Quasi-Monte Carlo for an Integrand with a Singularity along a Diagonal in the Square

Quasi-Monte Carlo f or an Integrand with a Singularity along a Diago nal in the Square Kinjal Basu and Art B. Owen Abstract Quasi-Monte Carlo method s are designed for integrand s of bou nded vari- ation, and this excludes singular integrands. Se vera l methods are known for inte- grands that become singular on the boundary of the unit cube [ 0 , 1 ] d or at isolated possibly u nknown points within [ 0 , 1 ] d . Here we co n sider functio n s on the square [ 0 , 1 ] 2 that may beco m e sing u lar a s the point appro aches the diagon al line x 1 = x 2 , and we study three quadratu re meth ods. The first method splits the square into two triangles separated by a region aroun d the line of sing ularity , an d applies recently developed tr ia n gle QMC rules to the two trian g ular parts. For fu nctions with a sin- gularity ‘no worse than | x 1 − x 2 | − A is’ for 0 < A < 1 that method yields an erro r of O (( log ( n ) / n ) ( 1 − A ) / 2 ) . W e also con sider meth ods extending the integrand into a region contain ing the singularity and show that method will n ot improve u pon us- ing two triang les. Finally , we conside r transform ing the integrand to have a more QMC-friendly sing u larity along the bo undary of the squ are. This then leads to error rates o f O ( n − 1 + ε + A ) when co mbined with some corn er-a voiding Halton points or with rando m ized QMC but it req uires some stronger assumption s on th e origin al singular integrand. 1 Intr o duction Quasi-Monte Carlo (QMC) in tegration is d esigned for integrands of bou nded vari- ation in the sense of Ha r dy and Krause ( BVHK). Such integrands must necessarily be bounded . Singu lar integrands cann o t be BVHK; they cannot even be Riemann Kinjal Basu LinkedI n Inc. e-mail: kbasu@link edin.com Art B. Owen Stanford Univ ersity e-mail: o wen@stanfor d.edu 1 2 Kinjal Basu and Art B. Owen integrable. It is kn own sinc e [6] a n d [3] that for any integrand f on [ 0 , 1 ] d that is not Riemann integrable, th ere exists a seq uence x x x i ∈ [ 0 , 1 ] d for which the star dis- crepancy D ∗ n ( x x x 1 , . . . , x x x n ) → 0 as n → ∞ while ( 1 / n ) ∑ n i = 1 f ( x x x i ) fails to con verge to R [ 0 , 1 ] d f ( x x x ) d x x x . W e ar e in terested in problems wher e the sing ularity arises along a manifold in [ 0 , 1 ] d . For motiv ation, see the engineerin g applications by Mishra and Gu pta in [1 0] and several o ther paper s. Apart from a few remar ks, we focus solely on the pro blem where there is a singu larity alo n g the line x 1 = x 2 in [ 0 , 1 ] 2 . It is p o ssible for QMC integration to succeed on unb o unded integrands. Sobol’ [15] noticed this when colleag u es used his me th ods on such problems. He explain ed it in terms of QMC p oints that av oid a hyper b olic region ar ound the lower bound - ary of the u nit cu be wh e re th e in tegrands became sing ular . Kling er [9] shows that Halton po in ts and some d ig ital nets a void a cubic a l region around the o rigin. Halton points (after the zero’th) av oid hyp e rbolic regions aroun d the bounda r y faces o f the unit cube at a rate suitab le to g et error bounds for QMC [ 13]. Certa in Kro n ecker sequences av oid hyperb olic r egions arou nd th e b oundar y of th e cube [8]. In all of these examples, avoiding the sing ularity shou ld be u n derstood as u sing po ints that approa c h it, but n ot too quickly , as the number n of fun ction ev aluations increases. For p lain Monte Carlo, th e lo cation of th e sing ularity is not important. One only needs to consider the first two moments o f the integrand. Becau se QMC exploits mild smoothn ess of the integrand, the natu re of th e sing ularity m atters. Reference [14] conside r s rando m ized QMC (RQMC) meth ods for integrands with po int sing u- larities at unkn own locations. In RQMC, the integrand is evaluated at points th at, in- dividually , ar e u niformly distributed on [ 0 , 1 ] d and this already implies a singularity av oida nce proper ty via the Borel-Cantelli lemma. If R f ( x x x ) 2 d x x x < ∞ th en scramb le d nets yield an unbiased estimate o f µ = R f ( x x x ) d x x x with RMSE o ( n − 1 / 2 ) [11]. The analyses in [ 13] and [14] em ploy an extension ˜ f of f from a set K = K n ⊂ [ 0 , 1 ] d to [ 0 , 1 ] d . The extension satisfies ˜ f ( x x x ) = f ( x x x ) fo r x x x ∈ K . N ow the quadratu r e error is 1 n n ∑ i = 1 f ( x x x i ) − Z [ 0 , 1 ] d f ( x x x ) d x x x = 1 n n ∑ i = 1 f ( x x x i ) − 1 n n ∑ i = 1 ˜ f ( x x x i ) + 1 n n ∑ i = 1 ˜ f ( x x x i ) − Z [ 0 , 1 ] d ˜ f ( x x x ) d x x x + Z [ 0 , 1 ] d ˜ f ( x x x ) d x x x − Z [ 0 , 1 ] d f ( x x x ) d x x x . If all of the po ints satisfy x x x i ∈ K , then the first term dr ops ou t a n d we find that     1 n n ∑ i = 1 f ( x x x i ) − Z [ 0 , 1 ] d f ( x x x ) d x x x     ≤     1 n n ∑ i = 1 ˜ f ( x x x i ) − Z [ 0 , 1 ] d ˜ f ( x x x ) d x x x     + Z − K | ˜ f ( x x x ) − f ( x x x ) | d x x x , where − K = [ 0 , 1 ] d \ K . T he extension used in [13] and [ 14] is d ue to Sobo l’ [1 5]. It is particularly well suited to a Koksma-Hlawka bou n d for the first term above as ˜ f has low variation. Quasi-Monte Carlo for an Integrand with a Singularity along a Di agonal in the Square 3 In our case, we can isolate the singularity in the set { x x x | | x 1 − x 2 | < ε } . A set K ⊂ [ 0 , 1 ] d is Sobol’ - extensible to [ 0 , 1 ] d with anc h or c c c if for every x x x ∈ K the rectang le ∏ d j = 1 [ min ( x j , c j ) , max ( x j , c j )] ⊂ K . In our case, th e set { x x x | | x 1 − x 2 | ≥ ε } in wh ich f is b ounded is n ot Sobol’ extensible. Th e extension ˜ f used in [13] and [ 14] cann ot be defined for this p r oblem. Section 2 presents a strategy of avoiding a region n ear the singular ity and inte- grating over two triangula r regions u sing the method from [1]. Th e erro r is then a sum of two quadratu re error s and one truncation erro r . W e con sider f unctions where the singularity is not more sev ere than that in | x 1 − x 2 | − A where 0 < A < 1. Sec- tion 3 shows that the truncation error in this appr oach is O ( ε − A ) and the quadra ture error is O ( ε − A − 1 log ( n ) / n ) using the points fr o m [1] an d a K ok sma-Hlawka bo und from [4]. Th e r esult is that we can attain a much better quadrature error boun d of O (( log ( n ) / n ) ( 1 − A ) / 2 ) . Section 4 shows that an appr o ach b ased on finding an exten- sion ˜ f of f would not yield a be tter rate f or th is p roblem. Section 5 tr ansforms th e problem so that each triangular region b ecomes th e image o f a u n it square , with the singularity now on the bounda r y o f the squa r e. The sing ularity m ay be too se vere for QMC. However , with an addition al assump tion on the n a ture of the singu lar- ity it is possible to attain a qua drature e rror of O ( n − 1 + ε + A ) . Section 6 summarizes the finding s and relates them to QMC-f r iendliness as d iscussed b y sev e ral au thors, including Ian Sloan in his work with Xiaoq un W a n g. 2 Backgr ound In the con text of a Festschrift for Ian Sloan, we presume that the r eader is familiar with quasi-Monte Carlo , discrep a ncy and variation. Mod ern appr oaches to QMC and discrepancy are covered in [ 7]. See [1 2] for an o utline of variation for QMC including variation in th e sen ses of V itali an d of Hardy and Krause. W e will use a n o tion o f functions that are singular but no t to o badly singular . Definition 1. The functio n f d efined on [ 0 , 1 ] 2 has a diag onal singu larity no worse than | x 1 − x 2 | − A for 0 < A < 1, if | f ( x x x ) | ≤ B | x 1 − x 2 | − A     ∂ f ( x x x ) ∂ x j     ≤ B | x 1 − x 2 | − A − 1 , j ∈ { 1 , 2 } , and     ∂ 2 f ( x x x ) ∂ x j ∂ x k     ≤ B | x 1 − x 2 | − A − 2 , j , k ∈ { 1 , 2 } (1) all hold for some B < ∞ . W e take A > 0 in order to allow a sin g ularity and A < 1 because f mu st be integrable. Smaller values of A describe easier cases to handle. T he value of A to use for a given integrand may be evident fro m its a nalytical form. If A < 1 / 2 then f 2 is integrable. Definition 1 is mod eled o n some pr e vious notions: 4 Kinjal Basu and Art B. Owen Definition 2. The fun ction f ( x x x ) d efined on [ 0 , 1 ] d has a lower edg e sing ularity no worse th an ∏ d j = 1 x − A j j , for constants 0 < A j < 1, if | ∂ u f ( x x x ) | ≤ B d ∏ j = 1 x − A j − 1 j ∈ u j , holds for some B < ∞ an d all u ⊆ { 1 , 2 , . . . , d } . Definition 3. The fu nction f ( x x x ) d efined on [ 0 , 1 ] d has a point singu larity no worse than k x x x − z z z k − A , for z z z ∈ [ 0 , 1 ] d , if | ∂ u f ( x x x ) | ≤ B k x x x − z z z k − A −| u | holds for some B < ∞ an d all u ⊆ { 1 , 2 , . . . , d } . Definition 2 is one of several con ditions in [13] fo r singularities that ar ise as x x x ap- proach e s the bou ndary of the unit cube. Definition 3 is used in [ 14] fo r isolated point singularities. De fin ition 1 is mor e string ent than Definition s 2 and 3 are, because it imposes a con straint on p artial der ivati ves taken twice with r espect to x 1 or x 2 . T o estimate µ = R [ 0 , 1 ] 2 f ( x x x ) d x x x we will sample po ints x x x i ∈ [ 0 , 1 ] 2 . The poin ts we use will av oid a region n ear the sing ularity by sampling only within S ε =  x x x ∈ [ 0 , 1 ] 2   | x 1 − x 2 | ≥ ε  where 0 < ε < 1. The set S ε is the union of two disjoint triangles: T u ε =  x x x ∈ [ 0 , 1 ] 2   x 2 ≥ x 1 + ε  , and T d ε =  x x x ∈ [ 0 , 1 ] 2   x 2 ≤ x 1 − ε  . W e let − S ε denote the set [ 0 , 1 ] 2 \ S ε . As r e m arked in the introdu ction, the set T u ∪ T d is not Sobol’ extensible to [ 0 , 1 ] 2 . W e will ch o ose po ints x x x i , u ∈ T u ε for i = 1 , . . . , n an d estimate µ ε , u = R T u ε f ( x x x ) d x x x by ˆ µ ε , u = vol ( T u ε ) n n ∑ i = 1 f ( x x x i , u ) . Using a similar estimate fo r T d ε we arrive at our estimate of µ , ˆ µ ε = ˆ µ ε , u + ˆ µ ε , d . Our err or then consists of two quad rature errors an d a trun cation error an d it satisfies the bound | ˆ µ ε − µ | ≤     ˆ µ ε , u − Z T u ε f ( x x x ) d x x x     +     ˆ µ ε , d − Z T d ε f ( x x x ) d x x x     +     Z − S ε f ( x x x ) d x x x     . (2) Quasi-Monte Carlo for an Integrand with a Singularity along a Di agonal in the Square 5 3 Err or bounds W e sh ow in Prop osition 1 b elow that the truncation error bo u nd | R − S ε f ( x x x ) d x x x | is O ( ε 1 − A ) as ε → 0. W e will use th e construction from [1] and the K oksma-Hlawka inequality from [4] to provide an upp er bou nd for the integration error over T u ε . That bound grows as ε → 0 and so to trade them off we will tun e the way ε depend s on n . Proposition 1. Und e r the re gularity cond itio ns (1) ,     Z − S ε f ( x x x ) d x x x     ≤ 2 B ε 1 − A 1 − A . Pr oof. W e take the absolute value inside the integral an d obtain Z − S ε | f ( x x x ) | d x x x ≤ Z − S ε B | x 1 − x 2 | − A d x x x ≤ B Z 1 0 2 Z ε 0 x − A 2 d x 2 d x 1 from which the conc lu sion fo llows. ⊓ ⊔ Next we tur n to the quadr ature er r ors over T u ε . Of course, T d ε is similar . The K o k sma-Hlawka boun d in [4] has | ˆ µ ε , u − µ ε , u | ≤ D ∗ T u ε ( x x x 1 , u , . . . , x x x n , u ) V T u ε ( f ) where D ∗ T u , ε and V T u ε are measures of discrep a n cy and variation suited to the triangle. Basu an d Owen [1] p r ovide a constructio n in which D ∗ T u ε = O ( log ( n ) / n ) , the best possible rate. Brandolini et al. [4, p. 46] p rovide a b ound fo r V T u ε , th e variation on the simp lex as specialized to the triangle. T o translate their bou n d into our setting, we intro d uce the notation f rs = ∂ r + s f / ∂ r x 1 ∂ s x 2 . Specializing their bou nd to the do main T u ε we find that the variation is O  | f ( 0 , 1 ) | + | f ( 0 , ε ) | + | f ( 1 − ε , 1 ) | + Z 1 ε | f ( 0 , x 2 ) | d x 2 + Z 1 − ε 0 | f ( x 1 , 1 ) | d x 1 + Z 1 − ε 0 | f ( x 1 , x 1 + ε ) | d x 1 + Z 1 ε | f 01 ( 0 , x 2 ) | d x 2 + Z 1 − ε 0 | f 10 ( x 1 , 1 ) | d x 1 (3) + Z 1 − ε 0 | f 10 ( x 1 , x 1 + ε ) | d x 1 + Z 1 − ε 0 | f 01 ( x 1 , x 1 + ε ) | d x 1 + Z T u ε | f ( x x x ) | + | f 01 ( x x x ) | + | f 10 ( x x x ) | + | f 20 ( x x x ) | + | f 02 ( x x x ) | + | f 11 ( x x x ) | d x x x  as ε → 0 . Th e implied constant in (3) includes their unknown co nstant C 2 , the recip- rocals o f ed ge len g ths of T u ε , the rec ip rocal of the area of T u ε , some small integers and some factors inv olvin g √ 2 ( 1 − ε ) , the length of the hypo te n use of T u ε . 6 Kinjal Basu and Art B. Owen Proposition 2. Let f satisfy the re gularity conditio n (1) . Then the trapezoidal vari- ation of f over T u ε satisfies V T u ε ( f ) = O ( ε − A − 1 ) as ε → 0 . Pr oof. Under conditio n (1), | f ( 0 , 1 ) | + Z 1 ε | f ( 0 , x 2 ) | d x 2 + Z 1 − ε 0 | f ( x 1 , 1 ) | d x 1 + Z T u ε | f ( x x x ) | = O ( 1 ) . Next | f ( 0 , ε ) | + | f ( 1 − ε , 1 ) | + Z 1 − ε 0 | f ( x 1 , x 1 + ε ) | d x 1 = O ( ε − A ) and Z 1 ε | f 01 ( 0 , x 2 ) | d x 2 + Z 1 − ε 0 | f 10 ( x 1 , 1 ) | d x 1 = O ( ε − A ) as well. Continuing thro ugh the term s, we find that Z 1 − ε 0 | f 10 ( x 1 , x 1 + ε ) | d x 1 + Z 1 − ε 0 | f 01 ( x 1 , x 1 + ε ) | d x 1 = O ( ε − A − 1 ) . The remainin g terms are integrals of absolute partial derivati ves of f over T u ε . They are domina te d by integrals of second derivati ves and those terms obey the boun d Z 1 − ε 0 Z 1 x 1 + ε B 2 | x 1 − x 2 | − A − 2 d x 2 d x 1 = O ( ε − A − 1 ) . ⊓ ⊔ Theorem 1. Under the r e g ularity con ditions (1) , we may choose ε ∝ p log ( n ) / n and get | ˆ µ − µ | = O  log ( n ) n  ( 1 − A ) / 2  . (4) Pr oof. From Propositions 1 and 2 we g et | ˆ µ − µ | = O  ε 1 − A + log ( n ) n ε − 1 − A  . T aking ε to be a p ositi ve multiple of p log ( n ) / n yields the result. ⊓ ⊔ The choice of ε ∝ p log ( n ) / n optimizes the up per b ound in (4 ). Quasi-Monte Carlo for an Integrand with a Singularity along a Di agonal in the Square 7 4 Extension based approaches Another approa c h to this pr oblem is to construc t a fu nction ˜ f where ˜ f ( x x x ) = f ( x x x ) for x x x ∈ S ε and app ly QMC to ˜ f . The fu nction ˜ f can smoothly br idge the gap betwee n T u ε and T d ε . W ith suc h a f unction, the quad r ature error satisfies     1 n n ∑ i = 1 ˜ f ( x x x i ) − Z [ 0 , 1 ] 2 f ( x x x ) d x x x     ≤ D ∗ n ( x x x 1 , . . . , x x x n ) V H K ( ˜ f ) + Z − S ε | f ( x x x ) − ˜ f ( x x x ) | d x x x (5) where V H K is total variation in the sense o f Hardy and Krause. Our regu larity condition (1 ) allows for f to take the value ε − A along the line x 2 = x 1 − ε and to take th e value − ε − A along x 2 = x 1 + ε . By placing squares of side 2 ε along the main diagonal we then find that the V itali variation of an extension ˜ f is at lea st ⌊ ( 2 ε ) − 1 ⌋ 2 ε − A ∼ ε − 1 − A . Therefo re th e Hardy -Krause variation of ˜ f grows at least this quickly for som e of the function s f that satisfy (1). More generally , for singular function s along a linear ma nifold M with in [ 0 , 1 ] d , and no worse than dist ( x x x , M ) − A , an extension over the r egion within ε of M cou ld hav e a variation lower bou nd growing as fast as ε − ( d − 1 ) − A . This result is muc h less fa vorable than the one fo r isolated point singularities [14]. For integrands on [ 0 , 1 ] d no worse than k x x x − z z z k − A , where z z z ∈ [ 0 , 1 ] d , Sobol’ s low variation extensio n yields a fun ction ˜ f that agr ees with f for k x x x − z z z k ≥ ε > 0 having V H K ( ˜ f ) = O ( ε − A ) . Here we see that n o extension can have suc h low variation for this type of singu larity . Owen [1 3] considers fu nctions with singu larities along th e lower boundar y o f [ 0 , 1 ] d that are n o worse than ∏ d j = 1 x − A j j . Sobo l’ s extension from the region wher e ∏ j x j ≥ ε has variation O ( ε − max A j ) wh en the A j are d istinct (othe rwise log arithmic factors en ter). So that pr oblem with singularities along the bound ary also has a m ore accurate extension th an can be obtained for singular ities alon g the dia g onal. No extension ˜ f from S ε to [ 0 , 1 ] 2 can yie ld a bound (5 ) with a b etter rate than O (( log n / n ) ( 1 − A ) / 2 ) . T o sh ow this we first c larify on e of the rules we impose o n extensions. When we extend f from x x x ∈ S to values of x x x 6∈ S we do not a llow the construction of ˜ f to depend on f ( x x x ) f or x x x 6∈ S . That is, we cannot peek o utside the set we are exten ding from . Some such rule must be necessary or we could tri vially get 0 error from an extension based on an oracle th at uses the value of µ to de fin e ˜ f . W ith our rule, any two functions f 1 and f 2 with f 1 ( x x x ) = f 2 ( x x x ) on S ε have the same extension ˜ f . Fro m the triangle inequality , max j = 1 , 2  Z − S ε | ˜ f ( x x x ) − f j ( x x x ) | d x x x  ≥ 1 2 Z − S ε | f 1 ( x x x ) − f 2 ( x x x ) | d x x x . Now let f 1 ( x x x ) = ( −| x 1 − x 2 | − A , x 2 − x 1 > 0 | x 1 − x 2 | − A , x 2 − x 1 < 0 , 8 Kinjal Basu and Art B. Owen and f 2 ( x x x ) =      | x 1 − x 2 | − A , x 2 − x 1 > 0 φ ( x 2 − x 1 ) , 0 > x 2 − x 1 ≥ − ε | x 1 − x 2 | − A , − ε > x 2 − x 1 , for a quad ratic poly nomial φ with φ ( − ε ) = ε − A , φ ′ ( − ε ) = − A ε − A − 1 , and φ ′′ ( − ε ) = A ( A + 1 ) ε − A − 2 . Both f 1 and f 2 satisfy (1) and R − S ε | f 1 ( x x x ) − f 2 ( x x x ) | d x x x is larger tha n a constan t times ε 1 − A . T hat is the same rate as the truncation error f rom Pr o posi- tion 1 and the q uadrature er ror from this app roach also attains the same rate as the error in Proposition 2. As a result, we conclu d e that ev en if we co uld construct th e best exten sion ˜ f , it would not lead to a b ound with a better r ate tha n the one in Theorem 1. 5 T ransf ormation Here we consider applyin g a chan ge of variable to m ove the singularity fro m the diagonal to an ed ge o f the u nit squ a re. W e f o cus on integratin g f ( x x x ) over T u = { ( x 1 , x 2 ) ∈ [ 0 , 1 ] 2 | 0 ≤ x 1 ≤ x 2 ≤ 1 } for f with a sing ularity n o worse than | x 1 − x 2 | − A . The same strategy an d sam e c o n vergen c e rate hold on T d = { ( x 1 , x 2 ) ∈ [ 0 , 1 ] 2 | 0 ≤ x 2 ≤ x 1 ≤ 1 } . Using a standa r d ch a n ge o f variable we have Z T u f ( x x x ) d x x x = 1 2 Z 1 0 Z 1 0 f (( 1 − u 1 ) √ u 2 , √ u 2 ) d u u u , which we then write as 1 2 Z [ 0 , 1 ] 2 g ( u u u ) d u u u , for g ( u u u ) = f (( 1 − u 1 ) √ u 2 , √ u 2 ) . That is g ( u u u ) = f ( τ ( u u u )) for a transformation τ : [ 0 , 1 ] 2 → T u ⊂ [ 0 , 1 ] 2 giv en by τ 1 ( u u u ) = ( 1 − u 1 ) √ u 2 and τ 2 ( u u u ) = √ u 2 . The archetypal function with diagonal sin g ularity satisfying Definition 1 is f ( x x x ) = | x 1 − x 2 | − A . The correspo nding fun c tion g fo r this f is g ( u u u ) = | τ 1 ( u u u ) − τ 2 ( u u u ) | − A = u − A 1 u − A / 2 2 . W e see that the change of variable has produ ced an integrand with a singu la r ity no worse than u − A 1 u − A / 2 2 accordin g to Definition 2. T ak ing u u u i to be the Halton poin ts leads to a quadratu r e err or at rate O ( n − 1 + ε + A ) for any ε > 0, because Halton points (after th e zeroth one) avoid the orig in at a suitable rate [13, Corollary 5 .6]. For th is integrand g , rand omized qua si-M onte Carlo points for will attain the m ean erro r rate E ( | ˆ µ − µ | ) = O ( n − 1 + ε + A ) as shown in Theo rem 5 .7 o f [ 13]. Quasi-Monte Carlo for an Integrand with a Singularity along a Di agonal in the Square 9 W e initially thought that the conv ersion f rom a diago nal singu larity to a lo wer edge sing ularity n o worse than u − A 1 u − A / 2 2 would follow for o ther function s satisfying Definition 1. Unfo rtunately , that is not nec e ssarily the c a se. Let f be defined o n [ 0 , 1 ] 2 with a diagona l sing ularity no worse than | x 1 − x 2 | − A for 0 < A < 1. First, | g ( u u u ) | = | f (( 1 − u 1 ) √ u 2 , √ u 2 ) | ≤ B | u 1 u 1 / 2 2 | − A which fits Definition 2. Similarly , g 10 ( u u u ) = f 10 ( τ 1 ( u u u ) , τ 2 ( u u u )) ∂ τ 1 ( u u u ) ∂ u 1 = O ( | τ 1 − τ 2 | − A − 1 ) × u 1 / 2 2 = O ( u − A − 1 1 u − A / 2 2 ) which also fits Definition 2. Howev er , g 01 ( u u u ) = f 10 ( τ ( u u u )) ∂ τ 1 ( u u u ) ∂ u 2 + f 01 ( τ ( u u u )) ∂ τ 2 ( u u u ) ∂ u 2 =  f 10 ( τ ( u u u )) + f 01 ( τ ( u u u ))  1 2 u − 1 / 2 2 − f 10 ( τ ( u u u )) 1 2 u 1 u − 1 / 2 2 . (6) Now f 10 and f 01 appearin g in (6 ) are b oth O ( u − A − 1 1 u − A / 2 − 1 / 2 2 ) . T herefore the two terms ther e ar e O ( u − A − 1 1 u − A / 2 − 1 2 ) an d O ( u − A 1 u − A / 2 − 1 2 ) resp ecti vely . The first term is too large by a factor of u − 1 1 to suit Definition 2. W e would need ( f 01 + f 10 )( τ ( u u u )) to be only O ( u − A 1 u − A / 2 − 1 / 2 2 ) . Definition 1 is also not strong en ough for g 11 to be O ( u − A − 1 1 u − A / 2 − 1 2 ) as it would need to be u nder Defin ition 2. That definitio n yields only O ( u − A − 2 1 u − A / 2 − 1 2 ) without stro nger assum ptions. Th eorem 2 below gives a suf- ficient condition where f is a modulated version of | x 1 − x 2 | − A . Theorem 2. Let f ( x x x ) = | x 1 − x 2 | − A h ( x x x ) for x x x ∈ [ 0 , 1 ] 2 and 0 < A < 1 wher e h a n d its first two de rivatives ar e bounded . Then g ( u u u ) = f (( 1 − u 1 ) √ u 2 , √ u 2 ) satisfies Definition 2 with A 1 = A and A 2 = A / 2 . Pr oof. W e begin with g ( u u u ) = u − A 1 u − A / 2 2 h (( 1 − u 1 ) u 1 / 2 2 , u 1 / 2 2 ) = O ( u − 1 1 u − A / 2 2 ) by boun dedness of h . Next because u 1 is not in the second argu m ent to h , g 10 ( u u u ) = − Au − A − 1 1 u − A / 2 2 h ( τ ( u u u )) + u − A 1 u − A / 2 2 h 10 ( τ ( u u u )) ∂ τ 1 ( u u u ) / ∂ u 1 = − Au − A − 1 1 u − A / 2 2 h ( τ ( u u u )) − u − A 1 u − A / 2 + 1 / 2 2 h 10 ( τ ( u u u )) = O ( u − A − 1 1 u − A / 2 2 ) as requir e d. Similarly , 10 Kinjal Basu and Art B. Owen g 01 ( u u u ) = − ( A / 2 ) u − A 1 u − A / 2 − 1 2 h ( τ ( u u u )) + u − A 1 u − A / 2 2  h 10 ( τ ( u u u ))( 1 − u 1 ) + h 01 ( τ ( u u u ))  ( 1 / 2 ) u − 1 / 2 2 = O ( u − A 1 u − A / 2 − 1 2 ) as requir e d. Finally g 11 ( u u u ) equa ls ( A 2 / 2 ) u − A − 1 1 u − A / 2 − 1 2 h ( τ ( u u u )) − ( A / 2 ) u − A 1 u − A / 2 − 1 2 h 10 ( τ ( u u u ))( − u 1 / 2 2 ) − ( A / 2 ) u − A − 1 1 u − A / 2 − 1 / 2 2  h 10 ( τ ( u u u ))( 1 − u 1 ) + h 01 ( τ ( u u u ))  + ( u − A 1 u − A / 2 − 1 / 2 2 / 2 )  − h 10 ( τ ( u u u )) + ( 1 − u 1 ) h 20 ( τ ( u u u ))( − u 1 / 2 2 ) + h 11 ( τ ( u u u ))( − u 1 / 2 2 )  = O ( u − A − 1 1 u − A / 2 − 1 2 ) as requir e d. ⊓ ⊔ 6 Discussion W e fin d that for an integrand with a singu larity ‘n o worse than | x 1 − x 2 | − A ’ alon g the line x 1 = x 2 we c a n get a QMC e stimate with error O (( log ( n ) / n ) ( 1 − A ) / 2 ) by splitting the sq uare in to two triangles and ignor ing a region in betwe en them . The same meth od applies to singular ities alo n g the o ther diago nal o f [ 0 , 1 ] 2 . Moreover , the result extends to singularities alon g other lines intersecting the square . On e can partition the squ are into rectang les, o f which one h as the sing ularity alo ng the d i- agonal while the o thers have no singularity , and th en integrate f over each of tho se rectangles. That result d oes not directly extend to singularities along a linear ma nifold in [ 0 , 1 ] d for d ≥ 3. The rea son is that the QMC result for integration in the trian gle from [1] has not be e n extended to the simplex. In a p ersonal co mmuncatio n , Dimitry Bilyk told us that such an extension would imp ly a coun terexample to the Littlew ood conjecture , which is widely b eliev e d to be tr ue. Basu and Owen [2] present some algorithm s for RQMC over simplices, but they co me with out a K oksma-Hlawka bound that would be require d for limiting argum ents using sequen c e s of simplices. The rate O (( log ( n ) / n ) ( 1 − A ) / 2 ) is a bit disappointin g. W e do much better b y tra ns- forming the p roblem to place the sing ularity a lo ng the bo undary o f a square re- gion, for then we c an attain O ( n − 1 + ε + A ) , un der a stronger assumption that f is our prototy p ical singular fun ction | x 1 − x 2 | − A possibly modula ted by a functio n h with bound ed second deriv atives on [ 0 , 1 ] 2 . A s a resu lt we find that the r e is someth ing to be gained by enginee r ing QMC-f riendly singularities in m uch the same way tha t benefits of QMC-friendly discon tinuities have been foun d valuable by W ang an d Sloan [16]. Quasi-Monte Carlo for an Integrand with a Singularity along a Di agonal in the Square 11 Acknowled gements This work was supported by the US National Science Foundation under grants DMS-14073 97 and DMS-1521145. W e t hank two anon ymous rev iewe rs for helpful com- ments. Refer ences 1. Basu, K., Owen, A. B.: Low discrepanc y constru ctions in the triangle. 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