NIntegrate::slwcon, NIntegrate::eincr and NIntegrate::inumri

Question

I have the following code to evaluate the double integral of which the integrand contains summations.

a = 1.1798710705178221747709894550582188967078238417029640017201995589\
3552025364542807472418914015859414022112280461408148970194496741533076\
882157607415223127492391019655558195361182929703459`158.\
94383771058207*^12;
b = 2.26562500;
c = 1.81505148861869`*^-11;
d = 23.662621294619998242971428558039280847039065920403134745951781161\
6604691248947208857758189787681405721656941282488226858308051995862227\
03295794982365258929783937779449526906107493517940605539567`157.\
52081186501843;
f = 3.9762816693358413444242383928332521659243792714105963808597582172\
1677895928488587525623222082345413892578490996192857648555810218065386\
055620034160228548784193009119494778657256763`157.54308182919476*^-13;
g = 3.1306512217343699049816678338611844042940126102861569115946355055\
2379739233549991684436932580097276616360046758705039003579253070583704\
87306284924487233677956546263347280851687424`154.55339124819713*^-283;
h = 0.4663677854613281996964274099654771725449995576718401982649626901\
4979290048497524493727931929716644258598652635846126429619061653535457\
7552234258390027879122494699655811617457309249292766647653`158.\
10068151409945;
k = 1.7400621010179746539715325376531540776673407694803313887835074772\
7621820290231417060167325520004426731623808979659285973341841054088991\
377348701663084581319054270584192856482987849`158.6501729599253*^-13;
L = 12;
p = 1.7267086550014064377504396122758581898174212801982663320383980364\
4189130935666171318482175845230982997098739340274603421154239660049599\
7742129919272319043201530223139294218906027174185411646792`156.\
9155606199771*^-19;
m = {193.1974795021252244647940778330697265346831202988930438449952777\
2307847128160223047093925434704114309184915713826376925945458184231453\
466263054478004310746486964933973801043322`148.30510799763482, \
-1.9831415769102246753169203737597186440257929720747405282387824878992\
6107613692269553118408251584641941543185280916783314001387339467872143\
8413408725537146318896037644145`148.23545499521762*^14, 
   9.27130443465927962680463415680935518438025048895563739033406430779\
2196012034144989441708086252518980765588393670969227402958335184629953\
81754195556168420354856191202632726995515843`148.1802144267804*^25, \
-2.5800039865768300579393918422060974207019853041395228305982709910784\
5385543803983822395718755270344540849858353480555085709766499883993525\
5692481465734198491137803554398276`148.1336524600536*^37, 
   4.74051682958982877757253604628388253288384781482056602069781764081\
9228970772949442973000148032178390675200697268768942345714056423740914\
80354727904019873987134563343319805735897818842`148.09340643440623*^\
48, -6.045696336170963761979351988393444721239513638248359564313546590\
9718715772107181860271630911859080048890550740293482441710924532226834\
88785001561949827887583104602354991010789`148.05796717887392*^59, 
   5.48169129849067708749576724703553417939408966217748199411635112509\
3975998785626154086737477706943924734870295242554891109862338634996706\
52881122759735494841981860519139551755923247573972`148.02635144833272*^\
70, -3.558859598535019932696238577795555409895936844852263611105033721\
0375255058472565008180720204152692875351619018996231358541261075464815\
50993035152346687208320263348602766130493464`147.99786882797923*^81, 
   1.64170225303118668483536395284823903373718786459586837660122032491\
6095608079033229473848757844731529141963765405064759781082803855127965\
2691286574811194198800983167974295`147.97199851695797*^92, \
-5.2480764106579160571712590401723827187713866001169686217276690968556\
0720450830001550831782374704062930304936342651979107670945180568135634\
934777787696977134688463363076848049109484`147.94833381460728*^102, 
   1.10376762837378400256596151856877143314949611822933817186964398056\
6513468174587221258709945187779831909161277645871738123234551311287274\
78355158886780258679136222861840141`147.92655298710872*^113, \
-1.3717147483659623561567398067035113741497860704561052158773749470395\
4451131015780543446068263484385374754209550914285685014429230059159675\
6883122631756455319026263237240593696821711`147.9063994508292*^123, 
   7.62038321127089845343261988865253586612439689357390721954186078231\
7745858168312353448416091122119241968300105088661006951809593285174538\
28785418554487594420138798900387611`147.8876665311791*^132};
n = {1.355573243087520514288089562497260829417934974879258545226394709\
1581632390251949362448991535838854387828992200011014212046924115116342\
36774361520751754411719411168290762941843303838260921034729`154.\
7068137372351, \
-1.0245928740767489003861056984284775651988428602209840685472503196463\
8795930234592796835719935424571882137914261342592202874852760076112574\
416630451012360208651254471215128194043516110717`153.94630606168067*^\
13, 8.9274486445716723149532799214464202934486639009370371697876732811\
7309338540881124740225222195651232772519644118676062290969039522944619\
182163951555475885638710858185564149165684843`153.80921876999915*^25, \
-3.6759676683395569226361903550968200824615587798139936187422973093672\
7524993405149619110906669196324634865139505465691286936359213505175950\
82648234290038347371835858805085681`153.70951619258835*^38, 
   8.46890235524130159449486145084290474315429938343042316420562067986\
2851660903090850781750760926753333681142269036924244448156733221457952\
7284650316472932006583589556818127479776452351575`153.63641768743574*^\
50, -1.183963727932234048419892914574504119691001960215443363374579536\
6221799023716788793836145690916125278181883509910251692260914506526151\
991183549806618069041478638356123410370047646`153.58049547400296*^63, 
   1.05061217672427573937875620745709379002627336725498531970829595877\
3024659127091166260173635393986216201350663927669330845944472454001554\
7653381517564746234073087737793092403156290468681009911`153.\
53584270924242*^75, \
-6.0584630947562597856380562413442419518448978885525607665447920472335\
7004600791968921362280443938576083533065571634125571548632808959929629\
790577441227915620375110110148244720464105958`153.4990714580894*^86, 
   2.28274980013156097133562209543324362618017404654189411106310354580\
1316698809067147502956432483851929920100534744472639286623024080167418\
0951137035977035828949374744764910684041`153.46810663882243*^98, \
-5.5393276975669543411312359391170032422891499208542799478529688449103\
0657737312682644055014461777520413955788265776706075897708285932473509\
487959913970225597988826267142550230756354142547`153.44158425404342*^\
109, 8.289929266459807127028509719838011396311076683133447742366507269\
1426222477306422877783169874953638427457706706377785690078513120930488\
09383654486564976662124022080955027406652237`153.41855789644956*^120, \
-6.9209387452305809520831335905017476070978872825599662420473808654194\
8931991860905633911584754872433560379615186289074673651242807441129188\
233340512050802389488222073074566318608837938629357`153.\
39834390904957*^131, 
   2.45300100486846411742403054500030433066570931665654433746550128485\
1080592792054839789289213596711427047642684395418250860055631043057336\
48562402527422622838036053732378834470177372`153.3804328940518*^142};
NIntegrate[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(98\)]\(
\*FractionBox[\(
\*SuperscriptBox[\((a*x + b)\), \(i\)]*
     Exp[\(-\((a*x + b)\)\)]\), \(i!\)] \(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(L\)]
\*FractionBox[\(
\*SuperscriptBox[\((y + x - c)\), \(d\)] Exp[\(-\((y + x - c)\)\)/
         f] \((m[\([i + 1]\)] 
\*SuperscriptBox[\((y + x - c)\), \(i\)])\)\), \(g\)] \(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(L\)]
\*FractionBox[\(
\*SuperscriptBox[\(y\), \(h\)] Exp[\(-y\)/k] \((n[\([i + 1]\)] 
\*SuperscriptBox[\(y\), \(i\)])\)\), \(p\)]\)\)\)\), {x, c, 
  Infinity}, {y, 0, Infinity}, 
 Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 4000}, 
 MinRecursion -> 10, MaxRecursion -> 30]

However, I get two error messages, which are

1. NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

2. NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 4000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 0.9999763600200141 and 0.0000461755863201869 for the integral and error estimates.

I have tried to increase MaxErrorIncreases to 20000. However, the above two error messages still exist. Why these errors happen? Is there a method to fix these issues?

Update:

NIntegrate[\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(L\)]\(
\*FractionBox[\(
\*SuperscriptBox[\((y + x - c)\), \(d\)] Exp[\(-\((y + x - c)\)\)/
       f] \((m[\([\)\(i + 1\)\(]\)] 
\*SuperscriptBox[\((y + x - c)\), \(i\)])\)\), \(g\)] \(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 0\), \(L\)]
\*FractionBox[\(
\*SuperscriptBox[\(y\), \(h\)] Exp[\(-y\)/k] \((n[\([\)\(i + 1\)\(]\)] 
\*SuperscriptBox[\(y\), \(i\)])\)\), \(p\)]\)\)\), {x, c, 
  Infinity}, {y, 0, Infinity}, 
 Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 4000}, 
 MinRecursion -> 10, MaxRecursion -> 30]
(*0.999976*)
(*NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 4000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 0.9999763600200167` and 0.000046175586322940094` for the integral and error estimates.*)

Combine with the poisson pdf:

NIntegrate[
 SetPrecision[Sum[((a*x + b)^j*Exp[-(a*x + b)])/j!, {j, 0, 98}], 
   64] SetPrecision[
   Sum[(((y + x - c)^d*
         Exp[-(y + x - c)/f]*(m[[i + 1]]*(y + x - c)^i))/g)*
     Sum[(y^h*Exp[-y/k]*(n[[i + 1]]*y^i))/p, {i, 0, L}], {i, 0, L}], 
   64], {x, c, Infinity}, {y, 0, Infinity}, MinRecursion -> 3, 
 WorkingPrecision -> 32]
(*1.0000000030812388724654833722125*)
(*
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 19 recursive bisections in y near {x,y} = {1.733749426269622122802915010608978488837265315801884727791202406575965983248912207*10^-9,1.610037995921788499566677793782828831793685299506806564866521408352787684409611295*10^-6}. NIntegrate obtained 1.000000003081238872465483372212524803405472370564982575528771716325065565992601043`82. and 2.937642115074185767546718140441621526496175969025481265504160565200333312980601489`82.*^-7 for the integral and error estimates.
*)

There are two errors after I combine with the poisson pdf. I have tried to increase the working precision and increase the MaxRecursion value. However, the errors still exist.

Update: The double integral of the same integrand and different intergral limits:

NIntegrate[
 SetPrecision[Sum[((a*x + b)^j*Exp[-(a*x + b)])/j!, {j, 0, 98}], 
   32] SetPrecision[
   Sum[(((y + x - c)^d*
         Exp[-(y + x - c)/f]*(m[[i + 1]]*(y + x - c)^i))/g)*
     Sum[(y^h*Exp[-y/k]*(n[[i + 1]]*y^i))/p, {i, 0, L}], {i, 0, L}], 
   32], {x, -Infinity, c}, {y, -(x - c), Infinity}, MinRecursion -> 3,
  WorkingPrecision -> 16]
(*
NIntegrate::inumri: The integrand (2.7182818284590450000000000000000^(-2.265625000000000-1.179871070517822*10^12 x)+2.7182818284590450000000000000000^(-2.265625000000000-1.179871070517822*10^12 x) (2.2656250000000000000000000000000+1.1798710705178220000000000000000*10^12 x)+<<48>>+<<49>>) (<<1>>) has evaluated to Overflow, Indeterminate, or Infinity for all sampling points in the region with boundaries {{0,0.1250000000000000},{0,0.1250000000000000}}.
*)

Why this warning message appears and how to fix this problem?

Answer 1

Use higher working precision:

NIntegrate[SetPrecision[
     Sum[(((y + x - c)^d*Exp[-(y + x - c)/f]*
               (m[[i + 1]]*(y + x - c)^i))/g)*
         Sum[(y^h*Exp[-y/k]*(n[[i + 1]]*y^i))/p, 
           {i, 0, L}], {i, 0, L}], 32], 
   {x, c, Infinity}, {y, 0, Infinity}, 
   MinRecursion -> 3, WorkingPrecision -> 16]
(*  0.9999995560228801  *)