{
"cells": [
{
"cell_type": "markdown",
"id": "f7c31d5f-9f44-452a-8919-44a219e65154",
"metadata": {},
"source": [
"Math 420/620 Final Question 1"
]
},
{
"cell_type": "markdown",
"id": "bb9ad9c6-be4a-4071-a40d-83d1e176ad1a",
"metadata": {},
"source": [
"1. Weather forecasting software has to produce a forecast within\n",
"4 hours to meet the quality of service agreed upon by a government\n",
"contractor.\n",
"Currently the program takes 5 hours to run on\n",
"an Origin 2000 parallel computer consisting of 16 processors.\n",
"The contractor is looking for ways to speed the\n",
"computation."
]
},
{
"cell_type": "markdown",
"id": "7f246380-dc80-4580-8d9f-73c8d0e8eacf",
"metadata": {},
"source": [
"(i) About 60 percent of the time all 16 processors work in parallel,\n",
"but since not all the work can run\n",
"in parallel, 40 percent of the time only 1 processor\n",
"is active.\n",
"By Amdahl's law\n",
"$$\n",
" M(P/16+1-P) = 5\\,\\hbox{hours}\n",
"$$\n",
"where $M$ is how long the program would run using only\n",
"one processor and $P$ represents the percent of the total computation\n",
"that is parallelizable. Thus,\n",
"$$\n",
" MP/16=3\\,\\hbox{hours}\\qquad\\hbox{and}\\qquad M(1-P)=2\\,\\hbox{hours}.\n",
"$$\n",
"Solve for $M$ and $P$."
]
},
{
"cell_type": "markdown",
"id": "1f4f1122-d559-4b4a-8968-a5dca07e564e",
"metadata": {},
"source": [
"Since $MP/16=3$ then $MP=48$. It follows that\n",
"$$\n",
" M(1-P)=M-MP=M-48=2\n",
"$$\n",
"Consequently, $M=50$.\n",
"\n",
"Now, substitute back into the first equation to solve for $P$. Thus,\n",
"$$\n",
" P=48/M=48/50=24/25=0.96.\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "546c2afc-4279-466c-952b-e0bc46474598",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.96"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"M=50; P=24/25"
]
},
{
"cell_type": "markdown",
"id": "e61822be-3541-414c-a156-7ed8c735120f",
"metadata": {},
"source": [
"(ii) How many more processors are needed to meet the 4 hour deadline?"
]
},
{
"cell_type": "markdown",
"id": "4c29eea1-cea8-4222-b991-a10a7aa78caf",
"metadata": {},
"source": [
"Consider a parallel computer with $n$ total processors.\n",
"We must solve for the least $n$ such that\n",
"$$\n",
" M(P/n+1-P)\\le 4.\n",
"$$\n",
"Thus,\n",
"$$\n",
" MP/n\\le 4+MP-M\n",
"$$\n",
"or\n",
"$$\n",
" n\\ge {MP\\over (4+MP-M)}={48\\over 4+48-50}=24.\n",
"$$\n",
"Since a total of $n=24$ processors would be sufficient, then $8$ more\n",
"processors are needed to upgrade the current $16$ processor\n",
"computer to $24$."
]
},
{
"cell_type": "markdown",
"id": "0cf75726-6de9-4140-aea5-36a7a3fc99b5",
"metadata": {},
"source": [
"(iii) There was an accident and the old computer destroyed. A new\n",
"computer will be purchased with\n",
"$n$ processors that each run at a speed $s$. Here $s=200$ MHz\n",
"corresponds to the processors in the old computer. Explain why\n",
"$$\n",
" (200/s) M(P/n+1-P)=4\\,\\hbox{hours}\n",
"$$\n",
"ensures the new computer will produce a forecast in $4$ hours.\n"
]
},
{
"cell_type": "markdown",
"id": "de3d4af3-b9ee-4b05-999c-1c5b121ae5b3",
"metadata": {},
"source": [
"Increasing the speed of the processors proportionately\n",
"increases the speed at which the weather forecast is finished\n",
"while holding the number of processors fixed. For example,\n",
"doubling the speed means the time taken is only half. Since\n",
"the time taken by the original computer was given by Amdahl's\n",
"law as\n",
"$$\n",
" T_0=M(P/n+1-P),\n",
"$$\n",
"faster processors will complete the work in time\n",
"$$\n",
" T(s)=(200/s) T_0=(200/s)M(P/n+1-P).\n",
"$$\n",
"The condition above comes by taking $T(s)=4$."
]
},
{
"cell_type": "markdown",
"id": "bcf9578c-7a65-4cb2-be59-f864c52aabf8",
"metadata": {},
"source": [
"(iv) Processors cost\n",
"$c(s)=As^3$\n",
"where $A$ is a constant and $s$ is processor speed.\n",
"Assuming the processors in the old computer cost \\\\$24000,\n",
"solve for $A$."
]
},
{
"cell_type": "markdown",
"id": "902bfee9-06d6-45f1-92a8-59697a3e7ab1",
"metadata": {},
"source": [
"We want $c(200)=24000$. Thus $A200^3=24000$, which implies\n",
"$$\n",
" A={24000\\over 200^3}=0.003.\n",
"$$\n",
"We remark that $A$ is a dimensional constant with units \\\\$/MHz$^3$. The\n",
"power dissipation in integrated circuits scales as $sV(s)^2$ where experience\n",
"suggests voltage $V$ must be scaled linearly with frequency. Thus, the\n",
"expected power dissipation of a faster processor scales at least as $s^3$.\n",
"\n",
"The actual pricing of processors will be related to supply and demand constraints\n",
"as well as engineering constraints, so while $c(s)=As^3$ may hold for $s$ near 200, many\n",
"assumptions have been made."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "516eec4a-3d0c-4c94-857a-e474b3c5662a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.003"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A=24000/200^3"
]
},
{
"cell_type": "markdown",
"id": "9de79d86-680e-4a6b-b111-8723916984fb",
"metadata": {},
"source": [
"(v) Suppose the processors are the main expense.\n",
"Then the total cost to build a computer with $n$ processors\n",
"is $C(n,s)=nAs^3$.\n",
"What values of $n$, $s$ and $C(n,s)$ correspond to the\n",
"cheapest computer that can produce a forecast in $4$ hours?"
]
},
{
"cell_type": "markdown",
"id": "2bcf0ff8-3ae9-4cac-a64f-1adb83b04c2c",
"metadata": {},
"source": [
"Our goal is to minimize $C(n,s)$ subject to the constraint\n",
"$$\n",
" (200/s) M(P/n+1-P)=4.\n",
"$$\n",
"This can be done using Lagrange multipliers; however, since\n",
"the constraint is easily solved for $s$, we instead substitute\n",
"$$\n",
" s=50 M (P/n+1-P)\n",
"$$\n",
"into the objective function to obtain\n",
"$$\n",
" C(n)=nA 50^3 M^3 (P/n+1-P)^3.\n",
"$$\n",
"Now search for the $n$ which minimizes $C(n)$."
]
},
{
"cell_type": "markdown",
"id": "1fc552c0-3e12-446e-8830-504e948ed207",
"metadata": {},
"source": [
"Now solve $C'(n)=0$ and if $n$ is not an integer\n",
"check $\\lfloor n\\rfloor$ and $\\lceil n\\rceil$ to determined which\n",
"is the least. Begin by plotting $C$ as a function of $n$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "31fcef9c-2a24-4215-bf38-1c3ddefa1a00",
"metadata": {},
"outputs": [],
"source": [
"using Plots"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "10b22e9c-5cd8-4661-83df-2b94cfa3ace3",
"metadata": {},
"outputs": [
{
"data": {
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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
""
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"s(n)=50*M*(P/n+1-P)\n",
"C(n)=n*A*s(n)^3\n",
"scatter(C,16:64,legend=:false,\n",
" xlabel=\"processors\",ylabel=\"cost\")"
]
},
{
"cell_type": "markdown",
"id": "0bf09c48-8923-4c20-b161-6a994d3e2691",
"metadata": {},
"source": [
"From the graph it appears the optimal number of processors is\n",
"somewhere between 40 and 60."
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "73eb0ebd-4ee0-4637-a8ae-0140bfc4ebf0",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"D (generic function with 1 method)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"using Symbolics\n",
"D(f,x)=expand_derivatives(Differential(x)(f))"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "37c9f243-b396-46c7-9772-6bd79ca2b4cd",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1-element Vector{Num}:\n",
" x"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"@variables x"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "885c4822-eedb-4a34-ab71-5d0669c74974",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"f (generic function with 1 method)"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"fn=D(C(x),x)\n",
"fs=\"f(x)=\"*string(fn)\n",
"eval(Meta.parse(fs))"
]
},
{
"cell_type": "markdown",
"id": "d28688fd-31c8-4102-8493-29943424ac0b",
"metadata": {},
"source": [
"Now set $f(x)=C'(x)$ and use Newton's method\n",
"$$\n",
" x_{k+1}=x_{k}-{f(x_k)\\over f'(x_k)}\n",
"$$\n",
"to solve for the $x$ where $C'(x)=0$."
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "b72a0962-8fbe-4446-a136-22272746f5c9",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"df (generic function with 1 method)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dfn=D(f(x),x)\n",
"dfs=\"df(x)=\"*string(dfn)\n",
"eval(Meta.parse(dfs))"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "26a1c5e7-665f-4c50-95d7-b70603ef4322",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"47.99999999999996"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xk=50.0\n",
"for k=1:10\n",
" xk=xk-f(xk)/df(xk)\n",
"end\n",
"xopt=xk"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "e38b15ed-d441-4d3c-b872-bb7c05f829a4",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"486000.00000000146"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Copt=C(48)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "8c3ecfc4-8512-4849-8d5a-6aa08a88aa3f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"486071.9782707112"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C(47)"
]
},
{
"cell_type": "markdown",
"id": "5eef81af-cc62-4d2c-bd51-d69afa1a70cf",
"metadata": {},
"source": [
"Since $C(48)P)"
]
},
{
"cell_type": "markdown",
"id": "702f9ce6-6726-4b90-bf1f-50ca9be390fd",
"metadata": {},
"source": [
"The sensitivity is negative, which reflects the fact that\n",
"if the amount of parallel work in the weather forcasting\n",
"computation is more than estimated the cost will go down.\n",
"If the amount of aprallel work is less than estimated the\n",
"cost will go up. Note also the fact that $S(C,P)\\approx -47$ is\n",
"large in magnitude reflects that the optimal cost depends strongly\n",
"on the estimate for the amount of parallel work."
]
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