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Chemistry Mathematics Physics Wolfram Language

Speed up this curve fitting?

Sean Bingus

Posted 5 years ago

Here is a dataset: data = {{0., 0.}, {1.10^-7, 3.3916610^9}, {2.10^-7, 3.8690810^9}, {3.10^-7, 3.6884510^9}, {4.10^-7, 3.4463310^9}, {5.10^-7, 2.8058110^9}, {6.10^-7, 2.4431110^9}, {7.10^-7, 1.8015910^9}, {8.10^-7, 1.4821510^9}, {9.10^-7, 1.2566410^9}, {1.10^-6, 1.0792710^9}, {1.110^-6, 9.3184610^8}, {1.210^-6, 7.9532710^8}, {1.310^-6, 7.0938910^8}, {1.410^-6, 5.8629910^8}, {1.510^-6, 4.8392610^8}, {1.610^-6, 4.7669210^8}, {1.710^-6, 4.9901510^8}, {1.810^-6, 3.9184610^8}, {1.910^-6, 4.1429610^8}, {2.10^-6, 3.8001610^8}, {2.110^-6, 3.9194310^8}, {2.210^-6, 4.0877810^8}, {2.310^-6, 4.0333410^8}, {2.410^-6, 3.8926810^8}, {2.510^-6, 3.8327310^8}, {2.610^-6, 4.0407610^8}, {2.710^-6, 4.0726210^8}, {2.810^-6, 4.8271510^8}, {2.910^-6, 4.6942910^8}, {3.10^-6, 4.7203510^8}, {3.110^-6, 4.6575510^8}, {3.210^-6, 5.3988110^8}, {3.310^-6, 5.162210^8}, {3.410^-6, 4.9653310^8}, {3.510^-6, 5.9333210^8}, {3.610^-6, 5.9804610^8}, {3.710^-6, 6.0224710^8}, {3.810^-6, 6.0182510^8}, {3.910^-6, 6.1047510^8}, {4.10^-6, 5.6312610^8}, {4.110^-6, 6.3777310^8}, {4.210^-6, 6.8110710^8}, {4.310^-6, 7.1657910^8}, {4.410^-6, 6.5039110^8}, {4.510^-6, 7.5605910^8}, {4.610^-6, 7.1689510^8}, {4.710^-6, 6.8283310^8}, {4.810^-6, 7.8969810^8}, {4.910^-6, 7.0005210^8}, {5.10^-6, 6.7852410^8}, {5.110^-6, 7.2838910^8}, {5.210^-6, 8.2253610^8}, {5.310^-6, 7.9163710^8}, {5.410^-6, 7.3536910^8}, {5.510^-6, 7.3263110^8}, {5.610^-6, 8.6758110^8}, {5.710^-6, 8.2345710^8}, {5.810^-6, 7.7928810^8}, {5.910^-6, 8.9514210^8}, {6.10^-6, 8.404710^8}, {6.110^-6, 8.139510^8}, {6.210^-6, 8.3974310^8}, {6.310^-6, 8.2192310^8}, {6.410^-6, 8.0507210^8}, {6.510^-6, 8.1871910^8}, {6.610^-6, 9.2368810^8}, {6.710^-6, 9.4336710^8}, {6.810^-6, 8.7708610^8}, {6.910^-6, 9.1496710^8}, {7.10^-6, 8.2894110^8}, {7.110^-6, 9.0782710^8}, {7.210^-6, 8.3151510^8}, {7.310^-6, 9.0956110^8}, {7.410^-6, 9.5578510^8}, {7.510^-6, 9.8529510^8}, {7.610^-6, 9.3368310^8}, {7.710^-6, 9.6890710^8}, {7.810^-6, 9.6154310^8}, {7.910^-6, 9.0401610^8}, {8.10^-6, 8.7189410^8}, {8.110^-6, 9.4033110^8}, {8.210^-6, 1.0492310^9}, {8.310^-6, 8.6498810^8}, {8.410^-6, 1.0548110^9}, {8.510^-6, 9.8546410^8}, {8.610^-6, 1.0518910^9}, {8.710^-6, 1.0035910^9}, {8.810^-6, 9.7272110^8}, {8.910^-6, 9.0402510^8}, {9.10^-6, 9.0835210^8}, {9.110^-6, 1.0410910^9}, {9.210^-6, 1.0311610^9}, {9.310^-6, 1.0255510^9}, {9.410^-6, 9.5093310^8}, {9.510^-6, 1.0108810^9}, {9.610^-6, 9.8703210^8}, {9.710^-6, 1.0077210^9}, {9.810^-6, 1.0881810^9}, {9.910^-6, 1.0604810^9}, {0.00001, 9.9996710^8}, {0.0000101, 9.1828910^8}, {0.0000102, 1.0364410^9}, {0.0000103, 9.5654710^8}, {0.0000104, 1.0022810^9}, {0.0000105, 9.804710^8}, {0.0000106, 1.0117310^9}, {0.0000107, 1.0231510^9}, {0.0000108, 1.0185810^9}, {0.0000109, 1.0517410^9}, {0.000011, 1.0414210^9}, {0.0000111, 9.0013510^8}, {0.0000112, 9.8081710^8}, {0.0000113, 1.0735210^9}, {0.0000114, 1.0538410^9}, {0.0000115, 9.247610^8}, {0.0000116, 8.9959610^8}, {0.0000117, 1.0515210^9}, {0.0000118, 9.9339710^8}, {0.0000119, 1.0743810^9}, {0.000012, 9.6050910^8}, {0.0000121, 1.0068910^9}, {0.0000122, 8.8617710^8}, {0.0000123, 9.4226410^8}, {0.0000124, 1.0719210^9}, {0.0000125, 8.9979410^8}, {0.0000126, 1.071310^9}, {0.0000127, 1.0298110^9}, {0.0000128, 1.0515410^9}, {0.0000129, 1.0309110^9}, {0.000013, 1.0363310^9}, {0.0000131, 9.34410^8}, {0.0000132, 9.7813710^8}, {0.0000133, 1.0235710^9}, {0.0000134, 8.9167910^8}, {0.0000135, 1.0230310^9}, {0.0000136, 9.1548210^8}, {0.0000137, 9.9534810^8}, {0.0000138, 9.7477810^8}, {0.0000139, 9.3804810^8}, {0.000014, 9.0471810^8}, {0.0000141, 1.040510^9}, {0.0000142, 8.8892410^8}, {0.0000143, 1.001210^9}, {0.0000144, 9.3317710^8}, {0.0000145, 9.434110^8}, {0.0000146, 9.7110510^8}, {0.0000147, 9.474610^8}, {0.0000148, 8.4360510^8}, {0.0000149, 9.0643610^8}, {0.000015, 9.6933810^8}, {0.0000151, 9.8495510^8}, {0.0000152, 8.5972510^8}, {0.0000153, 1.0114310^9}, {0.0000154, 9.725910^8}, {0.0000155, 9.4917410^8}, {0.0000156, 8.7511310^8}, {0.0000157, 9.1533610^8}, {0.0000158, 8.3455710^8}, {0.0000159, 8.1997110^8}, {0.000016, 9.3040710^8}, {0.0000161, 9.6715210^8}, {0.0000162, 8.9488210^8}, {0.0000163, 8.049810^8}, {0.0000164, 9.2344110^8}, {0.0000165, 9.4501310^8}, {0.0000166, 8.7210910^8}, {0.0000167, 8.6161410^8}, {0.0000168, 8.0960210^8}, {0.0000169, 9.5701910^8}, {0.000017, 8.8369910^8}, {0.0000171, 9.0085410^8}, {0.0000172, 8.1407610^8}, {0.0000173, 9.1616410^8}, {0.0000174, 9.0774810^8}, {0.0000175, 7.770210^8}, {0.0000176, 8.628610^8}, {0.0000177, 7.6843510^8}, {0.0000178, 9.0076510^8}, {0.0000179, 8.5646610^8}, {0.000018, 8.1808510^8}, {0.0000181, 8.2170410^8}, {0.0000182, 7.9181110^8}, {0.0000183, 8.064810^8}, {0.0000184, 7.551410^8}, {0.0000185, 8.2689310^8}, {0.0000186, 7.9568510^8}, {0.0000187, 7.626710^8}, {0.0000188, 8.4151110^8}, {0.0000189, 7.7311610^8}, {0.000019, 8.8589410^8}, {0.0000191, 8.2347510^8}, {0.0000192, 7.403410^8}, {0.0000193, 8.3964810^8}, {0.0000194, 7.3931510^8}, {0.0000195, 7.9287210^8}, {0.0000196, 7.7060510^8}, {0.0000197, 7.3937110^8}, {0.0000198, 7.7049210^8}, {0.0000199, 7.397610^8}, {0.00002, 7.8313410^8}, {0.0000201, 8.0903310^8}, {0.0000202, 7.3880110^8}, {0.0000203, 7.2187210^8}, {0.0000204, 8.400910^8}, {0.0000205, 7.9886810^8}, {0.0000206, 7.4626110^8}, {0.0000207, 8.0632510^8}, {0.0000208, 7.8941610^8}, {0.0000209, 8.0302710^8}, {0.000021, 7.4358110^8}, {0.0000211, 7.2488810^8}, {0.0000212, 7.833310^8}, {0.0000213, 7.4407810^8}, {0.0000214, 7.6524410^8}, {0.0000215, 7.95210^8}, {0.0000216, 7.2825110^8}, {0.0000217, 7.2317110^8}, {0.0000218, 6.8090910^8}, {0.0000219, 7.924610^8}, {0.000022, 6.8074110^8}, {0.0000221, 7.1594410^8}, {0.0000222, 7.2289610^8}, {0.0000223, 6.5363610^8}, {0.0000224, 7.0338110^8}, {0.0000225, 7.6707410^8}, {0.0000226, 6.8675710^8}, {0.0000227, 6.603510^8}, {0.0000228, 6.3665410^8}, {0.0000229, 7.3363910^8}, {0.000023, 7.3158910^8}, {0.0000231, 6.9840910^8}, {0.0000232, 6.4425910^8}, {0.0000233, 6.3548310^8}, {0.0000234, 6.3763410^8}, {0.0000235, 6.4818810^8}, {0.0000236, 7.3159210^8}, {0.0000237, 6.3466410^8}, {0.0000238, 6.7322710^8}, {0.0000239, 7.1032610^8}, {0.000024, 6.0269310^8}, {0.0000241, 6.7191710^8}, {0.0000242, 6.4644410^8}, {0.0000243, 6.3742310^8}, {0.0000244, 6.6304410^8}, {0.0000245, 6.0825410^8}, {0.0000246, 6.3215510^8}, {0.0000247, 6.2208610^8}, {0.0000248, 6.253210^8}, {0.0000249, 6.3654210^8}, {0.000025, 5.7938110^8}, {0.0000251, 6.6616410^8}, {0.0000252, 5.7011710^8}, {0.0000253, 6.1120910^8}, {0.0000254, 6.0527710^8}, {0.0000255, 5.9362410^8}, {0.0000256, 6.1606710^8}, {0.0000257, 5.9901810^8}, {0.0000258, 6.5787210^8}, {0.0000259, 6.3008510^8}, {0.000026, 6.6579910^8}, {0.0000261, 6.5541310^8}, {0.0000262, 5.9211310^8}, {0.0000263, 6.4456410^8}, {0.0000264, 6.1002910^8}, {0.0000265, 5.937110^8}, {0.0000266, 5.6537710^8}, {0.0000267, 5.9088310^8}, {0.0000268, 6.1163710^8}, {0.0000269, 6.2321210^8}, {0.000027, 6.1780510^8}, {0.0000271, 5.3874110^8}, {0.0000272, 5.8361810^8}, {0.0000273, 6.004110^8}, {0.0000274, 5.5633310^8}, {0.0000275, 5.1371310^8}, {0.0000276, 5.1214810^8}, {0.0000277, 5.3096510^8}, {0.0000278, 5.6060510^8}, {0.0000279, 5.227210^8}, {0.000028, 5.5112710^8}, {0.0000281, 5.0416810^8}, {0.0000282, 5.1417810^8}, {0.0000283, 5.9445410^8}, {0.0000284, 5.7872110^8}, {0.0000285, 5.2178810^8}, {0.0000286, 5.524110^8}, {0.0000287, 5.9020810^8}, {0.0000288, 5.4209910^8}, {0.0000289, 5.0966510^8}, {0.000029, 5.4598110^8}, {0.0000291, 5.2699610^8}, {0.0000292, 5.074810^8}, {0.0000293, 5.7147910^8}, {0.0000294, 5.3123210^8}, {0.0000295, 5.5956310^8}, {0.0000296, 5.5697710^8}, {0.0000297, 5.6137210^8}, {0.0000298, 5.044210^8}, {0.0000299, 5.435710^8}, {0.00003, 4.8613210^8}, {0.0000301, 5.45610^8}, {0.0000302, 5.3463110^8}, {0.0000303, 5.493610^8}, {0.0000304, 5.3413610^8}, {0.0000305, 5.2464510^8}, {0.0000306, 4.5458510^8}, {0.0000307, 4.5809110^8}, {0.0000308, 4.8766410^8}, {0.0000309, 4.5528610^8}, {0.000031, 4.4755110^8}, {0.0000311, 4.7243710^8}, {0.0000312, 4.4543510^8}, {0.0000313, 4.9761310^8}, {0.0000314, 4.6567110^8}, {0.0000315, 4.7939110^8}, {0.0000316, 4.4283110^8}, {0.0000317, 4.36810^8}, {0.0000318, 4.6595710^8}, {0.0000319, 4.5019610^8}, {0.000032, 4.3816210^8}, {0.0000321, 4.8776810^8}, {0.0000322, 4.5338110^8}, {0.0000323, 4.5752910^8}, {0.0000324, 4.8679410^8}, {0.0000325, 4.2318610^8}, {0.0000326, 4.6886210^8}, {0.0000327, 4.4475810^8}, {0.0000328, 4.5332710^8}, {0.0000329, 4.1854410^8}, {0.000033, 4.1030910^8}, {0.0000331, 4.5022410^8}, {0.0000332, 4.3754410^8}, {0.0000333, 4.6228210^8}, {0.0000334, 4.2952510^8}, {0.0000335, 3.9701510^8}, {0.0000336, 4.6963110^8}, {0.0000337, 4.689510^8}, {0.0000338, 4.7117610^8}, {0.0000339, 4.6241310^8}, {0.000034, 4.4657910^8}, {0.0000341, 4.3125810^8}, {0.0000342, 3.9724110^8}, {0.0000343, 4.3150410^8}, {0.0000344, 4.2007210^8}, {0.0000345, 4.3569510^8}, {0.0000346, 4.1858710^8}, {0.0000347, 4.2107410^8}, {0.0000348, 4.3261310^8}, {0.0000349, 3.8404210^8}, {0.000035, 3.8643910^8}, {0.0000351, 4.2959910^8}, {0.0000352, 4.1132110^8}, {0.0000353, 4.0248110^8}, {0.0000354, 4.4094310^8}, {0.0000355, 3.8182410^8}, {0.0000356, 4.3779410^8}, {0.0000357, 3.8332610^8}, {0.0000358, 3.7290410^8}, {0.0000359, 3.7361110^8}, {0.000036, 4.081810^8}, {0.0000361, 4.1422410^8}, {0.0000362, 4.0855710^8}, {0.0000363, 3.5736610^8}, {0.0000364, 3.72210^8}, {0.0000365, 3.5147810^8}, {0.0000366, 4.2427310^8}, {0.0000367, 3.9813810^8}, {0.0000368, 3.9887310^8}, {0.0000369, 4.0725910^8}, {0.000037, 3.482810^8}, {0.0000371, 3.5917510^8}, {0.0000372, 4.0144210^8}, {0.0000373, 4.0254510^8}, {0.0000374, 3.9700110^8}, {0.0000375, 3.6969510^8}, {0.0000376, 4.0108710^8}, {0.0000377, 3.4688410^8}, {0.0000378, 3.4066510^8}, {0.0000379, 3.8178210^8}, {0.000038, 3.8526710^8}, {0.0000381, 3.9367710^8}, {0.0000382, 3.416810^8}, {0.0000383, 3.4074810^8}, {0.0000384, 3.2800110^8}, {0.0000385, 3.7376110^8}, {0.0000386, 3.6115810^8}, {0.0000387, 3.8756110^8}, {0.0000388, 3.63310^8}, {0.0000389, 3.436710^8}, {0.000039, 3.2900610^8}, {0.0000391, 3.6326310^8}, {0.0000392, 3.6432110^8}, {0.0000393, 3.1861610^8}, {0.0000394, 3.3788710^8}, {0.0000395, 3.7196510^8}, {0.0000396, 3.6427910^8}, {0.0000397, 3.3865510^8}, {0.0000398, 3.5399810^8}, {0.0000399, 3.6995710^8}, {0.00004, 3.6659910^8}, {0.0000401, 3.4527310^8}, {0.0000402, 3.0592710^8}, {0.0000403, 3.3931510^8}, {0.0000404, 3.2074510^8}, {0.0000405, 3.3991510^8}, {0.0000406, 3.0263510^8}, {0.0000407, 3.250810^8}, {0.0000408, 3.5214510^8}, {0.0000409, 3.4310410^8}, {0.000041, 3.5595910^8}, {0.0000411, 3.0549210^8}, {0.0000412, 2.9411210^8}, {0.0000413, 3.142110^8}, {0.0000414, 3.1645510^8}, {0.0000415, 3.4840210^8}, {0.0000416, 3.4053510^8}, {0.0000417, 3.3844210^8}, {0.0000418, 3.262210^8}, {0.0000419, 3.3434510^8}, {0.000042, 3.1507610^8}, {0.0000421, 3.1872610^8}, {0.0000422, 3.0850410^8}, {0.0000423, 3.0675410^8}, {0.0000424, 3.360210^8}, {0.0000425, 3.357810^8}, {0.0000426, 3.0432210^8}, {0.0000427, 3.0400510^8}, {0.0000428, 2.923910^8}, {0.0000429, 2.7770310^8}, {0.000043, 3.0085510^8}, {0.0000431, 3.0311810^8}, {0.0000432, 2.7317810^8}, {0.0000433, 3.1012510^8}, {0.0000434, 3.2463410^8}, {0.0000435, 2.8550710^8}, {0.0000436, 3.025310^8}, {0.0000437, 2.9468210^8}, {0.0000438, 2.6745910^8}, {0.0000439, 2.7822210^8}, {0.000044, 2.7822810^8}, {0.0000441, 2.6813310^8}, {0.0000442, 2.8089710^8}, {0.0000443, 3.113310^8}, {0.0000444, 2.6748810^8}, {0.0000445, 2.9536910^8}, {0.0000446, 3.087210^8}, {0.0000447, 2.7809910^8}, {0.0000448, 2.8845310^8}, {0.0000449, 2.6408710^8}, {0.000045, 2.5368610^8}, {0.0000451, 2.7825910^8}, {0.0000452, 2.5554710^8}, {0.0000453, 3.0358810^8}, {0.0000454, 2.6550210^8}, {0.0000455, 2.9476810^8}, {0.0000456, 2.8347410^8}, {0.0000457, 2.8112810^8}, {0.0000458, 2.6305510^8}, {0.0000459, 2.571910^8}, {0.000046, 2.8638310^8}, {0.0000461, 2.511410^8}, {0.0000462, 2.6870510^8}, {0.0000463, 2.4760610^8}, {0.0000464, 2.6590710^8}, {0.0000465, 2.4273710^8}, {0.0000466, 2.7510910^8}, {0.0000467, 2.8719510^8}, {0.0000468, 2.7699410^8}, {0.0000469, 2.4129210^8}, {0.000047, 2.5880210^8}, {0.0000471, 2.3739610^8}, {0.0000472, 2.5468610^8}, {0.0000473, 2.320310^8}, {0.0000474, 2.355610^8}, {0.0000475, 2.5219110^8}, {0.0000476, 2.7322410^8}, {0.0000477, 2.5272310^8}, {0.0000478, 2.6619410^8}, {0.0000479, 2.3152210^8}, {0.000048, 2.2754310^8}, {0.0000481, 2.7142110^8}, {0.0000482, 2.5785210^8}, {0.0000483, 2.6112510^8}, {0.0000484, 2.3386610^8}, {0.0000485, 2.4218110^8}, {0.0000486, 2.6669510^8}, {0.0000487, 2.6417810^8}, {0.0000488, 2.3522510^8}, {0.0000489, 2.6558810^8}, {0.000049, 2.4889110^8}, {0.0000491, 2.4546810^8}, {0.0000492, 2.2897210^8}, {0.0000493, 2.616610^8}, {0.0000494, 2.3909810^8}, {0.0000495, 2.3978610^8}, {0.0000496, 2.2152110^8}, {0.0000497, 2.1879710^8}, {0.0000498, 2.2024810^8}, {0.0000499, 2.1302910^8}, {0.00005, 2.3802210^8}} I am trying to fit this dataset to a system of ODEs. Here are some constants: ( init initital values ) k1 = 10.10^-10; (* Cs^++Cs\[Rule]Subsuperscript[Cs, 2, +] ) k2 = 2.10^-14; (* Cs^++Subscript[H, 2]\[Rule]CsH^++H ) (k3=1.0000000000000000000010^-7; ( \!\( \SubsuperscriptBox[\(Cs\), \(2\), \(+\)] + e\)\[Rule]Cs+Cs(6s) )) k5 = 1.10^7; ( Cs* \[Rule]Cs(6s)+hv' ) kp = 8.10^-10; (* NOT SURE Cs\[Rule]Cs(6s)+hv ) f = 0.95; (* ) kr1 = 1.10^6; (* Cs(7p)\[Rule]Cs(6s)+hv ) nH2 = 1.510^18; (* ) Here is the ODE and fit Code: tmax = 5.1010^-5; ode = { ( Cs(6s) ) Derivative[1][n0][t] == kr1nhv[t] - k1n0[t]n1[t] + k5n2[t] + k3(nCsH[t] + nCs2[t])ne[t] + kpnhv[t]^2, (* Cs(7p) )Derivative[1][nhv][t] == -(kr1 + 2kpnhv[t])nhv[t], (* edensity ) Derivative[1][ne][t] == -k3(nCs2[t] + nCsH[t])ne[t] + kpfnhv[t]^2, ( Subsuperscript[Cs, 2, +] ) Derivative[1][nCs2][t] == k1n1[t]n0[t] - k3ne[t]nCs2[t], ( Cs^* ) Derivative[1][n2][t] == k3nCs2[t]ne[t] - k5n2[t] + kp(1 - f)nhv[t]^2, (* CsH^+ ) Derivative[1][nCsH][t] == k2n1[t]nH2 - k3nCsH[t]ne[t], ( Cs^+ ) Derivative[1][n1][t] == fkpnhv[t]^2 - (k1n0[t] + k2nH2)n1[t], n0[0] == 4.10^13,( Cs(6s) ) n1[0] == 0., nhv[0] == 4.10^13, ne[0] == 0., nCs2[0] == 0., nCsH[0] == 0., n2[0] == 0. }; paramSOL = ParametricNDSolve(Value)[ ode, {n0, nhv, n1, n2, nCs2, nCsH, ne}, {t, 0., tmax}, {k3}, (Method\[Rule]{NMinimize,Method\[Rule]"DifferentialEvolution"},) Method -> "StiffnessSwitching", AccuracyGoal -> 10 ] {n0Func, n1Func, nhvFunc, neFunc, nCs2Func, nCsHFunc, n2Func} = {n0, n1, nhv, ne, nCs2, nCsH, n2} /. paramSOL model[k3_][t_] := n2Func[k3][t] fit = NonlinearModelFit[ random, {n2Func[k3][t], 0.510.^-8 < k3 < 1.410.^-5}, {k3}, t]; // Timing With this, the calculation is finding k3. I simulated the data and gave it some randomness, and then took away the k3 value to have the computer find it. It takes about 120 seconds, which is fine for 1 parameter, but when I take away a second parameter to have it fit 2 parameters, it takes hours. I would like to be able to solve for 3 parameters in a reasonable amount of time. Parallelization does not seem to help (i mean this is fitting only 1 parameter so that makes sense), and I have tried some different settings, but overall nothing I have done has significantly increased the speed of the optimization procedure. Advice?

Here is a dataset:

data = {{0., 0.}, {1.*10^-7, 3.39166*10^9}, {2.*10^-7, 
      3.86908*10^9}, {3.*10^-7, 3.68845*10^9}, {4.*10^-7, 
      3.44633*10^9}, {5.*10^-7, 2.80581*10^9}, {6.*10^-7, 
      2.44311*10^9}, {7.*10^-7, 1.80159*10^9}, {8.*10^-7, 
      1.48215*10^9}, {9.*10^-7, 1.25664*10^9}, {1.*10^-6, 
      1.07927*10^9}, {1.1*10^-6, 9.31846*10^8}, {1.2*10^-6, 
      7.95327*10^8}, {1.3*10^-6, 7.09389*10^8}, {1.4*10^-6, 
      5.86299*10^8}, {1.5*10^-6, 4.83926*10^8}, {1.6*10^-6, 
      4.76692*10^8}, {1.7*10^-6, 4.99015*10^8}, {1.8*10^-6, 
      3.91846*10^8}, {1.9*10^-6, 4.14296*10^8}, {2.*10^-6, 
      3.80016*10^8}, {2.1*10^-6, 3.91943*10^8}, {2.2*10^-6, 
      4.08778*10^8}, {2.3*10^-6, 4.03334*10^8}, {2.4*10^-6, 
      3.89268*10^8}, {2.5*10^-6, 3.83273*10^8}, {2.6*10^-6, 
      4.04076*10^8}, {2.7*10^-6, 4.07262*10^8}, {2.8*10^-6, 
      4.82715*10^8}, {2.9*10^-6, 4.69429*10^8}, {3.*10^-6, 
      4.72035*10^8}, {3.1*10^-6, 4.65755*10^8}, {3.2*10^-6, 
      5.39881*10^8}, {3.3*10^-6, 5.1622*10^8}, {3.4*10^-6, 
      4.96533*10^8}, {3.5*10^-6, 5.93332*10^8}, {3.6*10^-6, 
      5.98046*10^8}, {3.7*10^-6, 6.02247*10^8}, {3.8*10^-6, 
      6.01825*10^8}, {3.9*10^-6, 6.10475*10^8}, {4.*10^-6, 
      5.63126*10^8}, {4.1*10^-6, 6.37773*10^8}, {4.2*10^-6, 
      6.81107*10^8}, {4.3*10^-6, 7.16579*10^8}, {4.4*10^-6, 
      6.50391*10^8}, {4.5*10^-6, 7.56059*10^8}, {4.6*10^-6, 
      7.16895*10^8}, {4.7*10^-6, 6.82833*10^8}, {4.8*10^-6, 
      7.89698*10^8}, {4.9*10^-6, 7.00052*10^8}, {5.*10^-6, 
      6.78524*10^8}, {5.1*10^-6, 7.28389*10^8}, {5.2*10^-6, 
      8.22536*10^8}, {5.3*10^-6, 7.91637*10^8}, {5.4*10^-6, 
      7.35369*10^8}, {5.5*10^-6, 7.32631*10^8}, {5.6*10^-6, 
      8.67581*10^8}, {5.7*10^-6, 8.23457*10^8}, {5.8*10^-6, 
      7.79288*10^8}, {5.9*10^-6, 8.95142*10^8}, {6.*10^-6, 
      8.4047*10^8}, {6.1*10^-6, 8.1395*10^8}, {6.2*10^-6, 
      8.39743*10^8}, {6.3*10^-6, 8.21923*10^8}, {6.4*10^-6, 
      8.05072*10^8}, {6.5*10^-6, 8.18719*10^8}, {6.6*10^-6, 
      9.23688*10^8}, {6.7*10^-6, 9.43367*10^8}, {6.8*10^-6, 
      8.77086*10^8}, {6.9*10^-6, 9.14967*10^8}, {7.*10^-6, 
      8.28941*10^8}, {7.1*10^-6, 9.07827*10^8}, {7.2*10^-6, 
      8.31515*10^8}, {7.3*10^-6, 9.09561*10^8}, {7.4*10^-6, 
      9.55785*10^8}, {7.5*10^-6, 9.85295*10^8}, {7.6*10^-6, 
      9.33683*10^8}, {7.7*10^-6, 9.68907*10^8}, {7.8*10^-6, 
      9.61543*10^8}, {7.9*10^-6, 9.04016*10^8}, {8.*10^-6, 
      8.71894*10^8}, {8.1*10^-6, 9.40331*10^8}, {8.2*10^-6, 
      1.04923*10^9}, {8.3*10^-6, 8.64988*10^8}, {8.4*10^-6, 
      1.05481*10^9}, {8.5*10^-6, 9.85464*10^8}, {8.6*10^-6, 
      1.05189*10^9}, {8.7*10^-6, 1.00359*10^9}, {8.8*10^-6, 
      9.72721*10^8}, {8.9*10^-6, 9.04025*10^8}, {9.*10^-6, 
      9.08352*10^8}, {9.1*10^-6, 1.04109*10^9}, {9.2*10^-6, 
      1.03116*10^9}, {9.3*10^-6, 1.02555*10^9}, {9.4*10^-6, 
      9.50933*10^8}, {9.5*10^-6, 1.01088*10^9}, {9.6*10^-6, 
      9.87032*10^8}, {9.7*10^-6, 1.00772*10^9}, {9.8*10^-6, 
      1.08818*10^9}, {9.9*10^-6, 1.06048*10^9}, {0.00001, 
      9.99967*10^8}, {0.0000101, 9.18289*10^8}, {0.0000102, 
      1.03644*10^9}, {0.0000103, 9.56547*10^8}, {0.0000104, 
      1.00228*10^9}, {0.0000105, 9.8047*10^8}, {0.0000106, 
      1.01173*10^9}, {0.0000107, 1.02315*10^9}, {0.0000108, 
      1.01858*10^9}, {0.0000109, 1.05174*10^9}, {0.000011, 
      1.04142*10^9}, {0.0000111, 9.00135*10^8}, {0.0000112, 
      9.80817*10^8}, {0.0000113, 1.07352*10^9}, {0.0000114, 
      1.05384*10^9}, {0.0000115, 9.2476*10^8}, {0.0000116, 
      8.99596*10^8}, {0.0000117, 1.05152*10^9}, {0.0000118, 
      9.93397*10^8}, {0.0000119, 1.07438*10^9}, {0.000012, 
      9.60509*10^8}, {0.0000121, 1.00689*10^9}, {0.0000122, 
      8.86177*10^8}, {0.0000123, 9.42264*10^8}, {0.0000124, 
      1.07192*10^9}, {0.0000125, 8.99794*10^8}, {0.0000126, 
      1.0713*10^9}, {0.0000127, 1.02981*10^9}, {0.0000128, 
      1.05154*10^9}, {0.0000129, 1.03091*10^9}, {0.000013, 
      1.03633*10^9}, {0.0000131, 9.344*10^8}, {0.0000132, 
      9.78137*10^8}, {0.0000133, 1.02357*10^9}, {0.0000134, 
      8.91679*10^8}, {0.0000135, 1.02303*10^9}, {0.0000136, 
      9.15482*10^8}, {0.0000137, 9.95348*10^8}, {0.0000138, 
      9.74778*10^8}, {0.0000139, 9.38048*10^8}, {0.000014, 
      9.04718*10^8}, {0.0000141, 1.0405*10^9}, {0.0000142, 
      8.88924*10^8}, {0.0000143, 1.0012*10^9}, {0.0000144, 
      9.33177*10^8}, {0.0000145, 9.4341*10^8}, {0.0000146, 
      9.71105*10^8}, {0.0000147, 9.4746*10^8}, {0.0000148, 
      8.43605*10^8}, {0.0000149, 9.06436*10^8}, {0.000015, 
      9.69338*10^8}, {0.0000151, 9.84955*10^8}, {0.0000152, 
      8.59725*10^8}, {0.0000153, 1.01143*10^9}, {0.0000154, 
      9.7259*10^8}, {0.0000155, 9.49174*10^8}, {0.0000156, 
      8.75113*10^8}, {0.0000157, 9.15336*10^8}, {0.0000158, 
      8.34557*10^8}, {0.0000159, 8.19971*10^8}, {0.000016, 
      9.30407*10^8}, {0.0000161, 9.67152*10^8}, {0.0000162, 
      8.94882*10^8}, {0.0000163, 8.0498*10^8}, {0.0000164, 
      9.23441*10^8}, {0.0000165, 9.45013*10^8}, {0.0000166, 
      8.72109*10^8}, {0.0000167, 8.61614*10^8}, {0.0000168, 
      8.09602*10^8}, {0.0000169, 9.57019*10^8}, {0.000017, 
      8.83699*10^8}, {0.0000171, 9.00854*10^8}, {0.0000172, 
      8.14076*10^8}, {0.0000173, 9.16164*10^8}, {0.0000174, 
      9.07748*10^8}, {0.0000175, 7.7702*10^8}, {0.0000176, 
      8.6286*10^8}, {0.0000177, 7.68435*10^8}, {0.0000178, 
      9.00765*10^8}, {0.0000179, 8.56466*10^8}, {0.000018, 
      8.18085*10^8}, {0.0000181, 8.21704*10^8}, {0.0000182, 
      7.91811*10^8}, {0.0000183, 8.0648*10^8}, {0.0000184, 
      7.5514*10^8}, {0.0000185, 8.26893*10^8}, {0.0000186, 
      7.95685*10^8}, {0.0000187, 7.6267*10^8}, {0.0000188, 
      8.41511*10^8}, {0.0000189, 7.73116*10^8}, {0.000019, 
      8.85894*10^8}, {0.0000191, 8.23475*10^8}, {0.0000192, 
      7.4034*10^8}, {0.0000193, 8.39648*10^8}, {0.0000194, 
      7.39315*10^8}, {0.0000195, 7.92872*10^8}, {0.0000196, 
      7.70605*10^8}, {0.0000197, 7.39371*10^8}, {0.0000198, 
      7.70492*10^8}, {0.0000199, 7.3976*10^8}, {0.00002, 
      7.83134*10^8}, {0.0000201, 8.09033*10^8}, {0.0000202, 
      7.38801*10^8}, {0.0000203, 7.21872*10^8}, {0.0000204, 
      8.4009*10^8}, {0.0000205, 7.98868*10^8}, {0.0000206, 
      7.46261*10^8}, {0.0000207, 8.06325*10^8}, {0.0000208, 
      7.89416*10^8}, {0.0000209, 8.03027*10^8}, {0.000021, 
      7.43581*10^8}, {0.0000211, 7.24888*10^8}, {0.0000212, 
      7.8333*10^8}, {0.0000213, 7.44078*10^8}, {0.0000214, 
      7.65244*10^8}, {0.0000215, 7.952*10^8}, {0.0000216, 
      7.28251*10^8}, {0.0000217, 7.23171*10^8}, {0.0000218, 
      6.80909*10^8}, {0.0000219, 7.9246*10^8}, {0.000022, 
      6.80741*10^8}, {0.0000221, 7.15944*10^8}, {0.0000222, 
      7.22896*10^8}, {0.0000223, 6.53636*10^8}, {0.0000224, 
      7.03381*10^8}, {0.0000225, 7.67074*10^8}, {0.0000226, 
      6.86757*10^8}, {0.0000227, 6.6035*10^8}, {0.0000228, 
      6.36654*10^8}, {0.0000229, 7.33639*10^8}, {0.000023, 
      7.31589*10^8}, {0.0000231, 6.98409*10^8}, {0.0000232, 
      6.44259*10^8}, {0.0000233, 6.35483*10^8}, {0.0000234, 
      6.37634*10^8}, {0.0000235, 6.48188*10^8}, {0.0000236, 
      7.31592*10^8}, {0.0000237, 6.34664*10^8}, {0.0000238, 
      6.73227*10^8}, {0.0000239, 7.10326*10^8}, {0.000024, 
      6.02693*10^8}, {0.0000241, 6.71917*10^8}, {0.0000242, 
      6.46444*10^8}, {0.0000243, 6.37423*10^8}, {0.0000244, 
      6.63044*10^8}, {0.0000245, 6.08254*10^8}, {0.0000246, 
      6.32155*10^8}, {0.0000247, 6.22086*10^8}, {0.0000248, 
      6.2532*10^8}, {0.0000249, 6.36542*10^8}, {0.000025, 
      5.79381*10^8}, {0.0000251, 6.66164*10^8}, {0.0000252, 
      5.70117*10^8}, {0.0000253, 6.11209*10^8}, {0.0000254, 
      6.05277*10^8}, {0.0000255, 5.93624*10^8}, {0.0000256, 
      6.16067*10^8}, {0.0000257, 5.99018*10^8}, {0.0000258, 
      6.57872*10^8}, {0.0000259, 6.30085*10^8}, {0.000026, 
      6.65799*10^8}, {0.0000261, 6.55413*10^8}, {0.0000262, 
      5.92113*10^8}, {0.0000263, 6.44564*10^8}, {0.0000264, 
      6.10029*10^8}, {0.0000265, 5.9371*10^8}, {0.0000266, 
      5.65377*10^8}, {0.0000267, 5.90883*10^8}, {0.0000268, 
      6.11637*10^8}, {0.0000269, 6.23212*10^8}, {0.000027, 
      6.17805*10^8}, {0.0000271, 5.38741*10^8}, {0.0000272, 
      5.83618*10^8}, {0.0000273, 6.0041*10^8}, {0.0000274, 
      5.56333*10^8}, {0.0000275, 5.13713*10^8}, {0.0000276, 
      5.12148*10^8}, {0.0000277, 5.30965*10^8}, {0.0000278, 
      5.60605*10^8}, {0.0000279, 5.2272*10^8}, {0.000028, 
      5.51127*10^8}, {0.0000281, 5.04168*10^8}, {0.0000282, 
      5.14178*10^8}, {0.0000283, 5.94454*10^8}, {0.0000284, 
      5.78721*10^8}, {0.0000285, 5.21788*10^8}, {0.0000286, 
      5.5241*10^8}, {0.0000287, 5.90208*10^8}, {0.0000288, 
      5.42099*10^8}, {0.0000289, 5.09665*10^8}, {0.000029, 
      5.45981*10^8}, {0.0000291, 5.26996*10^8}, {0.0000292, 
      5.0748*10^8}, {0.0000293, 5.71479*10^8}, {0.0000294, 
      5.31232*10^8}, {0.0000295, 5.59563*10^8}, {0.0000296, 
      5.56977*10^8}, {0.0000297, 5.61372*10^8}, {0.0000298, 
      5.0442*10^8}, {0.0000299, 5.4357*10^8}, {0.00003, 
      4.86132*10^8}, {0.0000301, 5.456*10^8}, {0.0000302, 
      5.34631*10^8}, {0.0000303, 5.4936*10^8}, {0.0000304, 
      5.34136*10^8}, {0.0000305, 5.24645*10^8}, {0.0000306, 
      4.54585*10^8}, {0.0000307, 4.58091*10^8}, {0.0000308, 
      4.87664*10^8}, {0.0000309, 4.55286*10^8}, {0.000031, 
      4.47551*10^8}, {0.0000311, 4.72437*10^8}, {0.0000312, 
      4.45435*10^8}, {0.0000313, 4.97613*10^8}, {0.0000314, 
      4.65671*10^8}, {0.0000315, 4.79391*10^8}, {0.0000316, 
      4.42831*10^8}, {0.0000317, 4.368*10^8}, {0.0000318, 
      4.65957*10^8}, {0.0000319, 4.50196*10^8}, {0.000032, 
      4.38162*10^8}, {0.0000321, 4.87768*10^8}, {0.0000322, 
      4.53381*10^8}, {0.0000323, 4.57529*10^8}, {0.0000324, 
      4.86794*10^8}, {0.0000325, 4.23186*10^8}, {0.0000326, 
      4.68862*10^8}, {0.0000327, 4.44758*10^8}, {0.0000328, 
      4.53327*10^8}, {0.0000329, 4.18544*10^8}, {0.000033, 
      4.10309*10^8}, {0.0000331, 4.50224*10^8}, {0.0000332, 
      4.37544*10^8}, {0.0000333, 4.62282*10^8}, {0.0000334, 
      4.29525*10^8}, {0.0000335, 3.97015*10^8}, {0.0000336, 
      4.69631*10^8}, {0.0000337, 4.6895*10^8}, {0.0000338, 
      4.71176*10^8}, {0.0000339, 4.62413*10^8}, {0.000034, 
      4.46579*10^8}, {0.0000341, 4.31258*10^8}, {0.0000342, 
      3.97241*10^8}, {0.0000343, 4.31504*10^8}, {0.0000344, 
      4.20072*10^8}, {0.0000345, 4.35695*10^8}, {0.0000346, 
      4.18587*10^8}, {0.0000347, 4.21074*10^8}, {0.0000348, 
      4.32613*10^8}, {0.0000349, 3.84042*10^8}, {0.000035, 
      3.86439*10^8}, {0.0000351, 4.29599*10^8}, {0.0000352, 
      4.11321*10^8}, {0.0000353, 4.02481*10^8}, {0.0000354, 
      4.40943*10^8}, {0.0000355, 3.81824*10^8}, {0.0000356, 
      4.37794*10^8}, {0.0000357, 3.83326*10^8}, {0.0000358, 
      3.72904*10^8}, {0.0000359, 3.73611*10^8}, {0.000036, 
      4.0818*10^8}, {0.0000361, 4.14224*10^8}, {0.0000362, 
      4.08557*10^8}, {0.0000363, 3.57366*10^8}, {0.0000364, 
      3.722*10^8}, {0.0000365, 3.51478*10^8}, {0.0000366, 
      4.24273*10^8}, {0.0000367, 3.98138*10^8}, {0.0000368, 
      3.98873*10^8}, {0.0000369, 4.07259*10^8}, {0.000037, 
      3.4828*10^8}, {0.0000371, 3.59175*10^8}, {0.0000372, 
      4.01442*10^8}, {0.0000373, 4.02545*10^8}, {0.0000374, 
      3.97001*10^8}, {0.0000375, 3.69695*10^8}, {0.0000376, 
      4.01087*10^8}, {0.0000377, 3.46884*10^8}, {0.0000378, 
      3.40665*10^8}, {0.0000379, 3.81782*10^8}, {0.000038, 
      3.85267*10^8}, {0.0000381, 3.93677*10^8}, {0.0000382, 
      3.4168*10^8}, {0.0000383, 3.40748*10^8}, {0.0000384, 
      3.28001*10^8}, {0.0000385, 3.73761*10^8}, {0.0000386, 
      3.61158*10^8}, {0.0000387, 3.87561*10^8}, {0.0000388, 
      3.633*10^8}, {0.0000389, 3.4367*10^8}, {0.000039, 
      3.29006*10^8}, {0.0000391, 3.63263*10^8}, {0.0000392, 
      3.64321*10^8}, {0.0000393, 3.18616*10^8}, {0.0000394, 
      3.37887*10^8}, {0.0000395, 3.71965*10^8}, {0.0000396, 
      3.64279*10^8}, {0.0000397, 3.38655*10^8}, {0.0000398, 
      3.53998*10^8}, {0.0000399, 3.69957*10^8}, {0.00004, 
      3.66599*10^8}, {0.0000401, 3.45273*10^8}, {0.0000402, 
      3.05927*10^8}, {0.0000403, 3.39315*10^8}, {0.0000404, 
      3.20745*10^8}, {0.0000405, 3.39915*10^8}, {0.0000406, 
      3.02635*10^8}, {0.0000407, 3.2508*10^8}, {0.0000408, 
      3.52145*10^8}, {0.0000409, 3.43104*10^8}, {0.000041, 
      3.55959*10^8}, {0.0000411, 3.05492*10^8}, {0.0000412, 
      2.94112*10^8}, {0.0000413, 3.1421*10^8}, {0.0000414, 
      3.16455*10^8}, {0.0000415, 3.48402*10^8}, {0.0000416, 
      3.40535*10^8}, {0.0000417, 3.38442*10^8}, {0.0000418, 
      3.2622*10^8}, {0.0000419, 3.34345*10^8}, {0.000042, 
      3.15076*10^8}, {0.0000421, 3.18726*10^8}, {0.0000422, 
      3.08504*10^8}, {0.0000423, 3.06754*10^8}, {0.0000424, 
      3.3602*10^8}, {0.0000425, 3.3578*10^8}, {0.0000426, 
      3.04322*10^8}, {0.0000427, 3.04005*10^8}, {0.0000428, 
      2.9239*10^8}, {0.0000429, 2.77703*10^8}, {0.000043, 
      3.00855*10^8}, {0.0000431, 3.03118*10^8}, {0.0000432, 
      2.73178*10^8}, {0.0000433, 3.10125*10^8}, {0.0000434, 
      3.24634*10^8}, {0.0000435, 2.85507*10^8}, {0.0000436, 
      3.0253*10^8}, {0.0000437, 2.94682*10^8}, {0.0000438, 
      2.67459*10^8}, {0.0000439, 2.78222*10^8}, {0.000044, 
      2.78228*10^8}, {0.0000441, 2.68133*10^8}, {0.0000442, 
      2.80897*10^8}, {0.0000443, 3.1133*10^8}, {0.0000444, 
      2.67488*10^8}, {0.0000445, 2.95369*10^8}, {0.0000446, 
      3.0872*10^8}, {0.0000447, 2.78099*10^8}, {0.0000448, 
      2.88453*10^8}, {0.0000449, 2.64087*10^8}, {0.000045, 
      2.53686*10^8}, {0.0000451, 2.78259*10^8}, {0.0000452, 
      2.55547*10^8}, {0.0000453, 3.03588*10^8}, {0.0000454, 
      2.65502*10^8}, {0.0000455, 2.94768*10^8}, {0.0000456, 
      2.83474*10^8}, {0.0000457, 2.81128*10^8}, {0.0000458, 
      2.63055*10^8}, {0.0000459, 2.5719*10^8}, {0.000046, 
      2.86383*10^8}, {0.0000461, 2.5114*10^8}, {0.0000462, 
      2.68705*10^8}, {0.0000463, 2.47606*10^8}, {0.0000464, 
      2.65907*10^8}, {0.0000465, 2.42737*10^8}, {0.0000466, 
      2.75109*10^8}, {0.0000467, 2.87195*10^8}, {0.0000468, 
      2.76994*10^8}, {0.0000469, 2.41292*10^8}, {0.000047, 
      2.58802*10^8}, {0.0000471, 2.37396*10^8}, {0.0000472, 
      2.54686*10^8}, {0.0000473, 2.3203*10^8}, {0.0000474, 
      2.3556*10^8}, {0.0000475, 2.52191*10^8}, {0.0000476, 
      2.73224*10^8}, {0.0000477, 2.52723*10^8}, {0.0000478, 
      2.66194*10^8}, {0.0000479, 2.31522*10^8}, {0.000048, 
      2.27543*10^8}, {0.0000481, 2.71421*10^8}, {0.0000482, 
      2.57852*10^8}, {0.0000483, 2.61125*10^8}, {0.0000484, 
      2.33866*10^8}, {0.0000485, 2.42181*10^8}, {0.0000486, 
      2.66695*10^8}, {0.0000487, 2.64178*10^8}, {0.0000488, 
      2.35225*10^8}, {0.0000489, 2.65588*10^8}, {0.000049, 
      2.48891*10^8}, {0.0000491, 2.45468*10^8}, {0.0000492, 
      2.28972*10^8}, {0.0000493, 2.6166*10^8}, {0.0000494, 
      2.39098*10^8}, {0.0000495, 2.39786*10^8}, {0.0000496, 
      2.21521*10^8}, {0.0000497, 2.18797*10^8}, {0.0000498, 
      2.20248*10^8}, {0.0000499, 2.13029*10^8}, {0.00005, 2.38022*10^8}}

I am trying to fit this dataset to a system of ODEs. Here are some constants:

(* init initital values *)
k1 = 10.*10^-10; (* Cs^++Cs\[Rule]Subsuperscript[Cs, 2, +] *)
k2 = 2.*10^-14; (* Cs^++Subscript[H, 2]\[Rule]CsH^++H *)
(*k3=1.00000000000000000000*10^-7; (* \!\(
\*SubsuperscriptBox[\(Cs\), \(2\), \(+\)] + e\)\[Rule]Cs*+Cs(6s) *)*)


k5 = 1.*10^7; (* Cs* \[Rule]Cs(6s)+hv' *)
kp = 8.*10^-10; (* NOT SURE Cs*\[Rule]Cs(6s)+hv *)
f = 0.95; (*  *)
kr1 = 1.*10^6; (* Cs(7p)\[Rule]Cs(6s)+hv *)
nH2 = 1.5*10^18; (*  *)

Here is the ODE and fit Code:

tmax = 5.*10*10^-5;
ode =
  {
   (* Cs(6s) *)
   Derivative[1][n0][t] == 
    kr1*nhv[t] - k1*n0[t]*n1[t] + k5*n2[t] + 
     k3*(nCsH[t] + nCs2[t])*ne[t] + kp*nhv[t]^2,
   (* Cs(7p) *)Derivative[1][nhv][t] == -(kr1 + 2*kp*nhv[t])*nhv[t],
   (* edensity *)
   Derivative[1][ne][t] == -k3*(nCs2[t] + nCsH[t])*ne[t] + 
     kp*f*nhv[t]^2,
   (* Subsuperscript[Cs, 2, +] *)
   Derivative[1][nCs2][t] == k1*n1[t]*n0[t] - k3*ne[t]*nCs2[t],
   (* Cs^* *)
   Derivative[1][n2][t] == 
    k3*nCs2[t]*ne[t] - k5*n2[t] + kp*(1 - f)*nhv[t]^2,
   (* CsH^+ *)
   Derivative[1][nCsH][t] == k2*n1[t]*nH2 - k3*nCsH[t]*ne[t],
   (* Cs^+ *)
   Derivative[1][n1][t] == f*kp*nhv[t]^2 - (k1*n0[t] + k2*nH2)*n1[t],
    n0[0] == 4.*10^13,(* Cs(6s) *)
    n1[0] == 0.,
    nhv[0] == 4.*10^13,
    ne[0] == 0.,
    nCs2[0] == 0.,
    nCsH[0] == 0.,
    n2[0] == 0.
   };

paramSOL = ParametricNDSolve(*Value*)[
  ode,
  {n0, nhv, n1, n2, nCs2, nCsH, ne},
  {t, 0., tmax},
  {k3},
  (*Method\[Rule]{NMinimize,Method\[Rule]"DifferentialEvolution"},*)
  Method -> "StiffnessSwitching",
  AccuracyGoal -> 10
  ]
{n0Func, n1Func, nhvFunc, neFunc, nCs2Func, nCsHFunc, 
  n2Func} = {n0, n1, nhv, ne, nCs2, nCsH, n2} /. paramSOL
model[k3_][t_] := n2Func[k3][t]
fit = NonlinearModelFit[
    random, {n2Func[k3][t], 0.5*10.^-8 < k3 < 1.4*10.^-5}, {k3}, 
    t]; // Timing

With this, the calculation is finding k3. I simulated the data and gave it some randomness, and then took away the k3 value to have the computer find it. It takes about 120 seconds, which is fine for 1 parameter, but when I take away a second parameter to have it fit 2 parameters, it takes hours. I would like to be able to solve for 3 parameters in a reasonable amount of time.

Parallelization does not seem to help (i mean this is fitting only 1 parameter so that makes sense), and I have tried some different settings, but overall nothing I have done has significantly increased the speed of the optimization procedure. Fit Results for 1 parameter, the others predefined Advice?

POSTED BY: Sean Bingus

3 Replies

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Neil Singer

Neil Singer, AC Kinetics, Inc.

Posted 5 years ago

Sean, The problem I see is substituting the k’s into the parametric expression is very expensive. I think it resolves the ndsolve each time it’s evaluated because the original differential equations are in the parametric expression. I’ll try to think about a way to speed that up. Regards

POSTED BY: Neil Singer

Sean Bingus

Posted 5 years ago

Thanks Neil, Isn't that the only way to attack the problem, though? My estimation was that was the only way it could possibly work. I would be very interested to hear about a workaround for this! The idea here, as you have probably guessed, is that I have a bunch of experimental data from radiative processes (COVID lockdown... old data... graduate student, haha), And I want to be able to fit it to these equations. This is just procedurally generated random data, but I think that I can do some of the fitting by hand for good initial guesses, put those into the solver with reasonable wiggle room in each parameter, go to bed and start the routine, and wake up with a fine tuned version of the hand fitting. The point is that it doesn't have to be lightning speed, but this is dummy data which I assume should be really fast. Maybe 10 minutes or less? Also, I have been wondering if Mathematica deals with scientific notation very well. The numbers span about 16 orders of magnitude, and I expect my data to have nearly an order of magnitude of uncertainty. It's never been an issue for me, but a lot of the stuff I've done for fits in Mathematica has been smaller values in matrices.

POSTED BY: Sean Bingus

Sean Bingus

Posted 5 years ago

Update, I have taken out all of the equations involving CsH, but it still seems really slow to fit to 2 rate constants. I am running a single iteration of the fit now and it has been at least 2 minutes!

POSTED BY: Sean Bingus

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