A random walk from molecular dynamics to data science
|
|
|
- Roderick Lawson
- 5 years ago
- Views:
Transcription
1 A random walk from molecular dynamics to data science Benedict Leimkuhler University of Edinburgh The Maxwell Institute for Mathematical Sciences and The Alan Turing Institute
2 Molecular Dynamics (MD) Biomolecules Materials cytochrome C folding: Elber and Cardenas Quasi-continuum method: molecular dynamics with coarse-graining for indentation studies. Ortiz, Phillips and Tadmor
3 Molecular Dynamics 2017: article in Drug Discovery Today Molecular dynamics-driven drug discovery: leaping forward with confidence developing a market-approved drug costs a staggering $2.6 billion Given that the dynamics of the covalent bonds involving hydrogen atoms are not crucial in biological problems, they are usually constrained using integration algorithms, such as SHAKE, RATTLE, Hence, a time-step value in the range of fs is possible and has been shown to be suitable for MD simulations of many biological systems." Examples of some 2017 simulations out of around published Ligand binding in the HIV-integrase enzyme Druggability of membrane bound Rab5 Fission fragment damage in nuclear fuel Patchy particle systems Nanoparticle cholestoral metabolism therapeutics Multiscale protein dynamics in antigen presentation Bottlebrush copolymers Modification of membrane structure by lithium Smectic C semiflexible polymers Graphene reinforced polymer nanocomposites Unsaturated Zwitterionic lipids Identifying the warfarin binding site
4 HIV-1 Virus Capsid (protective shield around virus) Molecular dynamics 64M atoms (including surroundings) Described by a potential energy function U(x) Chaotic, nonlinear dynamics, ever expanding scale Key questions are of a probabilistic nature
5 <latexit sha1_base64="rlexfyzr8cyk87qy3eae4nayum0=">aaacbnicbvdlssnafl2pr1pfuzcidbahbkoigi6lblxwsa9oqplmpu3qmstmtioldoxgx3hjqhg3fom7/8zpm4w2hhg4c8693htpkhcmton8w4wv1bx1jejmawt7z3fp3j9oqjivhdzizgpzdrcinew0oznmtj1iikxaassy3kz91ohkxeloxo8t6gvcj1ipeayn1lwpprzp5i2wtaas8ncgmk8kfe48kzpf1y47vwcgtezcnjqhr71rf3lhtfjbi004vqrjoon2myw1i5xosl6qailjepdpx9aic6r8bhbgbj0ajus9wjpnlpqpvzsyljqai8bucqwhatgbiv95nvt3rvymrumqautmg3oprzpg00xqycqlmo8nwuqysysiaywx0sa5kgnbxtx5mttpq65tde8uyrxrpi4ihmejvmcfs6jbldshaqqe4rle4c16sl6sd+tjxlqw8p5d+apr8wfywzgh</latexit> <latexit sha1_base64="rlexfyzr8cyk87qy3eae4nayum0=">aaacbnicbvdlssnafl2pr1pfuzcidbahbkoigi6lblxwsa9oqplmpu3qmstmtioldoxgx3hjqhg3fom7/8zpm4w2hhg4c8693htpkhcmton8w4wv1bx1jejmawt7z3fp3j9oqjivhdzizgpzdrcinew0oznmtj1iikxaassy3kz91ohkxeloxo8t6gvcj1ipeayn1lwpprzp5i2wtaas8ncgmk8kfe48kzpf1y47vwcgtezcnjqhr71rf3lhtfjbi004vqrjoon2myw1i5xosl6qailjepdpx9aic6r8bhbgbj0ajus9wjpnlpqpvzsyljqai8bucqwhatgbiv95nvt3rvymrumqautmg3oprzpg00xqycqlmo8nwuqysysiaywx0sa5kgnbxtx5mttpq65tde8uyrxrpi4ihmejvmcfs6jbldshaqqe4rle4c16sl6sd+tjxlqw8p5d+apr8wfywzgh</latexit> <latexit sha1_base64="rlexfyzr8cyk87qy3eae4nayum0=">aaacbnicbvdlssnafl2pr1pfuzcidbahbkoigi6lblxwsa9oqplmpu3qmstmtioldoxgx3hjqhg3fom7/8zpm4w2hhg4c8693htpkhcmton8w4wv1bx1jejmawt7z3fp3j9oqjivhdzizgpzdrcinew0oznmtj1iikxaassy3kz91ohkxeloxo8t6gvcj1ipeayn1lwpprzp5i2wtaas8ncgmk8kfe48kzpf1y47vwcgtezcnjqhr71rf3lhtfjbi004vqrjoon2myw1i5xosl6qailjepdpx9aic6r8bhbgbj0ajus9wjpnlpqpvzsyljqai8bucqwhatgbiv95nvt3rvymrumqautmg3oprzpg00xqycqlmo8nwuqysysiaywx0sa5kgnbxtx5mttpq65tde8uyrxrpi4ihmejvmcfs6jbldshaqqe4rle4c16sl6sd+tjxlqw8p5d+apr8wfywzgh</latexit> <latexit sha1_base64="rlexfyzr8cyk87qy3eae4nayum0=">aaacbnicbvdlssnafl2pr1pfuzcidbahbkoigi6lblxwsa9oqplmpu3qmstmtioldoxgx3hjqhg3fom7/8zpm4w2hhg4c8693htpkhcmton8w4wv1bx1jejmawt7z3fp3j9oqjivhdzizgpzdrcinew0oznmtj1iikxaassy3kz91ohkxeloxo8t6gvcj1ipeayn1lwpprzp5i2wtaas8ncgmk8kfe48kzpf1y47vwcgtezcnjqhr71rf3lhtfjbi004vqrjoon2myw1i5xosl6qailjepdpx9aic6r8bhbgbj0ajus9wjpnlpqpvzsyljqai8bucqwhatgbiv95nvt3rvymrumqautmg3oprzpg00xqycqlmo8nwuqysysiaywx0sa5kgnbxtx5mttpq65tde8uyrxrpi4ihmejvmcfs6jbldshaqqe4rle4c16sl6sd+tjxlqw8p5d+apr8wfywzgh</latexit> <latexit sha1_base64="5zgcieh/9te+m1b+d1lp/bb74sm=">aaacenicbvc9tsmwghtkxyl/auywiwqphagsharjbqtjkuhbqqmv4zitvtujbae1ivimllwkcwmistkx8ta4bqzoocns6e4+ff7otxivyrk+jdlk6tr6rnmzsrw9s7tn7h+0zzwktbwcs1h0fsqjoxfxffwmdbnbepcz6fij66nfesbc0ji6u5oeebwnihpsjjsw+my9cwwhqe7ytdauq+gmik5udml9dupojydfynw3q1bdmgeue7sgvvcg1te/3cdgkserwgxj2botrhkzeopirvkkm0qsidxca9ltnekcsc+bnztde60emiyffpgcm/x3ria4lbpu6yrhaigxvan4n9dlvxjpztrkukuipf8upgzqm6f9wiakghwbaikwopqvea+rqfjpfiu6bhvx5gxspmvyvso+pa82r4o6yuaihimasmefaiib0aiowoarpinx8gy8gs/gu/exj5amyuyq/ihx+qn+vpy4</latexit> <latexit sha1_base64="zohxt+42lb03snodu1wswi5h9ec=">aaab9xicbvbnswmxem3wr1q/qh69bitql2vxbd0wvxisyd+gu5zsntugzpmlma2u0v/hxymixv0v3vw3pu0etpxbwoo9gwbmhangblz32ymsrw9sbhw3szu7e/sh5cojllgzpqxjlvc6exldbjescrwe66saksqurb0ob2d+e8s04uo+wdhlqul6ksecerdsoz8ioh3wqk8jbee9cswtuxpgvellpijynhrllz9sneuybcqimv3ptsgyea2ccjyt+zlhkafd0mddsyvjmakm86un+mwqey6vtiubz9xfexosgdnoqtuzebiyzw8m/ud1m4ivgwmxaqzm0swiobmyfj5fgcougquxtorqze2tma6ijhrsucubgrf88ippxdq8t+bdx1bqn3kcrxsctleveegk1dedaqamokijz/sk3pwn58v5dz4wrqunnzlgf+b8/galo5jd</latexit> <latexit sha1_base64="zohxt+42lb03snodu1wswi5h9ec=">aaab9xicbvbnswmxem3wr1q/qh69bitql2vxbd0wvxisyd+gu5zsntugzpmlma2u0v/hxymixv0v3vw3pu0etpxbwoo9gwbmhangblz32ymsrw9sbhw3szu7e/sh5cojllgzpqxjlvc6exldbjescrwe66saksqurb0ob2d+e8s04uo+wdhlqul6ksecerdsoz8ioh3wqk8jbee9cswtuxpgvellpijynhrllz9sneuybcqimv3ptsgyea2ccjyt+zlhkafd0mddsyvjmakm86un+mwqey6vtiubz9xfexosgdnoqtuzebiyzw8m/ud1m4ivgwmxaqzm0swiobmyfj5fgcougquxtorqze2tma6ijhrsucubgrf88ippxdq8t+bdx1bqn3kcrxsctleveegk1dedaqamokijz/sk3pwn58v5dz4wrqunnzlgf+b8/galo5jd</latexit> <latexit sha1_base64="zohxt+42lb03snodu1wswi5h9ec=">aaab9xicbvbnswmxem3wr1q/qh69bitql2vxbd0wvxisyd+gu5zsntugzpmlma2u0v/hxymixv0v3vw3pu0etpxbwoo9gwbmhangblz32ymsrw9sbhw3szu7e/sh5cojllgzpqxjlvc6exldbjescrwe66saksqurb0ob2d+e8s04uo+wdhlqul6ksecerdsoz8ioh3wqk8jbee9cswtuxpgvellpijynhrllz9sneuybcqimv3ptsgyea2ccjyt+zlhkafd0mddsyvjmakm86un+mwqey6vtiubz9xfexosgdnoqtuzebiyzw8m/ud1m4ivgwmxaqzm0swiobmyfj5fgcougquxtorqze2tma6ijhrsucubgrf88ippxdq8t+bdx1bqn3kcrxsctleveegk1dedaqamokijz/sk3pwn58v5dz4wrqunnzlgf+b8/galo5jd</latexit> <latexit sha1_base64="zohxt+42lb03snodu1wswi5h9ec=">aaab9xicbvbnswmxem3wr1q/qh69bitql2vxbd0wvxisyd+gu5zsntugzpmlma2u0v/hxymixv0v3vw3pu0etpxbwoo9gwbmhangblz32ymsrw9sbhw3szu7e/sh5cojllgzpqxjlvc6exldbjescrwe66saksqurb0ob2d+e8s04uo+wdhlqul6ksecerdsoz8ioh3wqk8jbee9cswtuxpgvellpijynhrllz9sneuybcqimv3ptsgyea2ccjyt+zlhkafd0mddsyvjmakm86un+mwqey6vtiubz9xfexosgdnoqtuzebiyzw8m/ud1m4ivgwmxaqzm0swiobmyfj5fgcougquxtorqze2tma6ijhrsucubgrf88ippxdq8t+bdx1bqn3kcrxsctleveegk1dedaqamokijz/sk3pwn58v5dz4wrqunnzlgf+b8/galo5jd</latexit> Turning it into maths Physicists tell us that what they need to do is calculate Z '(x)dµ(x) observable function '( ) Gibbs measure just integration but with 180M variables!
6 <latexit sha1_base64="pohpw9humefyccb8iv+v21wdfp8=">aaack3icbvblswmxgmzwv62vqkcvwslus9kvqs9cqqoek9ghdjclm6ztajjdkmyxlpt/vphxpojbb179h2bbhmrrb4hjzdckm0heqnk2/wnlvlbx1jfym4wt7z3dvel+qvofscskgumwynaafgfukiammpf2janiasotyhid6a0rkyqg4kgpi+jx1be0rzhshvklnzcjpqic5db1xr5dd4rknkdlx1n4bv0qtj+4gdf4k85liss57kbgyg5+swrx7mnazedmqanmpu4xx91uignohmymkdvx7eh7czkaykbsghsreie8rh3smvagtpsxtlkm8mqwxdglptlcwwk770gqv2rma7ozjvolwkb+p3vi3bv0eiqiwbobpw/1ygz1clpiyjdkgjubg4cwpoavea+qrfibegumbgcx8jjonlucu+lcn5eqtvkdexaejkezooacvmedqimgwoajvib38ge9w2/wl/u9xc1zm88h+dpwzy/cwkbk</latexit> invariant measure <latexit sha1_base64="ufox4zita877+ikq+m6ienxmvh0=">aaab6nicbvdlsgnbeoynrxhfuy9ebopgkeykomegf48rzqosjcxozpmhm7pltk8qqj7biwdfvppf3vwbj8kenlggoajqprsrsqww6pvfxmftfwnzq7hd2tnd2z8ohx41bzizxhsskylpr9ryktrvoedj26nhvewst6lr7cxvpxfjraifczzyungbfrfgfj300fvzr1zxq/4czjueoalajnqv/nxtjyxtxcot1npo4kcytqhbwssflrqz5sllizrghuc1vdygk/mpu3lmld6je+nki5mrvycmvfk7vphrvbshdtmbif95nqzj63aidjoh12yxkm4kwytm/iz9ythdoxaemipcryqnqaemxtolf0kw/piqav5ua78a3f9wajd5heu4gvm4hwcuoaz3uicgmbjam7zcmye9f+/d+1i0frx85hj+wpv8av1ejdy=</latexit> <latexit sha1_base64="ypu/zz/rjuesmglf2kjx3atf4h4=">aaab+3icbvbns8naen3ur1q/yj16crahxkoigl6eohepfewhtdfstpt26wytdifsepjxvhhqxkt/xjv/xm2bg7y+ghi8n8pmpd/mtiftfxultfwnza3ydmvnd2//wdysdlsuselbjokr7plyuc4ebqmdtnuxpdj0oe36k9uz332iurfipeaauzfei8ecrjboytor9aknzrxmxnv544cjaflprnknew5rltgfqaeclc/8ggwjkoruaofyqb5jx+bmwaijnoavqajojmkej2hfu4fdqtxsfntunwplaawr1cxamqu/jzickpwgvu4mmyzvsjct//p6cqrxbszenaavzleoslgfktulwhoysqnwvbnmjno3wmsmjsag46roejzll1dj57zh2a3n/qlwvcnikknjdilqyegxqinuuau1euft9ixe0zurgy/gu/gxac0zxcwr+gpj8wdvk5ql</latexit> <latexit sha1_base64="zhzqntoszix3l2pm3wlwblnlzai=">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</latexit> <latexit sha1_base64="piajc1yww/awe6ss5xb0fgy0lp0=">aaacgxicbvbps8mwhe39o+e/qkcvwshmg6mvqs/c0ivhcxybrrwkabafjw1jutko/rpe/cpepcjiuu9+g7otb918epj47/1ifi9igjxksr6nhcwl5zxv0lp5fwnza9vc2w3kobwyodhmswghsbjgi+ioqhhpj4ighjdscgzxy7/1qiskcxsrrgnxoopftesxulrytstzbydh7vk0ojycf/dozztp77njo4dex9drrg6l3na3k1bnmgdoe7sgfvcg4zufbhjjljniyyak7nhworwmcuuxi3nztsvjeb6ghulogifopjdnnsvhovzc2i2fppgce/x3ria4lcme6crhqi9nvbh4n9djvffcy2iupipeeppqn2vqxxbcewypifixksyic6r/cnefcysvlross7bnv54nzzoabdxsm9nk/bkoowt2wqgoahucgtq4bg3gaawewtn4bw/gk/fivbsf0+icuczsgt8wvn4a71ue/g==</latexit> Sampling and MCMC Typical approach: using a Markov Chain with prescribed Example method: Metropolis Monte Carlo
7 <latexit sha1_base64="ipbhajzhpzzplun7nsgjggxlzim=">aaacdxicbvdltsjafj3ic/fvdelmipq0iwdbmojghojgjsywskcq6tdahokjm1mdifyag3/fjquncevenx/jaf0oejkbnjxzb+69x48zfdkyvrxmyura+kz2m7e1vbo7p+8fvewucexchlgi130kckmhcswvjnrjtldgm1lzbzdtv/zaukbrec9hmfec1atpl2ikldtwt1xjamiraphfdoayxrdlfg2z5cacnozcwtbbet4qwtpazwknja9svnr6v7mt4sqgocqmcdgwrvh6y8qlxyxmcs1ekbjhaeqrhqihcojwxrnvjvbukr3yjbiqumkz+ntijaihrogvogmk+2lrm4r/ey1edi+9mq3jrjiqzxd1ewzlbkfrwa7lbes2ugrhttwtepcrr1iqahmqbhvx5wvsduq2vblvzvpl6zsoldgcx8aanrgazxalksafgdycz/ak3rqn7uv71z7mrrktntkef6b9/gcqezy6</latexit> <latexit sha1_base64="u++gakoc+pkxbqglatnj+qg4cfq=">aaab8hicdvdltsjafj3ic/gfunqzkzjgwqblzy7oxiumfjbqyxqyymj02sxmjathk9y40bi3fo47/8ypykjgt3ktk3puzb33ecgjulnwh5fawv1b30hvzra2d3b3svshlrleahmhbywqhq9jwignjqkkku4ocpi9rtre5clx23desbrwazunieujeaddiphs0g25jc+c/p3prj/nwwalwq4xi9ay7vq5vk9iovsu2da0twuohfii2c++9wybjnzcfwziyq5thcqnkvaumzll9cjjqoqnaes6mnlke+ng84nn8eqrazgmhc6u4fz9phejx8qp7+loh6mx/o0l4l9en1ldmhtthkakclxyniwyvafmvocdkghwbkojwolqwyeei4gw0hlldahfn8l/satg2jqiq1kucb6miw2owdhiaxtuqqncgizwaay+eabp4nkqxqpxyrwuwlpgcuyq/idx9gl2hjaw</latexit> Example: Metropolis Monte Carlo for the Uneven Double Well Gaussian prior: rare events slow convergence too much measure in the right basin
8 <latexit sha1_base64="9tto/n8ofer8lgngebhgmxsljd8=">aaacjnicbzdnsgmxfiuz/lv/qi7dbitqectmexqjig5cvndaqmcomtrtg5nmmnyrlmgexo2v4safiulorzftr9dwcygh79xlck8qc67btj+tufmfxaxlldxc2vrg5lzxe6euo0rr5tjirkozem0el8wfdoi1y8vigajwco6urn7jgsnni3klw5j5ielj3uwugeht4nnqqrb3sge+x8eejieg2c0pdnhoar9ht98rskvzd5vcjawn7wljrtjjwrpcyuuj5vvrf1+9tkstkemggmjdcuwy/jqo4fswroalmswe3peeaxkpsci0n47xzpcbir3cjzq5evcy/p5isaj1maxmz0igr6e9efzpayxqpfntlumemksth7qjwbdhuwa4wxwjiizgekq4+sumfaiibznswytgtk88k+rvimnxnjut0svlhsck2kp7qiwcdiou0dwqirdr9iie0st6s56sf+vd+pi0zln5zc76u9bxn3xho/g=</latexit> Sampling using SDEs Ito-type Stochastic Differential Equation Brownian Dynamics Wiener measure describes a particle diffusing in a potential U at fixed temperature
9 Fokker-Planck Equation The stochastic paths of an Itō SDE sample the evolving distribution with probability = L L is the L 2 adjoint of L defined by (L g)(x) = lim t!0 E[g(x( t)) x(0) = x] g(x) t
10 <latexit sha1_base64="lnwzustj1+34cpnlf+gcgyowa1o=">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</latexit> <latexit sha1_base64="ur9aqlkrsnvuaj45yzb/1dknvmy=">aaab/hicbzdnssnafiun/tb6f+3szwar6sksikaboejgzqxtftpyjtpbdugke2ymygj1vdy4umstd+lot3hazqgtbwy+zr2xe+cemwdko863tbs8srq2xtgobm5t7+zae/snjrjjwaocc9kkialoiva00xxasqqsbhyaweh6um8+gfrmrhc6jcepysbifuajnlbxlnxkufqej/elhvvsxdm47tplp+pmhrfbzagmctw79lenj2gsqqqpj0q1xsfwfkakzptdunhjfmsejsga2gyjeolys+nxy3xknb7uc2lepphu/t2rkvcpnaxmz0j0um3xjuz/txai+xd+xqi40rdr2aj+wrewejie7jejvppuakgsmvsxhrjjqdz5fu0i7vyxf6fxwnwdqnt7vq5d5xeu0ae6rbxkonnuqzeojjxeuyqe0st6s56sf+vd+pi1lln5tan9kfx5a3o/k1o=</latexit> Brownian Dynamics so is a stationary density & geometric ergodicity under mild conditions on U Ergodic limit Just need a reliable way of computing numerical solutions of SDEs (on long intervals)!
11 SDE Numerics What is a suitable definition for error for an SDE? Examples: weak, strong, ergodic How to get high accuracy in SDE discretisation? How does discretisation affect convergence to the invariant distribution?
12 Euler-Maruyama Method Numerical Method discrete Brownian path Leimkuhler-Matthews Method [L. & Matthews, AMRX, 2013] [L., Matthews & Stoltz, IMA J. Num. Anal., 2015] [L., Matthews & Tretyakov, Proc Roy Soc A, 2014]
13 Uneven Double Well ergodic error small stepsize bimodal distribution large stepsize L-M E-M
14 Invariant measures of numerical methods Sampling Applications Charlie Matthews (UoE PhD ) 5 papers + a book and over 300 citations
15 Charlie Matthews (PhD 2015, Chicago) Langevin (BAOAB) Matthias Sachs (PhD 2017, Duke) Invariant measures Overdamped Limit Constrained LD Xiaocheng Shang (PhD 2015, Langevin/AdL Ergodicity Adaptive Langevin Gabriel Stoltz (ENPC Paris) Michael Tretyakov (Nottingham) Vincent Danos (ENS Paris)
16 the most recent citations for our 2016 geodesic integrators paper: physical chemistry biomolecules data science biomolecules molecular software data science physics stats & data science stats & data science Why are statisticians reading molecular dynamics papers?
17 Wind Turbines (with industry partner) Zofia Trstanova (postdoc) Anton Martinsson (PhD student)
18 Financial giant Goldman Sachs announces that it plans to invest $150 billion in clean energy projects and technology like solar and wind farms, energy efficiency upgrades for buildings, and power grid infrastructure. Problem: Goldman Sachs analysts lack of turbine knowledge Solution: Employ a specialist energy management firm Problem: Each windfarm may have hundreds of turbines each turbine produces data at 10s to 10m intervals on dozens of properties Large data problem [10K turbine years!] Solution 1: Solution 2: Hire engineers to stare at data Get a team of mathematicians to develop a data-driven model to analyse the data.
19 Ideal power output to windspeed diagram Example data set Learn: to categorise conditions: derating, icing, Goal: Automatic analysis of data streams for maintenance, performance analysis, planning
20 <latexit sha1_base64="6st+2uni49vtkzhpfrx8lltbni8=">aaab6hicbvbns8naej3ur1q/qh69lbbbu0le0gpri8cw7ae0owy2k3btzhn3n0ij/qvepcji1z/kzx/jts1bwx8mpn6bywzekaiujet+o4w19y3nrej2awd3b/+gfhju0ngqgdzzlglvcahgwsu2dtcco4lcgguc28h4dua3n1bphst7m0nqj+hq8pazaqzueoyxk27vnyosei8nfchr75e/eooyprfkwwtvuuu5ifezqgxnaqelxqoxowxmh9i1vniitz/nd52sm6smsbgrw9kqufp7iqor1pmosj0rnso97m3e/7xuasjrp+mysq1ktlgupokymmy+jgoukbkxsyqyxe2thi2ooszybeo2bg/55vxsuqh6btvrxfzqn3kcrtibuzghd66gbndqhyywqhigv3hzhpwx5935wlqwnhzmgp7a+fwb252m9q==</latexit> <latexit sha1_base64="do2tvhsgwpjem6ngq7vwez4dw5a=">aaacj3icbzdps8mwfmft+wvwx1wpxojd8dtaiuwpytclxwludtzs0jtbwtkkjkkwyv4bl/4rxgqv0ap/idlwudcfhhzzee/x8r5ryqjsrvtplzawv1bxyuv2xubw9o6zu9dwipoytlbgqnyipaijnlq01yx0uklqejfyfw2vjvm7eyivffxwj1isjkjpay9ipa0knytmmpsygey9hufqrz7vfd8uycz6m+jwfqdqayil7ivoxvztgivck0qffnemnrc/fjhlcneyiaw6npvqiedsu8zi2pyzrvkeh6hpukzylbav5nm9x/dikbj2hdshazilvztylcg1sijtmsa9upo5cfwv18107ztiku8zttiedepldgobj6bbmeqcnrszgbck5q8qd5bewbtrbwocn7/yomjxqp5b9w5oko3lwo4yoach4bh4oa4a4bo0qqtg8acewct4sx6tz+vd+pivlqyizx/8cevrg3jrofk=</latexit> Model Models may be artificial or not, but the distinction is often artificial! Ex: convolutional neural network
21 <latexit sha1_base64="rzlbq3g0+drgaqrply9ijnsmmqu=">aaacc3icbvc9tsmwght4leuvwmhitujqlypbsdbwsdawibarmihyxke16isu7sbvaxcwxowfayryeqe23ganzqatj1k63x2fpt8fnfgplovbwfvf2nzalu2ud/f2dw7no+ootfkbsrsnlbfogcrhncztrrujdhcerqej3wb0k/vdbyiktej7nehei9agpihfsgnjnysup7xx1kldl4ueqwtmgjmd5wl1rdq47ptvq2hnavejxzaqkndyzs+3n+a0irhcdenzsy2uvawjrtejs7kbssirhqeb6wkao4hil5tnmcezrfrhmaj9ygxn6u+ndevstqjat0zidewyl4v/eb1uhvdermoekhljxaewzvbhzoubfsoivmyicckc6r9cpeqcyaxrk+ss7oxiq6rz3rcthn13uw1ef3wuwcmogbqwwsvoglvqam2awsn4bq/gzxgyxox342mxumyuoyfgd4zph56emxs=</latexit> <latexit sha1_base64="qmjny7d3nwzcz0iuuxl57xkwnik=">aaaccnicbvdlssnafj34rpuvdelmtah1uxirdfmounelfewdmhgm00k7ddkjmxoxxk7d+ctuxcji1i9w5984abpq1gmxdufcy733+dgjulnwtze3v7c4tfxyka6urw9smlvbtrklapmgjlgk2j6shffogooqrtqxicj0gwn5g/pmb90riwner9uwjm6iepwgfcoljc/cc2jabj/chsiqdorus2nvht1cwky+96g2plnkvawx4cyxc1icoeqe+ev0i5yehcvmkjqd24qvmykhkgzkvhqsswkeb6hhoppyfblppunxrvbak10yreixv3cs/p5iusjlmpr1z4hux057mfif10lucoqmlmejihxpfgujgyqcws6wswxbig01qvhqfsvefsqqvjq9og7bnn55ljspkrzvsa+os7wzpi4c2ax7oaxscajq4aluqqng8aiewst4m56mf+pd+ji0zhn5za74a+pzb7p9mp8=</latexit> DATA PARAMETERS prior model LIKELIHOOD Bayes Theorem minimise U(q) to determine optimal parameters I m not really me Data Scientist Thomas Bayes, U of Edinburgh, Class of 1721
22 Wind Turbine Benchmarking With enough data, it is straightforward to replace the engineer with a convolutional neural network for detecting a specific condition, e.g. icing Problems: 1. Mixed data sets 2. Lack of transferability Unsupervised learning to learn classes from data Generalisation error Generative networks 3. Multimodality all require sampling!
23 Multimodality in NNs With Zofia Trstanova and Frederik Heber (Turing) P(red) isolated local min 9 parameter model, multimodal loss landscape with pretty severe trapping in upper well.
24 Sampling Metastable Systems Ways forward: 1. EQN. New types of diffusions that more efficiently explore the complex geometry; in particular nonreversible diffusions. w. J. Weare & C. Matthews, Chicago. 2. ISST. Use of temperature and other collective variables to enhance and guide sampling. w. A. Martinsson, J. Lu (Duke) & E. Vanden-Eijnden (NYU). 3. DM. Data-analytic tools based on diffusion maps to perform unsupervised manifold learning from simulated data in order to enhance exploration. w. Z. Trstanova, T. Lelievre (Paris) and R. Banisch (Berlin)
25
26
27
28 (building on work of Clementi, Kevrekides )
29
30 Thermodynamic Analytics Toolkit (TATi) Frederik Heber (Rutherford-Turing Fellow) Benedict Leimkuhler Zofia Trstanova (EPSRC PDRA - Edinburgh) Vision: Exploration tool for loss landscape in ANNs Sampling methods in TensorFlow purposes Develop understanding of NN properties Assist in analysis/redesign of NNs
31 The Edinburgh Environment Maxwell Institute Union of mathematicians at Edinburgh and Heriot-Watt Universities. Currently unifying graduate programmes under the banner Maxwell Institute Graduate School. International Centre for the Mathematical Sciences Runs workshops, supports small group research and coordinates knowledge exchange and public events. Centres for Doctoral Training e.g. MIGSAA Main mechanism for funding PhD students and coordinating training across the Maxwell Institute 3 proposed in 2018: MAC-MIGS (Modelling, analysis and computation), GLAMS (Glasgow-Maxwell School in algebraic methods), and Data Science & AI (a joint venture with Informatics)
32 Thomas Bayes Centre
33 Why Edinburgh? amazing skyline: access to the highlands fringe festival folk music still part of the EU but.
34 The real reason Scottish Boletus Harvest (Porcini/Ceps)
35 Thank you for listening!
Thermostatic Controls for Noisy Gradient Systems and Applications to Machine Learning
Thermostatic Controls for Noisy Gradient Systems and Applications to Machine Learning Ben Leimkuhler University of Edinburgh Joint work with C. Matthews (Chicago), G. Stoltz (ENPC-Paris), M. Tretyakov
From Physical Modelling to Big Data Analytics: Examples and Challenges
From Physical Modelling to Big Data Analytics: Examples and Challenges Ben Leimkuhler University of Edinburgh EPSRC/NSF ERC ATI Summit on Big Data in the Physical Sciences Outline Part I: Challenges Ideas
Gaussian Process Approximations of Stochastic Differential Equations
Gaussian Process Approximations of Stochastic Differential Equations Cédric Archambeau Centre for Computational Statistics and Machine Learning University College London c.archambeau@cs.ucl.ac.uk CSML
Effective dynamics for the (overdamped) Langevin equation
Effective dynamics for the (overdamped) Langevin equation Frédéric Legoll ENPC and INRIA joint work with T. Lelièvre (ENPC and INRIA) Enumath conference, MS Numerical methods for molecular dynamics EnuMath
Monte Carlo methods for sampling-based Stochastic Optimization
Monte Carlo methods for sampling-based Stochastic Optimization Gersende FORT LTCI CNRS & Telecom ParisTech Paris, France Joint works with B. Jourdain, T. Lelièvre, G. Stoltz from ENPC and E. Kuhn from
Heat bath models for nonlinear partial differential equations Controlling statistical properties in extended dynamical systems
Heat bath models for nonlinear partial differential equations Controlling statistical properties in extended dynamical systems Ben Leimkuhler University of Edinburgh Funding: NAIS (EPSRC) & NWO MIGSEE
Manifold Monte Carlo Methods
Manifold Monte Carlo Methods Mark Girolami Department of Statistical Science University College London Joint work with Ben Calderhead Research Section Ordinary Meeting The Royal Statistical Society October
Introduction to Machine Learning CMU-10701
Introduction to Machine Learning CMU-10701 Markov Chain Monte Carlo Methods Barnabás Póczos & Aarti Singh Contents Markov Chain Monte Carlo Methods Goal & Motivation Sampling Rejection Importance Markov
Random Vibrations & Failure Analysis Sayan Gupta Indian Institute of Technology Madras
Random Vibrations & Failure Analysis Sayan Gupta Indian Institute of Technology Madras Lecture 1: Introduction Course Objectives: The focus of this course is on gaining understanding on how to make an
Gaussian processes for inference in stochastic differential equations
Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017
Stochastic differential equation models in biology Susanne Ditlevsen
Stochastic differential equation models in biology Susanne Ditlevsen Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential
Development of Stochastic Artificial Neural Networks for Hydrological Prediction
Development of Stochastic Artificial Neural Networks for Hydrological Prediction G. B. Kingston, M. F. Lambert and H. R. Maier Centre for Applied Modelling in Water Engineering, School of Civil and Environmental
Numerical methods in molecular dynamics and multiscale problems
Numerical methods in molecular dynamics and multiscale problems Two examples T. Lelièvre CERMICS - Ecole des Ponts ParisTech & MicMac project-team - INRIA Horizon Maths December 2012 Introduction The aim
Introduction to Computational Stochastic Differential Equations
Introduction to Computational Stochastic Differential Equations Gabriel J. Lord Catherine E. Powell Tony Shardlow Preface Techniques for solving many of the differential equations traditionally used by
A variational radial basis function approximation for diffusion processes
A variational radial basis function approximation for diffusion processes Michail D. Vrettas, Dan Cornford and Yuan Shen Aston University - Neural Computing Research Group Aston Triangle, Birmingham B4
LECTURE 15 Markov chain Monte Carlo
LECTURE 15 Markov chain Monte Carlo There are many settings when posterior computation is a challenge in that one does not have a closed form expression for the posterior distribution. Markov chain Monte
Introduction to Computer Simulations of Soft Matter Methodologies and Applications Boulder July, 19-20, 2012
Introduction to Computer Simulations of Soft Matter Methodologies and Applications Boulder July, 19-20, 2012 K. Kremer Max Planck Institute for Polymer Research, Mainz Overview Simulations, general considerations
Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks
Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks Panos Stinis (joint work with T. Hagge, A.M. Tartakovsky and E. Yeung) Pacific Northwest National Laboratory
Markov Chains and MCMC
Markov Chains and MCMC Markov chains Let S = {1, 2,..., N} be a finite set consisting of N states. A Markov chain Y 0, Y 1, Y 2,... is a sequence of random variables, with Y t S for all points in time
Applied Mathematics Methods Stream
Applied Mathematics Methods Stream Irene Kyza Discipline of Mathematics School of Science and Engineering University of Dundee September 2016 Overview: what are methods (for)? Most mathematical models
M4A42 APPLIED STOCHASTIC PROCESSES
M4A42 APPLIED STOCHASTIC PROCESSES G.A. Pavliotis Department of Mathematics Imperial College London, UK LECTURE 1 12/10/2009 Lectures: Mondays 09:00-11:00, Huxley 139, Tuesdays 09:00-10:00, Huxley 144.
Kurume University Faculty of Economics Monograph Collection 18. Theoretical Advances and Applications in. Operations Research. Kyushu University Press
Kurume University Faculty of Economics Monograph Collection 18 Theoretical Advances and Applications in Operations Research Managing Editor Kangrong Tan General Advisor Joe Gani Kyushu University Press
Computing ergodic limits for SDEs
Computing ergodic limits for SDEs M.V. Tretyakov School of Mathematical Sciences, University of Nottingham, UK Talk at the workshop Stochastic numerical algorithms, multiscale modelling and high-dimensional
Handbook of Stochastic Methods
C. W. Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Third Edition With 30 Figures Springer Contents 1. A Historical Introduction 1 1.1 Motivation I 1.2 Some Historical
Control Theory in Physics and other Fields of Science
Michael Schulz Control Theory in Physics and other Fields of Science Concepts, Tools, and Applications With 46 Figures Sprin ger 1 Introduction 1 1.1 The Aim of Control Theory 1 1.2 Dynamic State of Classical
Deep Relaxation: PDEs for optimizing Deep Neural Networks
Deep Relaxation: PDEs for optimizing Deep Neural Networks IPAM Mean Field Games August 30, 2017 Adam Oberman (McGill) Thanks to the Simons Foundation (grant 395980) and the hospitality of the UCLA math
Riemann Manifold Methods in Bayesian Statistics
Ricardo Ehlers ehlers@icmc.usp.br Applied Maths and Stats University of São Paulo, Brazil Working Group in Statistical Learning University College Dublin September 2015 Bayesian inference is based on Bayes
Monte Carlo Methods. Handbook of. University ofqueensland. Thomas Taimre. Zdravko I. Botev. Dirk P. Kroese. Universite de Montreal
Handbook of Monte Carlo Methods Dirk P. Kroese University ofqueensland Thomas Taimre University ofqueensland Zdravko I. Botev Universite de Montreal A JOHN WILEY & SONS, INC., PUBLICATION Preface Acknowledgments
Prerequisite: STATS 7 or STATS 8 or AP90 or (STATS 120A and STATS 120B and STATS 120C). AP90 with a minimum score of 3
University of California, Irvine 2017-2018 1 Statistics (STATS) Courses STATS 5. Seminar in Data Science. 1 Unit. An introduction to the field of Data Science; intended for entering freshman and transfers.
The parallel replica method for Markov chains
The parallel replica method for Markov chains David Aristoff (joint work with T Lelièvre and G Simpson) Colorado State University March 2015 D Aristoff (Colorado State University) March 2015 1 / 29 Introduction
Markov Chain Monte Carlo Method
Markov Chain Monte Carlo Method Macoto Kikuchi Cybermedia Center, Osaka University 6th July 2017 Thermal Simulations 1 Why temperature 2 Statistical mechanics in a nutshell 3 Temperature in computers 4
Recent Advances in Bayesian Inference Techniques
Recent Advances in Bayesian Inference Techniques Christopher M. Bishop Microsoft Research, Cambridge, U.K. research.microsoft.com/~cmbishop SIAM Conference on Data Mining, April 2004 Abstract Bayesian
STA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 7 Approximate
A tutorial on numerical transfer operator methods. numerical analysis of dynamical systems
A tutorial on transfer operator methods for numerical analysis of dynamical systems Gary Froyland School of Mathematics and Statistics University of New South Wales, Sydney BIRS Workshop on Uncovering
The Molecular Dynamics Method
The Molecular Dynamics Method Thermal motion of a lipid bilayer Water permeation through channels Selective sugar transport Potential Energy (hyper)surface What is Force? Energy U(x) F = d dx U(x) Conformation
V e h i c l e I C T A r e n a I n n o v a t i o n B a z a a r L i n d h o l m e n S c i e n c e P a r k
I ndustrial Mathematics Applied to M a c hine Learning, B ig Data Analytics and Data Science V e h i c l e I C T A r e n a I n n o v a t i o n B a z a a r L i n d h o l m e n S c i e n c e P a r k 2018-02-
Advanced Molecular Dynamics
Advanced Molecular Dynamics Introduction May 2, 2017 Who am I? I am an associate professor at Theoretical Physics Topics I work on: Algorithms for (parallel) molecular simulations including GPU acceleration
Everyday NMR. Innovation with Integrity. Why infer when you can be sure? NMR
Everyday NMR Why infer when you can be sure? Innovation with Integrity NMR Only NMR gives you definitive answers, on your terms. Over the past half-century, scientists have used nuclear magnetic resonance
Probabilistic Graphical Models Lecture 17: Markov chain Monte Carlo
Probabilistic Graphical Models Lecture 17: Markov chain Monte Carlo Andrew Gordon Wilson www.cs.cmu.edu/~andrewgw Carnegie Mellon University March 18, 2015 1 / 45 Resources and Attribution Image credits,
Computational Physics. J. M. Thijssen
Computational Physics J. M. Thijssen Delft University of Technology CAMBRIDGE UNIVERSITY PRESS Contents Preface xi 1 Introduction 1 1.1 Physics and computational physics 1 1.2 Classical mechanics and statistical
SMSTC: Probability and Statistics
SMSTC: Probability and Statistics Fraser Daly Heriot Watt University October 2018 Fraser Daly (Heriot Watt University) SMSTC: Probability and Statistics October 2018 1 / 28 Outline Probability and Statistics
Second Year Chemistry 2014
Second Year Chemistry 2014 Information Session A/Prof Siggi Schmid Second Year Coordinator Key Contacts Siggi Schmid Second Year Coordinator Room 315, e-mail: siegbert.schmid@sydney.edu.au Suzanne Kania
Probabilistic Graphical Models
10-708 Probabilistic Graphical Models Homework 3 (v1.1.0) Due Apr 14, 7:00 PM Rules: 1. Homework is due on the due date at 7:00 PM. The homework should be submitted via Gradescope. Solution to each problem
CS242: Probabilistic Graphical Models Lecture 7B: Markov Chain Monte Carlo & Gibbs Sampling
CS242: Probabilistic Graphical Models Lecture 7B: Markov Chain Monte Carlo & Gibbs Sampling Professor Erik Sudderth Brown University Computer Science October 27, 2016 Some figures and materials courtesy
Gillespie s Algorithm and its Approximations. Des Higham Department of Mathematics and Statistics University of Strathclyde
Gillespie s Algorithm and its Approximations Des Higham Department of Mathematics and Statistics University of Strathclyde djh@maths.strath.ac.uk The Three Lectures 1 Gillespie s algorithm and its relation
Second Year Chemistry 2012
Second Year Chemistry 2012 Information Session Dr Richard Payne Second Year Coordinator Key Contacts Richard Payne Second Year Coordinator Room 545, e-mail: richard.payne@sydney.edu.au Linda Kristian Assistant
Pattern Recognition and Machine Learning. Bishop Chapter 11: Sampling Methods
Pattern Recognition and Machine Learning Chapter 11: Sampling Methods Elise Arnaud Jakob Verbeek May 22, 2008 Outline of the chapter 11.1 Basic Sampling Algorithms 11.2 Markov Chain Monte Carlo 11.3 Gibbs
From cell biology to Petri nets. Rainer Breitling, Groningen, NL David Gilbert, London, UK Monika Heiner, Cottbus, DE
From cell biology to Petri nets Rainer Breitling, Groningen, NL David Gilbert, London, UK Monika Heiner, Cottbus, DE Biology = Concentrations Breitling / 2 The simplest chemical reaction A B irreversible,
Stat 535 C - Statistical Computing & Monte Carlo Methods. Lecture February Arnaud Doucet
Stat 535 C - Statistical Computing & Monte Carlo Methods Lecture 13-28 February 2006 Arnaud Doucet Email: arnaud@cs.ubc.ca 1 1.1 Outline Limitations of Gibbs sampling. Metropolis-Hastings algorithm. Proof
Titles and Abstracts
Titles and Abstracts Stability of the Nonlinear Filter for Random Expanding Maps Jochen Broecker A ubiquitous problem in science and engineering is to reconstruct the state of a hidden Markov process (the
Introduction to asymptotic techniques for stochastic systems with multiple time-scales
Introduction to asymptotic techniques for stochastic systems with multiple time-scales Eric Vanden-Eijnden Courant Institute Motivating examples Consider the ODE {Ẋ = Y 3 + sin(πt) + cos( 2πt) X() = x
Lecture 7 and 8: Markov Chain Monte Carlo
Lecture 7 and 8: Markov Chain Monte Carlo 4F13: Machine Learning Zoubin Ghahramani and Carl Edward Rasmussen Department of Engineering University of Cambridge http://mlg.eng.cam.ac.uk/teaching/4f13/ Ghahramani
Winter 2019 Math 106 Topics in Applied Mathematics. Lecture 9: Markov Chain Monte Carlo
Winter 2019 Math 106 Topics in Applied Mathematics Data-driven Uncertainty Quantification Yoonsang Lee (yoonsang.lee@dartmouth.edu) Lecture 9: Markov Chain Monte Carlo 9.1 Markov Chain A Markov Chain Monte
Homework 3 due Friday, April 26. A couple of examples where the averaging principle from last time can be done analytically.
Stochastic Averaging Examples Tuesday, April 23, 2013 2:01 PM Homework 3 due Friday, April 26. A couple of examples where the averaging principle from last time can be done analytically. Based on the alternative
Molecular dynamics simulation. CS/CME/BioE/Biophys/BMI 279 Oct. 5 and 10, 2017 Ron Dror
Molecular dynamics simulation CS/CME/BioE/Biophys/BMI 279 Oct. 5 and 10, 2017 Ron Dror 1 Outline Molecular dynamics (MD): The basic idea Equations of motion Key properties of MD simulations Sample applications
CONTENTS. Preface List of Symbols and Notation
CONTENTS Preface List of Symbols and Notation xi xv 1 Introduction and Review 1 1.1 Deterministic and Stochastic Models 1 1.2 What is a Stochastic Process? 5 1.3 Monte Carlo Simulation 10 1.4 Conditional
The Kramers problem and first passage times.
Chapter 8 The Kramers problem and first passage times. The Kramers problem is to find the rate at which a Brownian particle escapes from a potential well over a potential barrier. One method of attack
DUBLIN CITY UNIVERSITY
DUBLIN CITY UNIVERSITY SAMPLE EXAMINATIONS 2017/2018 MODULE: QUALIFICATIONS: Simulation for Finance MS455 B.Sc. Actuarial Mathematics ACM B.Sc. Financial Mathematics FIM YEAR OF STUDY: 4 EXAMINERS: Mr
Numerical Solutions of ODEs by Gaussian (Kalman) Filtering
Numerical Solutions of ODEs by Gaussian (Kalman) Filtering Hans Kersting joint work with Michael Schober, Philipp Hennig, Tim Sullivan and Han C. Lie SIAM CSE, Atlanta March 1, 2017 Emmy Noether Group
THE UNIVERSITY OF MANCHESTER PARTICULARS OF APPOINTMENT FACULTY OF ENGINEERING AND PHYSICAL SCIENCES SCHOOL OF PHYSICS AND ASTRONOMY
THE UNIVERSITY OF MANCHESTER PARTICULARS OF APPOINTMENT FACULTY OF ENGINEERING AND PHYSICAL SCIENCES SCHOOL OF PHYSICS AND ASTRONOMY JODRELL BANK CENTRE FOR ASTROPHYSICS CRYOGENICS ENGINEER IN NEXT GENERATION
Algorithms for Ensemble Control. B Leimkuhler University of Edinburgh
Algorithms for Ensemble Control B Leimkuhler University of Edinburgh Model Reduction Across Disciplines Leicester 2014 (Stochastic) Ensemble Control extended ODE/SDE system design target ensemble SDE invariant
Unravelling the nuclear pore complex
Unravelling the nuclear pore complex Ian Ford + Dino Osmanović, Tony Harker, Aizhan Bestembayeva, Bart Hoogenboom (UCL P&A/LCN) Ariberto Fassati (UCL Virology) Armin Kramer, Ivan Leshkovich (Univ Münster)
Random Walks A&T and F&S 3.1.2
Random Walks A&T 110-123 and F&S 3.1.2 As we explained last time, it is very difficult to sample directly a general probability distribution. - If we sample from another distribution, the overlap will
Basics of Statistical Mechanics
Basics of Statistical Mechanics Review of ensembles Microcanonical, canonical, Maxwell-Boltzmann Constant pressure, temperature, volume, Thermodynamic limit Ergodicity (see online notes also) Reading assignment:
2.3 Modeling Interatomic Interactions Pairwise Potentials Many-Body Potentials Studying Biomolecules: The Force
Contents 1 Introduction to Computational Meso-Bio-Nano (MBN) Science and MBN EXPLORER.... 1 1.1 Meso-Bio-Nano Science: A Novel Field of Interdisciplinary Research.... 1 1.1.1 Structure and Dynamics of
Numerical Integration of SDEs: A Short Tutorial
Numerical Integration of SDEs: A Short Tutorial Thomas Schaffter January 19, 010 1 Introduction 1.1 Itô and Stratonovich SDEs 1-dimensional stochastic differentiable equation (SDE) is given by [6, 7] dx
Approximation of Top Lyapunov Exponent of Stochastic Delayed Turning Model Using Fokker-Planck Approach
Approximation of Top Lyapunov Exponent of Stochastic Delayed Turning Model Using Fokker-Planck Approach Henrik T. Sykora, Walter V. Wedig, Daniel Bachrathy and Gabor Stepan Department of Applied Mechanics,
Speeding up Convergence to Equilibrium for Diffusion Processes
Speeding up Convergence to Equilibrium for Diffusion Processes G.A. Pavliotis Department of Mathematics Imperial College London Joint Work with T. Lelievre(CERMICS), F. Nier (CERMICS), M. Ottobre (Warwick),
CSC 2541: Bayesian Methods for Machine Learning
CSC 2541: Bayesian Methods for Machine Learning Radford M. Neal, University of Toronto, 2011 Lecture 3 More Markov Chain Monte Carlo Methods The Metropolis algorithm isn t the only way to do MCMC. We ll
Introduction to multiscale modeling and simulation. Explicit methods for ODEs : forward Euler. y n+1 = y n + tf(y n ) dy dt = f(y), y(0) = y 0
Introduction to multiscale modeling and simulation Lecture 5 Numerical methods for ODEs, SDEs and PDEs The need for multiscale methods Two generic frameworks for multiscale computation Explicit methods
Pattern Recognition and Machine Learning
Christopher M. Bishop Pattern Recognition and Machine Learning ÖSpri inger Contents Preface Mathematical notation Contents vii xi xiii 1 Introduction 1 1.1 Example: Polynomial Curve Fitting 4 1.2 Probability
Handbook of Stochastic Methods
Springer Series in Synergetics 13 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences von Crispin W Gardiner Neuausgabe Handbook of Stochastic Methods Gardiner schnell und portofrei
Stat 516, Homework 1
Stat 516, Homework 1 Due date: October 7 1. Consider an urn with n distinct balls numbered 1,..., n. We sample balls from the urn with replacement. Let N be the number of draws until we encounter a ball
What Can Physics Say About Life Itself?
What Can Physics Say About Life Itself? Science at the Interface of Physics and Biology Michael Manhart Department of Physics and Astronomy BioMaPS Institute for Quantitative Biology Source: UIUC, Wikimedia
Syllabus BINF Computational Biology Core Course
Course Description Syllabus BINF 701-702 Computational Biology Core Course BINF 701/702 is the Computational Biology core course developed at the KU Center for Computational Biology. The course is designed
Systematic Closure Approximations for Multiscale Simulations
Systematic Closure Approximations for Multiscale Simulations Qiang Du Department of Mathematics/Materials Sciences Penn State University http://www.math.psu.edu/qdu Joint work with C. Liu, Y. Hyon and
Deep learning / Ian Goodfellow, Yoshua Bengio and Aaron Courville. - Cambridge, MA ; London, Spis treści
Deep learning / Ian Goodfellow, Yoshua Bengio and Aaron Courville. - Cambridge, MA ; London, 2017 Spis treści Website Acknowledgments Notation xiii xv xix 1 Introduction 1 1.1 Who Should Read This Book?
Monte Carlo in Bayesian Statistics
Monte Carlo in Bayesian Statistics Matthew Thomas SAMBa - University of Bath m.l.thomas@bath.ac.uk December 4, 2014 Matthew Thomas (SAMBa) Monte Carlo in Bayesian Statistics December 4, 2014 1 / 16 Overview
Different approaches to model wind speed based on stochastic differential equations
1 Universidad de Castilla La Mancha Different approaches to model wind speed based on stochastic differential equations Rafael Zárate-Miñano Escuela de Ingeniería Minera e Industrial de Almadén Universidad
A Degree In Chemistry
A Degree In Chemistry Why chemistry? It is the central science Chemists contribute a lot to our quality of life New drugs Detecting disease Nanotechnology Batteries New materials Food production CHEMISTRY
MSc Dissertation topics:
.... MSc Dissertation topics: Omar Lakkis Mathematics University of Sussex Brighton, England November 6, 2013 Office Location: Pevensey 3 5C2 Office hours: Autumn: Tue & Fri 11:30 12:30; Spring: TBA. O
Chapter 2. Brownian dynamics. 2.1 Introduction
Chapter 2 Brownian dynamics 2.1 Introduction The equations of physics are deterministic rather than stochastic in nature. Stochastic differential equations are used to approximate reality. They are introduced
Molecular Dynamics Lecture 3
Molecular Dynamics Lecture 3 Ben Leimkuhler the problem of the timestep in MD constraints - SHAKE/RATTLE multiple timestepping stochastic methods for holonomic constraints stochastic multiple timestepping
J. Sadeghi E. Patelli M. de Angelis
J. Sadeghi E. Patelli Institute for Risk and, Department of Engineering, University of Liverpool, United Kingdom 8th International Workshop on Reliable Computing, Computing with Confidence University of
Density estimation. Computing, and avoiding, partition functions. Iain Murray
Density estimation Computing, and avoiding, partition functions Roadmap: Motivation: density estimation Understanding annealing/tempering NADE Iain Murray School of Informatics, University of Edinburgh
A Statistical Input Pruning Method for Artificial Neural Networks Used in Environmental Modelling
A Statistical Input Pruning Method for Artificial Neural Networks Used in Environmental Modelling G. B. Kingston, H. R. Maier and M. F. Lambert Centre for Applied Modelling in Water Engineering, School
Lecture 12: Detailed balance and Eigenfunction methods
Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),
Convex Optimization CMU-10725
Convex Optimization CMU-10725 Simulated Annealing Barnabás Póczos & Ryan Tibshirani Andrey Markov Markov Chains 2 Markov Chains Markov chain: Homogen Markov chain: 3 Markov Chains Assume that the state
Recommended Courses by ECE Topic Area Graduate Students
Recommended s by ECE Topic Area Graduate Students The course recommendations below do not represent full plans of study. The courses listed under each heading represent appropriate courses to take if you
STOCHASTIC PROCESSES IN PHYSICS AND CHEMISTRY
STOCHASTIC PROCESSES IN PHYSICS AND CHEMISTRY Third edition N.G. VAN KAMPEN Institute for Theoretical Physics of the University at Utrecht ELSEVIER Amsterdam Boston Heidelberg London New York Oxford Paris
Lecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
Introduction to Machine Learning CMU-10701
Introduction to Machine Learning CMU-10701 Markov Chain Monte Carlo Methods Barnabás Póczos Contents Markov Chain Monte Carlo Methods Sampling Rejection Importance Hastings-Metropolis Gibbs Markov Chains
Path Integrals. Andreas Wipf Theoretisch-Physikalisches-Institut Friedrich-Schiller-Universität, Max Wien Platz Jena
Path Integrals Andreas Wipf Theoretisch-Physikalisches-Institut Friedrich-Schiller-Universität, Max Wien Platz 1 07743 Jena 5. Auflage WS 2008/09 1. Auflage, SS 1991 (ETH-Zürich) I ask readers to report
Sampling-based probabilistic inference through neural and synaptic dynamics
Sampling-based probabilistic inference through neural and synaptic dynamics Wolfgang Maass for Robert Legenstein Institute for Theoretical Computer Science Graz University of Technology, Austria Institute
A mathematical framework for Exact Milestoning
A mathematical framework for Exact Milestoning David Aristoff (joint work with Juan M. Bello-Rivas and Ron Elber) Colorado State University July 2015 D. Aristoff (Colorado State University) July 2015 1
Elementary Applications of Probability Theory
Elementary Applications of Probability Theory With an introduction to stochastic differential equations Second edition Henry C. Tuckwell Senior Research Fellow Stochastic Analysis Group of the Centre for
Afternoon Meeting on Bayesian Computation 2018 University of Reading
Gabriele Abbati 1, Alessra Tosi 2, Seth Flaxman 3, Michael A Osborne 1 1 University of Oxford, 2 Mind Foundry Ltd, 3 Imperial College London Afternoon Meeting on Bayesian Computation 2018 University of
Seminar: Data Assimilation
Seminar: Data Assimilation Jonas Latz, Elisabeth Ullmann Chair of Numerical Mathematics (M2) Technical University of Munich Jonas Latz, Elisabeth Ullmann (TUM) Data Assimilation 1 / 28 Prerequisites Bachelor:
String method for the Cahn-Hilliard dynamics
String method for the Cahn-Hilliard dynamics Tiejun Li School of Mathematical Sciences Peking University tieli@pku.edu.cn Joint work with Wei Zhang and Pingwen Zhang Outline Background Problem Set-up Algorithms
Markov Chain Monte Carlo
1 Motivation 1.1 Bayesian Learning Markov Chain Monte Carlo Yale Chang In Bayesian learning, given data X, we make assumptions on the generative process of X by introducing hidden variables Z: p(z): prior