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We can use this to generate a 3D plot with values for \ \[Gamma]3 on the x-axis, time on the y-axis, and the value of the predicted \ \"a\" concentration on the z. 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What we would like is to seek parameter values \ that produce predictions which match the data. To do this , we create a \ function that takes parameter values, compares the model with these \ parameters to the data, and returns the difference. This \"objective function\ \" acts as a score for proposed parameter values. \ \>", "Text", CellChangeTimes->{{3.448220888489384*^9, 3.4482209915004*^9}, { 3.448221113910063*^9, 3.448221393777379*^9}, {3.448221427128022*^9, 3.448221557345419*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"evaluate", "[", RowBox[{ RowBox[{"proposed\[Gamma]2_", "?", "NumberQ"}], ",", " ", RowBox[{"proposed\[Gamma]3_", "?", "NumberQ"}]}], "]"}], ":=", "\[IndentingNewLine]", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "interpolatingFuncs", ",", "predictedA", ",", "predictedB", ",", "errorA", ",", "errorB", ",", "ode"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"ode", " ", "=", " ", RowBox[{"Join", "[", RowBox[{ RowBox[{"eqns", " ", "/.", " ", RowBox[{"{", RowBox[{ RowBox[{"\[Gamma]2", "\[Rule]", "proposed\[Gamma]2"}], ",", RowBox[{"\[Gamma]3", "\[Rule]", "proposed\[Gamma]3"}]}], "}"}]}], " ", ",", RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"a", "[", "0", "]"}], "==", "1"}], ",", RowBox[{ RowBox[{"b", "[", "0", "]"}], "==", "2"}]}], "}"}]}], " ", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"interpolatingFuncs", "=", RowBox[{ RowBox[{"Quiet", "[", RowBox[{"NDSolve", "[", RowBox[{"ode", ",", "vars", ",", RowBox[{"{", RowBox[{"t", ",", "0", ",", "100"}], "}"}], ",", RowBox[{"MaxSteps", "\[Rule]", "100000"}]}], "]"}], "]"}], "[", RowBox[{"[", "1", "]"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"predictedA", " ", "=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"Quiet", "[", RowBox[{ RowBox[{"(", RowBox[{"a", "/.", "interpolatingFuncs"}], ")"}], "[", "iii", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"iii", ",", "0", ",", "100", ",", ".1"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"predictedB", " ", "=", " ", RowBox[{"Table", "[", RowBox[{ RowBox[{"Quiet", "[", RowBox[{ RowBox[{"(", RowBox[{"b", "/.", "interpolatingFuncs"}], ")"}], "[", "iii", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"iii", ",", "0", ",", "100", ",", ".1"}], "}"}]}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"errorA", " ", "=", RowBox[{"Norm", "[", RowBox[{ RowBox[{"(", RowBox[{"predictedA", "-", " ", RowBox[{"(", RowBox[{"a", "/.", "data"}], ")"}]}], ")"}], ",", " ", "2"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"errorB", " ", "=", " ", RowBox[{"Norm", "[", RowBox[{ RowBox[{"(", RowBox[{"predictedB", "-", " ", RowBox[{"(", RowBox[{"b", "/.", "data"}], ")"}]}], ")"}], ",", " ", "2"}], "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"errorA", " ", "+", " ", "errorB"}]}]}], " ", "\[IndentingNewLine]", "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.448132779973414*^9, 3.448132832714129*^9}, { 3.448132868583523*^9, 3.448133028115499*^9}, {3.448133065573512*^9, 3.448133116879644*^9}, {3.448133167745678*^9, 3.448133169355941*^9}, { 3.44813324079771*^9, 3.448133288019096*^9}, {3.448133327929583*^9, 3.448133900081023*^9}, {3.448133992243649*^9, 3.448134016494375*^9}, { 3.448134337859189*^9, 3.448134357626451*^9}, {3.448134391203727*^9, 3.448134393364832*^9}, {3.448134426017901*^9, 3.448134435691311*^9}, { 3.448134480362441*^9, 3.44813451928538*^9}, 3.448134567910929*^9, { 3.448134641384129*^9, 3.448134697050075*^9}, {3.448134755479128*^9, 3.448134760236059*^9}, {3.44813481140214*^9, 3.448134838910192*^9}, { 3.448135066190669*^9, 3.448135067521736*^9}, {3.448135182020238*^9, 3.448135185579507*^9}, {3.448135319711552*^9, 3.448135333134918*^9}, 3.448136410773771*^9, {3.448137328753547*^9, 3.448137611130721*^9}, { 3.448137687923962*^9, 3.448137723477886*^9}, {3.448137783950349*^9, 3.448137821150608*^9}, {3.448137854441493*^9, 3.448137866907363*^9}, { 3.448210397564382*^9, 3.448210447362235*^9}, {3.448210484106827*^9, 3.448210497558088*^9}, {3.448210656514609*^9, 3.448210665773611*^9}, { 3.448210851443232*^9, 3.448210851885844*^9}, {3.448216440878598*^9, 3.448216457681646*^9}, 3.448221401658267*^9, {3.448223834244219*^9, 3.448223840746314*^9}, {3.44822390091412*^9, 3.44822393594965*^9}}], Cell["\<\ For problems with small dimension, we can plot this objective function. Here \ it looks like this. Smaller values are better matches with the data. But \ where is the lowest point?\ \>", "Text", CellChangeTimes->{{3.448220888489384*^9, 3.4482209915004*^9}, { 3.448221113910063*^9, 3.448221393777379*^9}, {3.448221427128022*^9, 3.448221609647074*^9}}, Background->RGBColor[0.87, 0.94, 1]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{"evaluate", "[", RowBox[{"i", ",", "j"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "0", ",", ".1"}], "}"}], ",", RowBox[{"{", RowBox[{"j", ",", "0", ",", ".3"}], "}"}]}], "]"}]], "Input", CellChangeTimes->{{3.448204331196026*^9, 3.448204333955971*^9}, { 3.448211100702495*^9, 3.448211120556867*^9}, {3.448211262861124*^9, 3.448211286071967*^9}, 3.448222292119919*^9}], Cell[BoxData[ Graphics3DBox[GraphicsComplex3DBox[CompressedData[" 1:eJx0vXc819///19UtEWZaQ+jISv7+bR39kz25jmsZDWMlJQR2iElTUVGhSSh yEpDSUqRthVK+Xl/Po/7/bx/Ppfv65/X5XV93S/353nczv2ccz/3czweK11Z 5h4c06ZN4507bRrn5L9P22snXrympjbMKXtJlcNGjUvIrsjwWCR9ocDf+8/U o2jA9VYEcBdHR9KvvUpqaFM/jLzBry/sFyOS3pfG4+DUn45cZZ/4dm67SHqk 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