kITNEmPxOWRXdMIbsBOx4PqjW/4U+C/odgBjUyi9ag+tmLNM8SUMgls/Y68FPs58yg4BLGlV5mIWX81rW/dqhYArafT3YZ0z1XiWisjTsK6aT6U5GzxNpREtapg4ThcAI4TOq0yFBraetT2Z9JGB/1c6HSDEHx6aA4kxQ1LYyhkTDAUY4cMG6E+yvlQ2YYR2dfxZkQ/F35P3SajeCg4V80ayfHMzvKZIfskn0SkENyQs5Ji7QJJw/kibyQV+/Re/bevPfp05I302yTX+F9fANuvJup

AAACB3icZZC9TsMwFIUdfkv5CzAyYFEhMVRVwgJslVgYC6K0Eq0qx72hFo4d2TeIKurIwquwMABi5RXYeBvcNgPQI1n6fO3r63OiVAqLQfDtzc0vLC4tl1bKq2vrG5v+1va11Znh0ORaatOOmAUpFDRRoIR2aoAlkYRWdHc2Pm/dg7FCqyscptBN2K0SseAMXann73UQHjCK806m+mDGz+SXEIMBxcGORj2/EtSCiegshAVUSKFGz//q9DXPElDIJbP2JgxS7ObMoOASRuVOZiFl/I7dwo1DxRKw3XxiZEQPXKVPY23cUkgn1d8dOUusHSZRlTpIGA4mgIOkSsc71Fra6rTt3ySMT7q5UGmGzth0UJxJipqOU6F9YYCjHDpg3Aj3V8oHzDCOLruyCyH8b3kWmke101p4cVSpnxRplMgu2SeHJCTHpE7OSYM0CSeP5Jm8kjfvyXvx3r2P6dU5r+jZIX/kff4AWTeaHw==sha1_base64="v+a23l4P+wz+jIo4bGE9UFDmHr4=">AAAB2HicZZDNSsNAFIUn9a/GqnXtZrAILkpJ3Kg7wY3LCsYW2lAmk0kzdH7CzI1QQl/AhRvFB3Pn2zhNu9D2wMB3z3C5956kENxCEPx4jZ3dvf2D5qF/1PKPT07brRerS0NZRLXQZpgQywRXLAIOgg0Lw4hMBBsks4fl/+CVGcu1eoZ5wWJJpopnnBJwVn/S7gS9oBbehnANHbTWpP09TjUtJVNABbF2FAYFxBUxwKlgC39cWlYQOiNTNnKoiGQ2ruo1F/jSOSnOtHFPAa7dvx0VkdbOZdLFDiSBvAbIZRcvK9Ba2O6qbWMSZLdxxVVRAlN0NSgrBQaNlzfjlBtGQcwdEGq42xXTnBhCwSXjuwzCzYu3Ibru3fXCpwA10Tm6QFcoRDfoHj2iPooQRSl6Qx9e7r17n6uoGt46szP0T97XLzmziYY=sha1_base64="29RMW3RfU21Fm7H/no6KYlrubtQ=">AAAB/HicZZC7TgMxEEVneYbwCrQUWCAkiijapQE6JBrKgAhBIlHkdWYTK157Zc8iolVKGn6FhgIQ30HH3+A8Ch5XsnQ81sz43jhT0lEYfgVz8wuLS8ullfLq2vrGZmVr7caZ3ApsCKOMvY25QyU1NkiSwtvMIk9jhc14cD5+b96jddLoaxpm2E55T8tECk6+1KnstggfKE6KVq67aMdjiitM0KIW6EajTmU/rIUTsf8QzWAfZqp3Kp+trhF5ipqE4s7dRWFG7YJbkkLhqNzKHWZcDHgP7zxqnqJrFxMjI3bgK12WGOuPJjap/uwoeOrcMI2rzEPKqT8B6qdVNr6RMcpVp21/NlFy0i6kznLyxqaLklwxMmycCutKi4LU0AMXVvq/MtHnlgvy2ZV9CNFfy/+hcVQ7rUWXIZRgB/bgECI4hjO4gDo0QMAjPMMrvAVPwUvwPk1rLpjFtg2/FHx8A7znmLk=sha1_base64="/pIHbKHoDESvn4ewHW0AhXWJ2aU=">AAACB3icZZC7TsMwFIYdrqXcAowMWFRIDFWVsFC2SiyMBRFAolXluCfUwrEj+wRRRR1ZeBUWBkCsvAIbb4PbZuDyS5Y+H/v4+P/jTAqLQfDlzczOzS8sVpaqyyura+v+xuaF1bnhEHEttbmKmQUpFEQoUMJVZoClsYTL+PZ4fH55B8YKrc5xmEE3ZTdKJIIzdKWev9NBuMc4KTq56oMZP1OcQQIGFAc7GvX8WtAIJqL/ISyhRkq1e/5np695noJCLpm112GQYbdgBgWXMKp2cgsZ47fsBq4dKpaC7RYTIyO65yp9mmjjlkI6qf7sKFhq7TCN69RBynAwARykdTreodbS1qdtfyZh0uwWQmU5OmPTQUkuKWo6ToX2hQGOcuiAcSPcXykfMMM4uuyqLoTwr+X/EB00jhrhaVBrNcs0KmSb7JJ9EpJD0iInpE0iwskDeSIv5NV79J69N+99enXGK3u2yC95H99Yl5od References Sparsity at poles: 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 Fourier Series represent smooth functions on the sphere with exponentials:

[1] R. M. Slevinsky. Fast and backward stable transforms between spherical harmonic expansions and 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 Fourier series, in press at Appl. Comput. Harmon. Anal., 2017. functions are grossly undersampled 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

1 1 Harmon.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 Anal., 2017. Let f(, ✓) be bi-periodic on [ ⇡, ⇡]2: in✓ im AAACInicZZDLbhMxFIY95RYCpYEu2VgklVJpEs2EBZBVpG5YsAgSuUiZaeTxnEmseOyRfQYpivIsbPoq3bCgKqyQeBicywKSIx3p82//Oj5/UkhhMQh+eycPHj56/KTytPrs+emLs9rLV0OrS8NhwLXUZpwwC1IoGKBACePCAMsTCaNkcbW5H30FY4VWX3BZQJyzmRKZ4AydNK19+ARIG1kzks6TMj/COSC7bNAEaCJaBRihU8GpVrQxaUWF8F3H151Gd1qrB+1gW/QYwj3Uyb7609rPKNW8zEEhl8zaSRgUGK+YQcElrKtRaaFgfMFmMHGoWA42Xm1XXNMLp6Q008a1QrpV/3WsWG7tMk986iBnON8CznOfbk6otbT+znYwCbP38UqookRQfDcoKyVFTTd50VQY4CiXDhg3wv2V8jkzjKNLtepCCA9XPoZhpx2+bQefO/Vedx9Hhbwmb0iThOQd6ZGPpE8GhJNv5Jb8IHfejffdu/d+7Z6eeHvPOfmvvD9/AY+JoO0= f(, ✓)= an,me e [2] K. M. Gorski, B. D. Wandelt, E. Hivon, and F. K. Hansen, The HEALPix Primer, (2010).

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 near the poles of the sphere 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_base64="1RltIJ8R28o4mKuhHm8shpV63kM=">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sha1_base64="Zgtp1clKcrBe0P22MaIeLZ4jzDs=">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sha1_base64="2yifVDN6WADDsC57vqBtzxE0UV4=">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 Unequal spacing: 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 n= m= [3] A. Townsend, H. Wilber, and G. B. Wright, Computing with functions in spherical and polar geometries I. The sphere, SIAM J. Sci. Comp., 38-4 (2016), 1 1 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 Pixels are misaligned and unequally spaced X X The Fast Spherical Harmonic Transform [1] quickly computes spherical harmonic pp.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 C403-C425. [4] D. Ruiz-Antol´ın and A. Townsend, A Nonuniform Fast Fourier Transform Based on Low Rank Approximation, SIAM J. Sci. Comp., 40-1 (2017), p. 529-547.

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 in longitude, making fast bivariate Fourier coecients from the bivariate Fourier coecients with a complexity of (N log N) 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 O

analysisAAAC5XicZVFNaxRBEJ0Zv+L6tdGjl8KN4CGE3QhRAmJAEY8R3CSws4SanpqdIj3dY3dNkmXYn+BNvHr3H/lv7J1dgiYFDY/qV92v3stqzV6Gwz9xcuv2nbv3Nu73Hjx89PhJf/PpkbeNUzRWVlt3kqEnzYbGwqLppHaEVabpODv7sLw/Pifn2ZqvMq9pWuHMcMEKJbRO+79TY9nkZARSoUvJinZs6FuDGnyNis1ssZ+mcMiXpD2gI6jYo+aZoRzQ5NCs2Hre8SkPZDagrZmxNDltQ4Vn4RUo0AtkfI6OUQg+hQWYXGCjQT337IGr2nrPQTlcsJS2EahsfqW1d9ofDHeGXcFNMFqDQbSuw9PNeJDmVjVV2E5p9H4yGtYybdEJK02LXtp4CqLPcEaTAA1W5KdtZ+oCXoZODoV14QR3uu6/Ey1W3s+rbBsCqFDKDkhZLVeWUqzVfns1du0nKd5OWzZ1I2TU6qOi0SAWlglBzo6UBENzRuU4aAVVokMlIcdemlORfmzTGt1VdGnwmDJ7ORlRNZ3oabuVZo3WJFuwSEsMUXS0cBuULFO+4FxKeAd73Nk6um7iTXC0uzN6vTP8sjs42F8bvBE9j15Er6JR9CY6iD5Hh9E4UnE/3ovfxwfJLPme/Eh+rqhJvJ55Fv1Xya+/55/pdw== impossible without modification (minus some precomputational cost). 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 [5] J. Keiner and D. Potts, Fast evaluation of quadrature formulae on the sphere, Math. Comp., 77 (2008), pp. 397–419.

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