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The Astrophysical Journal, 741:42 (8pp), 2011 November 1  C 2011.

doi:10.1088/0004-637X/741/1/42

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NEW DEVELOPMENTS ON INVERSE POLYGON MAPPING TO CALCULATE GRAVITATIONAL LENSING MAGNIFICATION MAPS: OPTIMIZED COMPUTATIONS ˜ 4 , O. Ariza3 , P. Lopez1 , C. Gonzalez-Morcillo5 , E. Mediavilla1,2 , T. Mediavilla3 , J. A. Munoz and J. Jimenez-Vicente6,7

1 Instituto de Astrof´ısica de Canarias, V´ıa L´ actea S/N, 38200 La Laguna, Tenerife, Spain Departamento de Astrof´ısica, Universidad de La Laguna, 38200 La Laguna, Tenerife, Spain 3 Departamento de Estad´ıstica e Investigaci´ on Operativa, Universidad de C´adiz, Avda Ram´on Puyol s/n, 11202, Algeciras, Cadiz, Spain 4 Departamento de Astronom´ıa y Astrof´ısica, Universidad de Valencia, 46100 Burjassot, Valencia, Spain 5 Escuela Superior de Informatica, Universidad de Castilla-La Mancha, Paseo de La Universidad 4, 13071 Ciudad Real, Spain 6 Dpto. de Fisica Teorica y del Cosmos, Campus de Fuentenueva, Universidad de Granada, 18071 Granada, Spain 7 Instituto Carlos I de Fisica Teorica y Computacion, Granada, Spain Received 2011 May 6; accepted 2011 July 29; published 2011 October 13 2

ABSTRACT We derive an exact solution (in the form of a series expansion) to compute gravitational lensing magnification maps. It is based on the backward gravitational lens mapping of a partition of the image plane in polygonal cells (inverse polygon mapping, IPM), not including critical points (except perhaps at the cell boundaries). The zeroth-order term of the series expansion leads to the method described by Mediavilla et al. The first-order term is used to study the error induced by the truncation of the series at zeroth order, explaining the high accuracy of the IPM even at this low order of approximation. Interpreting the Inverse Ray Shooting (IRS) method in terms of IPM, we explain the previously reported N −3/4 dependence of the IRS error with the number of collected rays per pixel. Cells intersected by critical curves (critical cells) transform to non-simply connected regions with topological pathologies like autooverlapping or non-preservation of the boundary under the transformation. To define a non-critical partition, we use a linear approximation of the critical curve to divide each critical cell into two non-critical subcells. The optimal choice of the cell size depends basically on the curvature of the critical curves. For typical applications in which the pixel of the magnification map is a small fraction of the Einstein radius, a one-to-one relationship between the cell and pixel sizes in the absence of lensing guarantees both the consistence of the method and a very high accuracy. This prescription is simple but very conservative. We show that substantially larger cells can be used to obtain magnification maps with huge savings in computation time. Key words: gravitational lensing: micro plane using inverse lens mapping, and the magnification of each source-plane pixel is made proportional to the number of points (“rays”) that hit the pixel. It is implicitly supposed that each ray carries the area of one of the primitive cells of the image-plane lattice.8 When a transformed cell intersects more than 1 pixel (typically when the ray hits close to the pixel boundaries), its area should be adequately apportioned among the intersected pixels. IRS, however, assigns the entire cell area to the pixel where the ray hits inducing noise and, hence, working properly only in the limit of many rays per pixel (very small cells at the image-plane lattice), typically of the order of hundreds (see Bate et al. 2010, for example). To solve this problem, Mediavilla et al. (2006) proposed a technique (based on inverse polygon mapping, IPM) for exactly apportioning the area of each cell among the image-plane pixels covered by the transformed cell. With this method, the number of cells9 can be drastically reduced (even to 1 per unlensed pixel or less) and the accuracy strongly improved. An example of the massive calculations of magnification maps that can be afforded with IPM with very basic computing resources can be found in Mediavilla et al. (2009), where ∼600 magnification maps of 2000 × 2000 pixels were evaluated. More than a closed algorithm, IPM is a technique applied by Mediavilla et al. (2006)

1. INTRODUCTION Quasar microlensing (Chang & Refsdal 1979, 1984; see also Kochanek 2004 and Wambsganss 2006) is one of the most powerful tools for studying the composition of matter in the halos of lens galaxies and the unresolved structure of active galactic nuclei (AGNs). The main element in studying quasar microlensing is the magnification map, which, for a random configuration of microlenses in the lens galaxy, gives the flux magnification as a function of source position. With a magnification map, we can simulate the variations induced by microlensing of the flux of a lensed quasar or AGN and make statistical estimates of the properties of the microlens distribution (e.g., Morgan et al. 2008; Pooley et al. 2009; Mediavilla et al. 2009) or concerning the unresolved source structure (accretion disk (e.g., Pooley et al. 2007; Anguita et al. 2008; Mosquera et al. 2009; Agol et al. 2009; Bate et al. 2008; Floyd et al. 2009; Morgan et al. 2010; Poindexter et al. 2008; Dai et al. 2010; Blackburne et al. 2011; Mediavilla et al. 2011) or BLR (e.g., Abajas et al. 2002, 2007; Lewis & Ibata 2004)). Although there are several techniques for computing magnification maps (Schneider et al. 1992; Lewis et al. 1993; Witt 1993; Fluke et al. 1999; Wyithe & Webster 1999), the classical method is inverse ray shooting (IRS; Kayser et al. 1986; Schneider & Weiss 1987; Wambsganss 1990; Bate et al. 2010; Thompson et al. 2010), often provided with a hierarchical tree code (e.g., Wambsganss 1999; Kochanek 2004; Bate et al. 2010; Garsden & Lewis 2010). In this method, a congruent lattice of points in the image plane is backwardly transported to the source

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The magnification of a source (a pixel in a magnification map, for instance) under the action of gravitational lensing is simply the sum of the areas of the images of the source divided by the area of the source (see Schneider et al. 1992, for example). 9 For a congruent lattice of primitive cells the number of cells (i.e., “rays”) is equal to the number of points in the lattice.

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to the computation of magnification maps susceptible of further improvement in at least two ways. On the one hand, it is based on a linear approximation of the lens equation and does not account for the magnification gradients within each cell. On the other hand, IPM needs to deal with cells intersected by critical curves that are reprocessed using IRS or other alternative treatments. The first objective of this paper, then, is to formalize and extend IPM beyond the approximation of Mediavilla et al. (2006) and give a rigorous treatment of the critical cells. From a practical point of view (see Bate et al. 2010), the main limitation of microlensing simulations of AGN is the calculation of statistically significant microlensing maps with both sufficient spatial resolution and size to simultaneously study the UV emitting region (∼1 light-days of size) and the BLR (∼100 lightdays of size).10 In this paper, we show how the huge savings in computing time that can be attained by optimizing the cell size according to our new developments on IPM can be decisive in performing these challenging calculations.

2.2. Taylor Expansion of the Magnification Let us consider in the source plane a simply connected region of uniform brightness, P. A point in this region, y, can have several images, xi , f −1 : P → Ii , y → xi .

According to Liouville’s theorem, the magnification of P is simply the ratio of the total area of the images divided by the area of the source (in dimensionless units, see Schneider et al. 1992),    SI 1 μP = i i =  dx 1 dx 2 , (5) 1 dy 2 SP dy Ii P i where i runs over each of the images of the source, Ii . This expression is therefore not very useful because to know the magnification at a given region of the source plane we need to know all the images of this region; that is, we need to solve the lens equation. To make operative Equation (5) we are going to give a suitable approach based on the partition of the image plane. Let us consider a collection of open sets Ui (cells) that  cover the image plane, Ui = R 2 , and are pairwise disjoints, Ui ∩ Uj = ∅, ∀ i, j . Let us consider the inverse lens mapping of this image-plane partition (tessellation):

2. MAGNIFICATION OF A SOURCE OF UNIFORM BRIGHTNESS FROM A PARTITION OF THE IMAGE PLANE 2.1. Mathematical Preliminaries In gravitational lensing a source should have at least one image but can have more than one. To avoid the problems of multi-valuation, the lens equation can be written as a surjective map (i.e., all the points in the source plane have at least one image) from the image plane, X, to the source plane, Y, f : X → Y, x → y = x − ∇ψ( x ).

f : Uk → Vk , x → y.

x → y,

If it is possible to do the partition leaving the critical points outside the open sets, Equation (6) defines diffeomorphisms and we can apply the change of variables theorem,   ∂(y 1 , y 2 ) −1 1 2 dy dy , (8) dx 1 dx 2 = 1 2 Uk ∩Ii Vk ∩P ∂(x , x )

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is a diffeomorphism, that is, f maps a small region in the image plane to a unique region in the source plane and this map is bijective, smooth, andinvertible. The inverse mapping of the critical points (det ∂∂ yx = 0) forms the caustic set at the source plane. On the other hand, a source-plane position, y, is the inverse image of N points (N  1) in the image plane. Thus, in an open neighborhood, V, of y, not including caustics, f −1 establishes N diffeomorphisms with disjoint open sets, Ui , in the image plane, f −1 : V → Ui ,

y → xi ,

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As far as each one of the points of the image plane is contained in one and only one of the sets of the partition, Equation (5) can be trivially rewritten in the following way:   1 μP =  dx 1 dx 2 . (7) 1 dy 2 dy Uk ∩Ii P i k

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Let us consider this map to be discontinuous only at a finite number of points and of class Ck (k times differentiable) elsewhere. Then, according to the inversefunction theorem, if  ∂ y ∂ y the Jacobian, ∂ x , is invertible (that is, det ∂ x = 0) at a point, x, then there is an open neighborhood U in the image plane containing x such that f : U → f (U ),

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and write μP =  P

  1 dy 1 dy 2 k Vk ∩P

∂(y 1 , y 2 ) −1 1 2 ∂(x 1 , x 2 ) dy dy ,

(9)

which, defining μ( y ) = |A|

(3)

where the xi are the N images of y. That is, f −1 maps a small region in the source plane into N regions in the image plane and every map is, again, bijective, smooth, and invertible. The topological properties of an image-plane region not including either points where f is discontinuous or critical points are then preserved by the inverse lens mapping. In particular, simply connected regions are mapped to simply connected regions.

can be written as μP =  P

−1

∂(y 1 , y 2 ) −1 , = ∂(x 1 , x 2 )

  1 μ( y ) dy 1 dy 2 . dy 1 dy 2 k Vk ∩P

(10)

(11)

The Taylor expansion of μ( y ) about the centroid of a given cell is  · ( μ( y ) = μ( y Vk ) + ∇μ y −  y Vk ) 1  + ∇(∇μ) · [( y −  y Vk )] + · · · , y −  y Vk ) ⊗ ( 2

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Supercomputing implementations of the IRS method have been used to deal with this kind of problems (Bate et al. 2010; Garsden & Lewis 2010).

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The Astrophysical Journal, 741:42 (8pp), 2011 November 1

where ⊗ is the tensor product and  1  y Vk = y dy 1 dy 2 SVk Vk

Mediavilla et al.

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is the centroid. SVk is the surface of the transformed cell, Vk . The  ∇(  ∇μ),...,  derivatives of μ, ∇μ, are evaluated at the centroid. Implementing the Taylor expansion (Equation (12)) in Equation (11), we obtain  

 1  · ( μP =  μ  y Vk + ∇μ y −  y Vk ) 1 2 dy dy V ∩P k P k   

 1  y Vk + · · · dy 1 dy 2 , + ∇( ∇μ) · y −  y Vk ⊗ y −  2 (14)

integrating we obtain μP =

 1    ·  y Vk μ  y Vk + ∇μ y Vk ∩P −  SP k

    1  y Vk Vk ∩P + · · · SVk ∩P . + ∇( ∇μ) · y −  y Vk ⊗ y −  2 (15)

Figure 1. Critical cells and straight line approximation to the critical curve corresponding to a binary lens. The critical cells are 0.1 Einstein radius in size. The critical curve divides each critical cell into two non-critical subcells.

This series expansion is an exact solution of μP , provided a partition without critical points within the cells is available. The values of μ and its derivatives at the centroid of a given cell can be calculated from the inverse lens mapping of the cell. It is usual to consider polygonal cells obtaining the transformed cells as the polygons defined by the transformation of the vertices of the cell.11 In Mediavilla et al. (2006), the transformation of the vertices and the center of a square centered cell  at the transformed center. The are used to compute μ and ∇μ transformations of a greater number of points per cell should be considered if higher order terms (higher derivatives of μ) are required.

the cell-averaged magnification can be written as the ratio between areas, SU μ Vk = k , (19) SVk and Equation (17) can be written as    1  SUk  SVk ∩P . (20) μ(1) = + ∇μ ·  y −  y Vk ∩P Vk P SP k SVk When the first-order term is neglected, a zeroth-order approximation is obtained:   1  SUk μ(0) SVk ∩P . = (21) P SP k SVk

2.3. First- and Zeroth-order Approximations If we limit the series expansion to first order, the cell-averaged magnification,  1 μ Vk = μ dy 1 dy 2 , (16) SVk Vk

This approximation, used by Mediavilla et al. (2006), allows very fast and accurate calculations. According to this equation, to estimate the magnification, μP , of a source-plane region of uniform brightness, P, we should add all the areas corresponding to the intersection of the mapped cells of the partition, Vk , with S P multiplied by the cell-averaged magnification, μ Vk = SUVk . k To do this we only need to compute the areas of Uk , Vk , P, and Vk ∩ P . Applying this procedure to each source-plane pixel, a zeroth-order magnification map is obtained (in practice, each cell is transformed by the inverse lens mapping and then apportioned among the pixels covered by it; see Mediavilla et al. 2006).

coincides with the value of the magnification at the centroid, μ( y Vk ), and the first-order approximation, μ(1) P , can be written as   1   ·  μ(1) μ Vk + ∇μ y Vk ∩P −  y Vk SVk ∩P . (17) P = SP k Taking into account that, according to the change of variables theorem,   ∂(y 1 , y 2 ) −1 1 2 1 2 dy dy dx dx = SUk = 1 2 Uk Vk ∂(x , x ) = μ Vk SVk , (18)

2.4. First-order Corrections: Truncation Error We can approximate the error we would obtain by truncating the series at zeroth order by evaluating the first-order term in the series expansion (Equation (20)):

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Notice, however, that straight lines are transformed into straight lines only in the linear approximation of lens mapping; in rigor, the transform of a polygon is not a polygon. Keeton (2001) shows the problems induced by the curvature of the transformed polygon sides in the search of solutions of the lens equation.

Δμ(0) P = 3

  1   ∇μ ·  y Vk ∩P −  y Vk SVk ∩P . SP k

(22)

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Figure 2. Large panel: inverse lens mapping of the non-critical subcells of Figure 1. Bottom small panel: detail of a caustic fold (notice the collapse of cells in the direction perpendicular to the caustic). Top small panel: detail of a caustic cusp (notice the distortions of the cells and their collapse in the direction perpendicular to the caustics that join at the cusp). See the text.

These first-order corrections are useful for estimating and improving the accuracy of the zeroth-order approximation. Specifically, notice that the contribution to the error of the k-esim cell is null if Vk ∩ P = Vk or if μ is constant within Vk . Thus, we can control the error by controlling the size of the cell according to the gradient of the magnification within the cell. On the other hand, by applying Equation (22) to each pixel we could evaluate a map of first-order corrections to the magnifications; adding this map to the zeroth-order map, a firstorder magnification map would be obtained.

If the size of the transformed cell, Vk , in the direction of the magnification gradient, is contained in the pixel size, the contribution to the error will then be zero. Inverse lens mapping in the neighborhood of a critical curve will act to reduce the dimension of the cell in the gradient direction and the error could be zero with an adequate selection of the size of the cell, Uk . For the particular case in which the size of the cell is selected as the size of the pixel in the absence of lens effect, this always happens. This demonstrates that the IPM method is naturally well conditioned and explains the very good results obtained with this method even with samplings of 1 cell per unlensed pixel, equivalent to a ray per unlensed pixel in the IRS method.

2.5. Accuracy Close to Critical Curves 2.6. Inverse Lens Mapping of Simply Connected Regions Including Critical Points or Points Where f is Discontinuous

Away from the critical curves, lens mapping can be seen as a linearly invertible transformation of the plane even for cells of relatively large size. However, close to critical curves the departures from linearity can be very strong. Let us consider a cell, Uk , located in the neighborhoods of a critical curve. Close to a fold, one (and only one) of the eigenvalues of the Jacobian matrix goes to zero and the inverse lens mapping collapses the cell in the direction perpendicular to the caustic (see Figures 1 and 2). In addition to the collapse the cell may also be rotated and sheared. Close to a cusp, the inverse lens transformation is clearly nonlinear; straight lines bend and parallelism is not preserved. Nevertheless, also in this case the cells collapse in the direction perpendicular to the caustics that join at the cusp (see Figure 2). According to the discussion of Section 2.4, the contribution of a cell to the error in the zeroth-order determination of the magnification of a pixel (see Equation (22)) depends on   ·  ∇μ y Vk ∩P −  y Vk .

The previous derivation of the Taylor expansion of the magnification is only valid if the cells are simply connected regions transformed by diffeomorphisms. However, at discontinuities and critical points, inverse lens mapping is not a diffeomorphism. Let us consider in the first case the transformation of a simply connected region crossed by a critical curve. In this case, the transformed region will not be simply connected. This can have significant consequences. For instance, the boundary of a region crossing a critical curve will not be transformed in the boundary of the transformed region. To consider another consequence, let us think of this region as separated into two simply connected disjoint subregions limited by the critical curve. The transformed subregions must then also be simply connected. However they can intersect. Both consequences (non-preservation of the boundary and auto-overlapping) demonstrate that dealing with regions crossed by a critical curve is not immediate

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each one of the intersecting cells falls in a given pixel (of each contiguous pair) is 1/2, the expected value of the number of centers of the intersecting cells (rays in the √ boundary) falling in √ a given pixel is ∼2 N and the variance ∼ N . Thus, the error in the magnification of a pixel with N rays will be 1

N4 Δμ(N ) 3 ∼ = N−4 , μ(N ) N

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which coincides with the experimental result found by Kayser et al. (1986) and Mediavilla et al. (2006). In the IRS, the number of rays shot per unlensed pixel is usually a few tens. This guarantees the good behavior of this method at the caustics (where the concentration of rays collected by pixel, N, is high). In contrast, there are regions of relatively low magnification where the cell area is not collapsed but expanded and the IRS will tend to be less accurate. Finally, IRS has the same problems as IPM in dealing with critical cells. The rays closest to the critical curves carry areas that include critical points and are transformed into non-simply connected regions in the source plane. Obviously, this is not taken into account in the IRS algorithm, which simply adds all the cell area to the pixel where the ray hits. In any case, the typical separation between rays (size of the cells) used in IRS is so small that the erroneous contribution of the critical cells passes unnoticed.

Figure 3. Detail of Figure 2 zooming the transformed subcells of one critical cell. One of the subcells is plotted with a solid line and the other with a dashed line except in the common side (the caustic). Notice the overlapping of the transformed subcells, or, in other words, the auto-overlapping of the transformed critical cell.

4. DEFINING A NON-CRITICAL PARTITION

(see Figure 3). On the other hand, in so far as the inverse lens mapping is continuous in the whole region, path connectedness is preserved. Thus, the two transformed subregions cannot be disconnected. In the second case, we consider a simply connected region including a point where f is discontinuous. Formally, we must avoid the discontinuity excluding the point and an infinitesimal neighborhood from the partition. In practice, this can be done simply by considering that the cell enclosing the point is small enough to be transformed outside the magnification map. In the third case, we can have a simply connected region including an inner closed critical curve. This can occur when we are defining a relatively large cell that includes a discontinuity. The sampling around a discontinuity should then be adapted to resolve the critical curve around it (of 1 Einstein radius of scale size). In practice, this can be relevant only for very small mass microlenses (see Section 4.3).

The basic steps in the computation of magnification maps using the IPM outlined in Mediavilla et al. (2006) are (1) partition of the image plane using a periodic lattice (conventional square centered cells, for instance), (2) transformation of the lattice with the lens equation, (3) critical cell identification (using the Δ|A|/|A| criterion; Δ|A| is the maximum variation of |A| within the cell boundaries), (4) critical cells reprocessing, and (5) apportioning of the cell area among the source-plane pixels covered by the transformed cells (that is, applying Equation (21)). Mediavilla et al. (2006) use IRS to reprocess critical cells (step (4)). Here we propose a more consistent alternative based on the definition of a non-critical partition. 4.1. Dividing the Critical Cells Using a Linear Approximation If the size of a critical cell is small enough, the critical curve included in the cell can be approximated by a straight line that will divide the critical cell into two polygonal non-critical subcells. As a first-order approximation, the variation of |A| with respect to its value at the cell centroid, |A|( x Uk ), is

3. IRS AS AN APPROXIMATION OF IPM: IRS ACCURACY IRS is like an IPM method in which the entire area of the image-plane cell is carried by the transformed center of the cell (the ray). Specifically, the IRS approximation consists in taking SVk ∩P = SVk in Equation (21) if the transformed center of the cell falls within the pixel and zero otherwise. This introduces fluctuations into the computation of the contribution of the mapped cells intersecting more than 1 pixel and, obviously, IRS can work properly only if the size of the transformed cell is much smaller than the pixel size. In this case, we can use Equation (22) to estimate the error of a magnification map computed using IRS. Let us consider a pixel collecting N rays (N transformed cell centers). According to Equation (22) the contribution to the error of the transformed cells inside to the pixel is zero. The number of rays (transformed cell centers) in the boundary corresponding to transformed cells that intersect adjacent pixels √ is ∼4 N . Assuming that the probability of that the center of

   |A|( x ) = |A|  x Uk + ∇|A| · x −  x Uk . The equation of the critical curve will then be    |A|  x Uk + ∇|A| · x −  x Uk = 0.

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The two non-critical polygonal subcells are found by calculating the intersections of this straight line with the cell sides. Reprocessing all the critical cells in this way, we construct a noncritical partition and give a consistent solution (as a first-order approximation) to the problem of calculating a magnification map. The first-order approximations to the critical curves in each cell will trace the critical curves, and its transformation under the 5

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Table 1 Magnification Map Parameters Cells Number

Cell Size (in Einstein Radii)

Cell Area (in pixels)

σ

3200 × 3200 1600 × 1600 800 × 800 400 × 400 200 × 200

0.0056 0.0113 0.0225 0.045 0.09

0.88 3.51 14.06 55.56 227.27

0.001 0.003 0.007 0.018 0.044

lens equation will give the caustic curves at the source plane that can be of great interest for many studies. Critical curves can be traced in a less accurate but simpler way by the (centers of the) critical cells. Notice, however, that caustics are not adequately traced by the inverse lens transformation of the (centers of the) critical cells as the transformed cells are not simply connected. In Figure 1, we show the critical cells corresponding to a binary lens (for this demonstration we have used very large cells of size 0.1 Einstein radii). Each critical cell is divided into two non-critical subcells by the local straight line approximation to the critical curve. In Figure 2 we present the transformed subcells in the source plane, which, in spite of the rough spatial resolution of the image-plane partition, are nicely disposed following the caustic curve.

Figure 4. Histograms of the magnification maps corresponding to cell sizes of 0.006 (continuous line) and 0.1 (dashed line) Einstein radii (see Table 1).

4.2. Processing Cells with Discontinuities

with IRS (which typically shoots hundreds of rays per pixel to achieve a significantly lower accuracy than IPM; see Mediavilla et al. 2006) but can be significantly relaxed in many applications. To explore the upper limit of cell size that can be reasonably adopted let us consider a typical problem, the computation of a magnification map of 6 Einstein radii of side for a distribution of 1 M microlenses with κ = 0.5 (γ = 0). If the map has 1000 × 1000 pixels the spatial resolution will be 0.006 Einstein radii per pixel, which, for a lensed quasar system with zl = 0.5 and zs = 2, corresponds to 0.12 light-days per pixel, enough to study the quasar accretion disk. We have computed magnification maps for different cell sizes ranging from a map that we will consider “exact” (with 4 cells per pixel in the absence of microlensing or lUk ∼ 0.0015) to a limiting case (lUk ∼ 0.1). The map parameters and the standard deviations with respect to the “exact” map are presented in Table 1. As can be seen in this table, the relative differences are less than 1% for lUk  0.02 reaching 4% in the limiting case, lUk ∼ 0.1. This difference is, in fact, irrelevant for estimating the probability density function of microlensing magnification (a typical application) that, according to Figure 4, shows no significant differences between the lUk ∼ 0.006 (approximately corresponding to the one-to-one relationship between cell and pixel sizes) and the lUk ∼ 0.1 cases. However, the number of cells to process diminishes by a factor 256. It is interesting to note that similar low dispersions with respect to the “exact” map are found even if one uses the plain IPM method without a noncritical partition (Mediavilla et a. 2006). This method is, then, good enough12 to be used with large cells to study magnification probability (although it should be used with small cells to study the environments of caustics). In many applications the magnification map is smoothed with source profiles of different sizes (e.g., to compare the effects of microlensing in the quasar continuum source and in the BLR). Thus, although for the smallest source considered the

We can use the cell size to avoid the discontinuities. If the cell size is small enough, any cell including a discontinuity will be mapped outside the magnification map. If Ly is the size of the magnification map and lUk is the cell size, both expressed in Einstein radius units for a solar mass microlens, the cell will be mapped outside the limits if Ly