PLURIPOTENTIAL NUMERICS

Transcription

1 PLURIPOTNTIAL NUMRICS FDRICO PIAZZON Abstract. We introduce numerical methods for the approximation of the main (global) quantities in Pluripotential Theory as the extremal plurisubharmonic function V of a compact L-regular set Cn, its transfinite diameter δ(), and the pluripotential equilibrium measure µ := ( dd c V ) n. The methods rely on the computation of a polynomial mesh for and numerical orthonormalization of a suitable basis of polynomials. We prove the convergence of the approximation of δ() and the uniform convergence of our approximation to V on all Cn ; the convergence of the proposed approximation to µ follows. Our algorithms are based on the properties of polynomial meshes and Bernstein Marov measures. Numerical tests are presented for some simple cases with R 2 to illustrate the performances of the proposed methods. Contents. Introduction 2 2. Preliminaries Pluripotential theory: some definitions Admissible meshes and Bernstein Marov measures 8 3. Approximating the extremal function 3.. Theoretical results 3.2. The SZF and SZF-BW algorithms Numerical Tests of SZF and SZF-BW 7 4. Approximating the transfinite diameter Theoretical result Implementation of the TD-GD algorithm Numerical test of the TD-GD algorithm Approximating the equilibrium measure 30 Acnowledgements 33 References 33 Date: April, Mathematics Subject Classification. MSC 6505 and MSC 4A0 and MSC 32U35 and MSC 42C05. Key words and phrases. Pluripotential theory, Orthogonal Polynomials, admissible meshes.

2 2 FDRICO PIAZZON. Introduction Let C be a compact infinite set. Polynomial interpolation of holomorphic functions on and its asymptotic are intimately related with logarithmic potential theory, i.e., the study of subharmonic function of logarithmic growth on C. This is a well established classical topic whose study goes bac to Bernstein, Feete, Leja, Szegö, Walsh and many others; we refer the reader to [37], [47] and [4] for an extensive treatment of the subject. To study polynomial interpolation on a given compact set one introduces the Vandermonde determinant (usually with respect to the monomial basis) and, for any degree N, tries to maximize its modulus among the tuples of + distinct points. Any such array of points is termed Fete array of order for. A primary interest on Feete arrays is that one immediately has the bound Λ (z 0,..., z ) := sup z I [ f ](z) sup ( + ) f C (), f 0 f for the Lebesgue constant Λ (i.e., the norm of the polynomial interpolation operator I ) for any Feete array of order for. On the other hand Feete arrays provide the lin of polynomial interpolation with potential theory. Indeed, the logarithmic energy cap() of a unit charge distribution on at equilibrium ( ) cap() := exp min log z ζ dµ(z)dµ(ζ) µ M () turns out to coincide with certain asymptotic of the modulus of the Vandermonde determinant computed at any sequence of Feete points and with respect to the monomial basis. Since the considered asymptotic is a geometric mean of mutual distances of Feete points, it is termed transfinite diameter of and denoted by δ(), δ() := lim Vdm(z 0,..., z ) dim P, for (z 0,..., z ) Feete array. The Fundamental Theorem of Logarithmic Potential Theory asserts that δ() = cap() = τ(), where τ() is the Chebyshev constant of and is defined by means of asymptotic of certain normalized monic polynomials. Moreover, provided δ() 0, for any sequence of arrays having the same asymptotic of the Vandermonde determinants as Feete points the sequence of uniform probability measures supported on such arrays converges wea star to the unique minimizer of the logarithmic energy minimization problem, that is the equilibrium measure µ of. The other fundamental object in this theory is the Green function with pole at infinity g (, ) for the domain C \ Ê, where Ê is the polynomial hull of. () g (z, ) := lim sup ζ z ( sup {u(ζ) : u L(C), u 0} ). Here L(C) is the Lelong class of subharmonic function of logarithmic growth, i.e., u(z) log z is bounded near infinity.

3 It turns out that, provided δ() 0, one has ( sup (2) g (z, ) = lim sup ζ z PLURIPOTNTIAL NUMRICS 3 { deg p log p(ζ), p P, p }). It follows that the Green function of is intimately related to polynomial inequalities and polynomial interpolation: for instance, one has the Bernstein Walsh Inequality (3) p(z) p exp(deg p g (z, )), p P and the Bernstein Walsh Theorem [47], that relates the rate of decrease of the error of best polynomial uniform approximation on of a function f hol(int ) C () to the possibility of extending f holomorphically to a domain of the form {g (z, ) < c}. Lastly, it is worth to recall that, using the fact that log is the fundamental solution of the Laplace operator in C, it is possible to show that g (z, ) = µ in the sense of distributions. When we move from the complex plane to the case of C n, n >, the situation becomes much more complicated. Indeed, one can still define Feete points and loo for asymptotic of Vandermonde determinants with respect to the graded lexicographically ordered monomial basis computed at these points, but this is no more related to a linear convex functional on the space of probability measures (the logarithmic energy) neither to a linear partial differential operator (the Laplacian) as in the case of C. During the last two decades (see for instance [25], [27]), a non linear potential theory in C n has been developed: Pluripotential Theory is the study of plurisubharmonic functions, i.e., functions that are upper semicontinuous and subharmonic along each complex line. Plurisubharmonic functions in this setting enjoy the role of subharmonic functions in C, while maximal plurisubharmonic functions replace harmonic ones; the geometric-analytical relation among the classes of functions being the same. It turned out that this theory is related to several branches of Complex Analysis and Approximation Theory, exactly as happens for Logarithmic Potential Theory in C. It was first conjectured by Leja that Vandermonde determinants computed at Feete points should still have a limit and that the associated probability measures sequences should converge to some unique limit measure, even in the case n >. The existence of the asymptotic of Vandermonde determinants and its equivalence with a multidimensional analogue of the Chebyshev constant was proved by Zaharjuta [48], [49]. Finally the relation of Feete points asymptotic with the equilibrium measure and the transfinite diameter in Pluripotential Theory (even in a much more general setting) has been explained by Berman Boucsom and Nymström very recently in a series of papers; [6], [5]. Indeed, the situation in the several complex variables setting is very close to the one of logarithmic potential theory, provided

4 4 FDRICO PIAZZON a suitable translation of the definitions, though the proof and the theory itself is much much more complicated. Since Logarithmic Potential Theory has plenty of applications in Analysis, Approximation Theory and Physics, many numerical methods for computing approximations to Greens function, transfinite diameters and equilibrium measure has been developed following different approaches as Riemann Hilbert problem [3], numerical conformal mapping [24], linear or quadratic optimization [39, 38] and iterated function systems [28]. On the other hand, to the best authors nowledge, there are no algorithms for approximating the corresponding quantities in Pluripotential Theory; the aim of this paper is to start such a study. This is motivated by the growing interest that Pluripotential Theory is achieving in applications during the last years. We mention, among the others, the quest for nearly optimal sampling in least squares regression [30, 42, 22], random polynomials [4, 50] and estimation of approximation numbers of a given function [46]. Our approach, first presented in the doctoral dissertation [32, Part II Ch. 6], is based on certain sequences, first introduced by Calvi and Levenberg [2], of finite subsets of a given compact set termed admissible polynomial meshes having slow increasing cardinality and for which a sampling inequality for polynomials holds true. The core idea of the present paper is inspired by the analogy of sequences of uniform probability measures supported on an admissible mesh with the class of Bernstein Marov measures (see for instance [32], [5] and [8]). Indeed, we use L 2 methods with respect to these sequences of measures: we can prove rigorously that our L 2 maximixation procedure leads to the same asymptotic as the L maximization that appears in the definitions of the transfinite diameter (or other objects in Pluripotential Theory), this is due to the sampling property of admissible meshes. On the other hand the slow increasing cardinality of the admissible meshes guarantees that the complexity of the computations is not growing too fast. We warn the reader that, though all proposed examples and tests are for real sets R n C n, our results hold in the general case C n. This choice has been made essentially for two reasons: first, the main examples for which we have analytical expression to compare our computation with are real, second, the case of R 2 is both computationally less expensive and easier from the point of view of representing the obtained results. The methods we are introducing in the present wor are suitable to be extended in at least three directions. First, one may consider weighted pluripotential theory (see for instance [2]) instead of the classical one: the theoretical results we prove here can be recovered in such a more general setting by some modifications. It is worth to mention that a relevant part of the proofs of our results rest upon this weighted theory even if is not presented in such a framewor, since we extensively use the results of the seminal paper [6]. However, in order to produce the same algorithms in the weighted framewor, one needs to wor with weighted polynomials, i.e., functions of the type p(z)w deg p (z) for a given weight function w and typically needs no more to be compact in this theory, these changes cause some

5 PLURIPOTNTIAL NUMRICS 5 theoretical difficulties in constructing suitable admissible meshes and may carry non trivial numerical issues as well. Second, we recall that pluripotential theory has been developed on certain lower dimensional sets of C n as sub-manifolds and affine algebraic varieties. If is a compact subset of an algebraic subset A of C n, then one can extend the pluripotential theory of the set A reg of regular points of A to the whole variety and use traces of global polynomials in C n to recover the extremal plurisubharmonic function V (z, A); see [40]. This last direction is probably even more attractive, due to the recent development of the theory itself especially when lies in the set of real points of A, see for instance [29] and [7]. Lastly, we mention an application of our methods that is ready at hand. Very recently polynomial spaces with non-standard degree ordering (e.g., not total degree nor tensor degree) start to attract a certain attention in the framewor of random sampling [22], Approximation Theory [46], and Pluripotential Theory [8]. For instance, one can consider spaces of polynomials of the form Pq := span{z α, α i N n, q(α) < }, where q is any norm or even, more generally, PP := span{zα, α i N n, α P} for any P R n + closed and star-shaped with respect to 0 such that N P = R n +. Since this spaces are being used only very recently, many theoretical questions, from the pluripotential theory point of view, arise. Our methods can be used to investigate conjectures in this framewor by a very minor modification of our algorithms. The paper is structured as follows. In Section 2 we introduce admissible meshes and all the definitions and the tools we need from Pluripotential Theory. Then we present our algorithms of approximation: for each of them we prove the convergence and we illustrate their implementation and their performances by some numerical tests; we stress that, despite the fact that we will consider only cases of R 2 for relevance and simplicity, our techniques are fine for general C n. We consider the extremal function V (Pluripotential Theory counterpart of the Green function, see (3) below) in Section 3, the transfinite diameter δ() in Section 4 and the pluripotential equilibrium measure µ := ( dd c V) n in Section 5. All the experiment are performed with the MATLAB software PPN pacage, see fpiazzon/software. 2. Preliminaries 2.. Pluripotential theory: some definitions. Let Ω C be any domain and u : Ω R { }, u is said to be subharmonic if u is upper semicontinuous and u(z) 2πr u(ζ)ds(ζ) for any r > 0 and z Ω such that B(z, r) Ω. z ζ =r A function u : Ω R { }, where Ω C n, is said to be plurisubharmonic if u is upper semicontinuous and is subharmonic along each complex line (i.e., each complex one dimensional affine variety); the class of such functions is usually denoted by PSH(Ω). It is worth to stress that the class of plurisubharmonic functions is strictly smaller than the class of subharmonic function on Ω as a domain in R 2n.

6 6 FDRICO PIAZZON The Lelong class of plurisubharmonic function with logarithmic growth at infinity is denoted by L(C n ) and u L(C n ) iff u PSH(C n ) is a locally bounded function such that u(z) log z is bounded near infinity. Let C n be a compact set. The extremal function V (also termed pluricomplex Green function) of is defined mimicing one of the possible definitions in C of the Green function with pole at infinity; see (). (4) (5) V (ζ) := sup{u(ζ) L(C n ), u 0}, V (z) := lim sup V (ζ). ζ z It turns out that the extremal function enjoys the same relation with polynomials of the Green function; precisely Sicia introduced [44] (6) (7) { } Ṽ (ζ) := sup deg p log p(ζ), p P, p, Ṽ (z) := lim sup V (ζ). ζ z and shown (the general statement has been proved by Zaharjuta) that Ṽ V and Ṽ = q.e. V. Here q.e. stands for quasi everywhere and means for each z C n \ P where P is a pluripolar set, i.e., a subset of the { } level set of a plurisubharmonic function not identically. In the case that V is also continuous, the set is termed L-regular. A remarable consequence of this equivalence is that the Bernstein Walsh Inequality (3) and Theorem (see [44]) hold even in the several complex variable setting simply replacing g (z, ) by V (z); see for instance [27]. In Pluripotential Theory the role of the Laplace operator is played by the complex Monge Ampere operator (dd c ) n. Here dd c = 2i, where u(z) := n u(z) j= z j dz j, u(z) := n u(z) j= z j d z j and (dd c u) n = dd c u dd c u dd c u. For u C 2 (C n, R) one has (dd c u) n = c n det[ 2 u/ z i z j ]dvol C n. The Monge Ampere operator extends to locally bounded plurisubharmonic functions as shown by Bedford and Taylor [4], [3], (dd c u) n being a positive Borel measure. The equation (dd c u) n = 0 on a open set Ω (in the sense of currents) characterize the maximal plurisubharmonic functions; recall that u is maximal if for any open set Ω Ω and any v PSH(Ω) such that v Ω u Ω we have v(z) u(z) for any z Ω. Given a compact set C n, two situations may occur: either V +, or is a locally bounded plurisubharmonic function. The first case is when is V pluripolar, it is too small for pluripotential theory. In the latter case ( dd c V) n = 0 in C n \ (i.e., V is maximal on such a set), in other words the positive measure ( ) dd c V n is supported on. Such a measure is usually denoted by µ and termed the (pluripotential) equilibrium measure of by analogy with the one dimensional case.

7 PLURIPOTNTIAL NUMRICS 7 Let us introduce the graded lexicographical strict order on N n. For any α, β N n we have α = β α β (8) α β if α < β or, (9) α j < β j where j := min{ j {, 2,..., n} : α j β j }. This is clearly a total (strict) well-order on N n and thus it induces a bijective map On the other hand the map (0) e : N n P(C n ) α : N N n. α e α (z) := z α zα 2 2 zα n () n is a isomorphism having the property that deg e α (z) = α. Thus, if we denote by e i (z) the i-th monomial function e α(i) (z), we have where P (C n ) = span{e i (z), i N } =: span M n, ( ) n + N := dim P (C n ) =. n From now on we will refer to M n as the graded lexicographically ordered monomial basis of degree. For any array of points {z,... z N } N we introduce the Vandermonde determinant of order as Vdm (z,... z N ) := det[e i (z j )] i, j=,...,n. If a set {z,..., z N } of points of satisfies Vdm (z,... z N ) = max ζ,...ζ N Vdm (ζ,... ζ N ) it is said to be a array of Feete points for. Clearly Feete points do not need to be unique. Zaharjuta proved in his seminal wor [48] that the sequence δ () := max Vdm (ζ,... ζ N ) n+ nn ζ,...ζ N does have limit and defined, by analogy with the case n =, the transfinite diameter of as (2) δ() := lim δ (). It turns out that the condition δ() = 0 characterize pluripolar subsets of C n as it characterize polar subset of C. Let {z () } N := {(z (),..., z() N )} N be a sequence of Feete points for the compact set, we consider the canonically associated sequence of uniform probability measures ν := N j=. N δ z () j

8 8 FDRICO PIAZZON Berman and Boucsom [5] showed that the sequence ν converges wea star to the pluripotential equilibrium measure µ as it happens in the case n =. We will use both this result (in Section 5) and a remarable intermediate step of its proof (in Section 3 and Section 4) termed Bergman Asymptotic, see (2) below Admissible meshes and Bernstein Marov measures. We recall that a compact set R n (or C n ) is said to be polynomial determining if any polynomial vanishing on is necessarily the null polynomial. Let us consider a polynomial determining compact set R n (or C n ) and let A be a subset of. If there exists a positive constant C such that for any polynomial p P (C n ) the following inequality holds (3) p C p A, then A is said to be a norming set for P (C n ). Let {A } be a sequence of norming sets for P (C n ) with constants {C }, suppose that both C and Card(A ) grow at most polynomially with (i.e., max{c, Card(A )} = O( s ) for a suitable s N), then {A } is said to be a wealy admissible mesh (WAM) for ; see 2 [2]. Observe that necessarily ( ) + n (4) Card A N := dim P (C n ) = = O( n ) since a (W)AM A is P (C n )-determining by definition. If C C, then {A } N is said an admissible mesh (AM) for ; in the sequel, with a little abuse of notation, we term (wealy) admissible mesh not only the whole sequence but also its -th element A. When Card(A ) = O( n ), following Kroó [26], we refer to {A } as an optimal admissible mesh, since this grow rate for the cardinality is the minimal one in view of equation (4). Wealy admissible meshes enjoy some nice properties that can be also used together with classical polynomial inequalities to construct such sets. For instance, WAMs are stable under affine mappings, unions and cartesian products and well behave under polynomial mappings. Moreover any sequence of interpolation nodes whose Lebesgue constant is growing sub-exponentially is a WAM. It is worth to recall other nice properties of (wealy) admissible meshes. Namely, they enjoy a stability property under smooth mapping and small perturbations both of and A itself; [34]. For a survey on WAMs we refer the reader to [6]. Wealy admissible meshes are related to Feete points. For instance assume a Feete triangular array {z () } = {(z (),..., z() N )} for is nown, then setting A := z ( log ) for all N we obtain an admissible mesh for ; []. Conversely, if we start with an admissible mesh {A } for it has been proved in [9] that it is possible to extract (by numerical linear algebra) a set z () := The results of [5] and [6] hold indeed in the much more general setting of high powers of a line bundle on a complex manifold. 2 The original definition in [2] is actually slightly weaer (sub-exponential growth instead of polynomial growth is allowed), here we prefer to use the present one which is now the most common in the literature.

9 {z () PLURIPOTNTIAL NUMRICS 9,..., z() N } A from each A such that the sequence {z () } is an asymptotically Feete sequence of arrays, i.e., (5) Vdm (z () n+,..., z() nn N ) δ() as happens for Feete points. By the deep result of Berman and Boucsom [6] (and some further refining, see [0]) it follows that the sequence of uniform probability measures {ν } canonically associated to z () converges wea star to the pluripotential equilibrium measure. In [6] authors pointed out, among other deep facts, the relevance of a class of measures for which a strong comparability of uniform and L 2 norms of polynomials holds, they termed such measures Bernstein Marov measures. Precisely, a Borel finite measure µ with support S µ is said to be a Bernstein Marov measure for if we have (6) lim sup sup p P \{0} p p L 2 µ /. Let us denote by {q j (z, µ)}, j =,..., N the orthonormal basis (obtained by Gram-Schmidt orthonormalizaion starting by M n) of the space P endowed by the scalar product of Lµ. 2 The reproducing ernel of such a space is K µ (z, z) := N j= q j(z, µ)q j (z, µ), we consider the related Bergman function N B µ (z) := Kµ (z, z) = q j (z, µ) 2. As a side product of the proof of the asymptotic of Feete points Berman Boucsom and Nyström deduces the so called Bergman Asymptotic (7) j= B µ N µ µ, for any positive Borel measure µ with support on and satisfying the Bernstein Marov property. Note that, by Parseval Identity, the property (6) above can be rewritten as (8) lim sup B µ /(2). Bernstein Marov measures are very close to the Reg class defined (in the complex plane) by Stahl and Toti and studied in C n by Bloom [8]. Recently Bernstein Marov measures have been studied by different authors, we refer the reader to [5] for a survey on their properties and applications. Our methods in the next sections rely on the fact admissible meshes are good discrete models of Bernstein Marov measures; let us illustrate this. From now on we assume C n to be a compact L-regular set and hence polynomial determining,

10 0 FDRICO PIAZZON we denote by µ the uniform probability measure supported on A = {z (),..., z() M }, i.e., µ := Card A Card A j= δ z (), j we denote by B (z) the function B µ (z) and by K (z, ζ) the function K µ (z, ζ). Assume that an admissible mesh {A } of constant C for the compact polynomial determining set C n is given. Now pic ẑ such that B (ẑ) = max B, we note that N B (ẑ) = c j q j (ẑ, µ) := p(ẑ), c j := q j (ẑ, µ). j= By Parseval inequality we have N p c j 2 j= Therefore we an write /2 max z N q j (z, µ) 2 j= /2 = B (ẑ) = B. thus B = p(ẑ) p C p A C B (ẑ) B /2, (9) B C 2 B A. On the other hand for any polynomial p P we have p C p A C Card A p L 2 µ. Recall that it follows by the definition of (wealy) admissible meshes that (C 2 Card A ) /(2). Thus, the sequence of probability measures associated to the mesh has the property (20) lim sup B /(2) lim sup(c Card A ) / =, which closely resembles (8). Conversely, assume {µ } to be a sequence of probabilities on with Card supp µ = O( s ) for some s, then we have p B p supp µ, p P. Therefore, if B = O( t ) for some t, the sequence of sets {supp µ } is a wealy admissible mesh for.

11 PLURIPOTNTIAL NUMRICS 3. Approximating the extremal function 3.. Theoretical results. In this section we introduce certain sequences of functions, namely u, v, ũ and ṽ, that can be constructed starting by a wealy admissible mesh, all of them having the property of local uniform convergence to V, provided is L-regular. Theorem 3.. Let C n be a compact L-regular set and {A } a wealy admissible mesh for, then, uniformly in C n, we have (2) (22) lim v := lim 2 log B = V, lim u := lim log K (, ζ) dµ (ζ) = V. Proof. We first prove (2), for we introduce F () := {p P : p } log Φ () (z) := sup { log p(z), p F () The sequence of function Φ () has been defined by Sicia and has been shown to converge to exp Ṽ (see equation (7)) for L-regular, moreover we have V Ṽ ; [44], see also [43]. Let us denote by F () 2 the family {p P : p L 2 µ } we notice that, due to the Parseval Identity, we have B (z) = sup p(z) 2. p F () 2 Let us pic p F () 2, we have p B p L 2 µ for the reason above, thus q := p B /2 F (). Hence It follows that }. log Φ () (z) log q(z) = log p(z) 2 log B, p F () 2. log Φ () (z) + 2 log B v (z). On the other hand, since µ is a probability measure, we have p p L 2 µ for any polynomial. Hence if p F () log Φ () (z). Therefore we have it follows that p F () 2. Thus v (z) log Φ () (z) + 2 log B v (z) log Φ () (z).

12 2 FDRICO PIAZZON Note that we have lim sup B /2 since {A } is wealy admissible (see equation (20)), hence we can conclude that locally uniformly we have ( V (z) lim inf log Φ () (z) ) 2 log B lim inf v (z) lim sup v (z) lim sup log Φ () (z) = V (z). This concludes the proof of (2), let us prove (22). It follows by Cauchy-Schwarz and Holder Inequalities and by B dµ = N that K (z, ζ) dµ (ζ) Thus it follows that N /2 N /2 q j (z, µ ) 2 q j (ζ, µ ) 2 dµ (ζ) j= j= B (ζ) B L 2 (z) N /2 B (z). µ (23) u (z) 2 log[n ] + v (z) uniformly in C n. On the other hand, for any p P we have p(z) = K (z, ζ); p(ζ) L 2 µ = K (z, ζ)p(ζ)dµ (ζ) p L µ K (z, ζ) dµ (ζ) p K (z, ζ) dµ (ζ), hence, using the definition of Sicia function, K µ (z, ζ) dµ p(z) (ζ) sup = (Φ () p P \{0} p ). Finally, using (23), we have log Φ () (z) u (z) v (z) + log N /(2), uniformly in C n, this concludes the proof of (22) since N /(2) and both v and log Φ () converge to V uniformly in Cn. It is worth to notice that both u and v are defined in terms of orthonormal polynomials with respect to µ, hence they can be computed with a finite number

13 of operations at any point z C n, indeed we have v (z) = log (24) = log Card A PLURIPOTNTIAL NUMRICS 3 Card A h= K (z, ζ) dµ (ζ) N q j (z, µ ) q j (ζ h, µ ). Also we note that Theorem 3. can be understood as a generalization of the original Sicia statement [44, Th. 4.2]. Indeed, if we tae A = {z (),..., z() N } a set of Feete points of order for we get q j (z, µ ) = N l j, (z), where l j, (z) is the j-th Lagrange polynomial, hence we have N N q N j (z, µ ) q j (ζ h, µ ) = N N q N j (z, µ )δ j h h= = N N h= j= j= h= q h (z, µ ) N = l h, (z) =: Λ A (z). h= Here Λ A (z) is the Lebesgue function of the interpolation points A. Therefore, for A being a Feete array of order, we have u (z) = log ( Λ A (z) ) /, this is precisely exp ( Φ (2) (z)) in the Sicia notation. In Section 5 we will deal with measures of the form ν := Bµ N µ for a Bernstein Marov measure µ for, or, more generally ν N µ, where the sequence {µ } has the property lim sup B µ /2 = ; we refer to such a sequence {µ } as a Bernstein Marov sequence of measures. Due to a modification of the Berman Boucsom and Nymstrom result, ν converges wea star to µ (see Proposition 5. below). Note that ν is still a probability measure since B (z) 0 for any z C n and dν = N B dµ = N N j= j= := Bµ q j (z) 2 L 2 µ =. Here we point out another (easier, but very useful to our aims) property of the weighted sequence ν : actually they are a Bernstein Marov sequence of measure, more precisely the following theorem holds. Theorem 3.2. Let C n be a compact L-regular set and {A } a wealy admissible mesh for. Let us set µ := Bµ N µ, where µ is the uniform probability measure on A. Then the following holds. i) For any N and any z C n (25) B µ (z) Thus lim sup B µ /2 =. N min B B µ (z) NBµ (z).

14 4 FDRICO PIAZZON ii) We have (26) lim ṽ := lim 2 log B µ = V, (27) lim ũ := lim log K µ (, ζ) d µ (ζ) = V, uniformly in C n. From now on we use the notations where B is as above B µ B := B µ (z), µ := B N µ, and µ will be clarified by the context. Proof. We prove (25), then lim sup B /2 = follows immediately by lim sup B /2 and lim N / =. The proof of (26) and (27) is identical to the ones of Theorem 3. so we do not repeat them. We simply notice that, for any sequence of polynomials {p } with deg p, we have p 2 = p Lµ 2 (z) 2 N dµ = B (z) p (z) 2 B (z) N dµ N max p (z) 2 B (z) B N dµ N = p 2. min B L 2 µ Now, for any z, we pic a sequence {p } such that it maximizes (for any ) the ratio ( q(z) q ) / among q P and we get L 2 µ B (z) /2 = p (z) p L 2 µ N N p (z) min B p L 2 µ B (z) /2. min B Here the last inequality follows by the definition of B. Note in particular that B (z) = + q 2 (z) 2 N +..., thus min B N = O( n ) and B /2 N /2 B /2 B /2 as. Remar 3.3. We stress that the upper bound (25) is in many cases quite rough, though sufficient to prove the convergence result (26). Indeed, since B dµ = N for any, it follows that min B is always larger than, but we warn the reader that B does not need to be larger than B in general. Hence the measure µ may be more suitable than µ for our approximation purposes The SZF and SZF-BW algorithms. The function V, at least for a regular set, can be characterized as the unique continuous solution of the following problem (dd c u) n = 0, in C n \ u 0, on u L(C n ). =

15 PLURIPOTNTIAL NUMRICS 5 It is rather clear that writing a pseudo-spectral or a finite differences scheme for such a problem is a highly non trivial tas, as one needs to deal with a unbounded computational domain C n \, with a positivity constraint on (dd c u) n, and with a prescribed growth rate at infinity (both encoded by u L(C n )). Here we present the SZF and SZF-BW algorithms (which stands for for Sicia Zaharjuta xtremal Function and Sicia Zaharjuta xtremal Function by Bergman weight) to compute the values of the functions u and v (see Theorem 3.) and the functions ũ and ṽ (see Theorem 3.2) respectively at a given set of points. In our methods both the growth rate and the plurisubharmonicity are encoded in the particular structure of the approximated solutions u or v, while the unboundedness of C n \ does not carry any issue since all the sampling points used to build the solutions lie on. Indeed, once the approximated solution is computed on a set of points and the necessary matrices are stored, it is possible to compute u or v on another set of points by few very fast matrix operations; this will be more clear in a while. To implement our algorithms we mae the following assumptions. Let C n be a compact regular set, for simplicity let us assume to be a real body (i.e., the closure of a bounded domain), but notice that this assumption is not restrictive neither from the theoretical nor from the computational point of view. We further assume that we are able to compute a wealy admissible mesh {A } = {z,..., z M } for with constants C, N. Note that an algorithmic construction of an admissible mesh is available in the literature for several classes of sets [36, 35, 33], since the study of (wealy) admissible meshes is attracting certain interest during last years. We assume n = 2 to hold the algorithm complexity growth. The implementation is based on the following choices. Let us fix a computational grid {ζ (),..., ζ(l ) } {ζ () 2,..., ζ(l 2) 2 } =: Ω C 2 with finite cardinality L := L L 2 and let us denote by Ω the (possibly empty) set Ω, while by Ω 0 the set Ω \ Ω. We will reconstruct the values u (ζ), v (ζ) ζ Ω, however we will test the performance of our algorithm (see Subsection 3.3 below) only in term of error on Ω 0. This choice is motivated as follows. First, we have to mention the fact that the point-wise error of u and v exhibits two rather different behaviours depending whether the considered point ζ lies on or not: the convergence for ζ is much slower. In contrast, for any regular set, the function V identically vanishes on, hence there is no point in trying to approximate it on. Note that the function V (ζ) 0 for any ζ Cn \, thus we can measure the error of u of v on Ω 0 C 2 \ both in the absolute and in the relative sense. Let L : C 2 C 2 be the invertible affine ( map, mapping ) A in the square [, ] 2 and defined by P i (z) := 2 b i a zi i a i+b i 2, ai := min z A z i, b i :=

16 6 FDRICO PIAZZON max z A z i. We denote by T j (z) the classical j-th Chebyshev polynomial and we set φ j (z) := T α ( j)(p (z))t α2 ( j)(p 2 (z)), j N, j > 0, where α : N N 2 is the one defined in (). The set TP := {φ j(z), j (+)(+2)/2} is a (adapted Chebyshev) basis of P (C 2 ). We will use the basis TP for computing and orthogonalizing the Vandermonde matrix of degree computed at A. This choice has been already fruitfully used, for instance in [9, 6], when stable computations with Vandermonde matrices are needed and is on the bacground of the widely used matlab pacage ChebFun2 [45] SZF Algorithm Implementation. The first step of SZF is the computation of the Vandermonde matrices with respect to the basis T P (28) (29) VT := (φ j (z i )) i=,...,m :=Card A, j=,...,(+)(+2)/2 WT := (φ j (ζ i )) i=,...,l, j=,...,(+)(+2)/2. Here VT is computed simply by the formula T h (x) = cos(h arccos(x)) while for WT we prefer to use the recursion algorithm to improve the stability of the computation since the points P(ζ i ), ζ i Ω, may in general lie outside of [, ] 2 or even be complex. The second step of the algorithm is the most delicate: we perform the orthonormalization of VT and store the triangular matrix defining the change of basis. More precisely the orthonormalization procedure is performed by applying the QR algorithm twice (following the so called twice is enough principle). First we apply the QR algorithm to VT and we store the obtained R, then we apply the QR algorithm to VT \ R, we obtain Q and R 2 that we store. Here \ is the matlab bacslash operator implementing the bacward substitution; this is much more stable than the direct inversion of the matrix R, which may be ill conditioned. Note that Q i, j = p j (z i ), where, since Q is an orthogonal matrix, M M p j (z) p h (z)dµ (z) = p j (z i ) p h (z i ) = (QQ T ) j,h = δ j,h. i= Therefore M Q i, j = q j (z i ). Step three. We compute the orthonormal polynomials evaluated at the points {ζ,... ζ L } = Ω. Again we prefer the bacslash operator rather than the matrix inversion to cope with the possible ill-conditioning of R and R 2 : we compute W := WT \ R \ R 2. Note that W i, j = p j (ζ i ) and thus q j (ζ i ) = W i, j M. We also compute the matrix K = Q W T, here we have K i,h = N j= p j (z i )p j (ζ h ) = N q j (z i )q j (ζ h ). M j=

17 PLURIPOTNTIAL NUMRICS 7 Step 4. Finally we have N (30) (v (ζ ),..., v (ζ L )) i=,...,l = V := 2 log M Wi, 2 j and M (3) (u (ζ ),..., u (ζ L )) i=,...,l = U := log K i,h j= i= i=,...,l h=,...,l SZF-BW Algorithm Implementation. First the step and step 2 of SZF algorithm are performed. Step 3. The Bergman weight σ = (B /N(z ),..., B (z M )/N) is computed by σ i = M N N j= Q 2 i, j. Then the weighted Vandermonde matrix V (w) i, j := σ i Q i, j is computed by a matrix product. Step 4. We compute the orthonormal polynomials by another orthonormalization by the QR algorithm: V (w) = Q (w) R (w). We get q j (z i ) = M σ i V (w) i, j =: Q i, j. We also compute the evaluation of q j s at Ω by q j (ζ i ) = WT \ R \ R 2 \ R (w) =: W i, j. Step 5. We perform the step 4 of the SZF algorithm, where Q and W are replaced by Q and W Numerical Tests of SZF and SZF-BW. The extremal function V can be computed analytically for very few instances as real convex bodies and sub-level sets of complex norms. For the unit real ball B Lundin (see for instance [25]) proved that (32) V B (z) VB (z) = log h( z + z, z ), 2 where h(ζ) := ζ + ζ 2 denotes the inverse of the Jouowsi function and maps conformally the set C \ [, ] onto C \ {z C : z }. Let be any real convex set, one can define the convex dual set as := {x R n : x, y, y }. Baran [, 2] proved, that if is a convex compact set symmetric with respect to 0 and containing a neighbourhood of 0, the following formula holds. (33) V (z) = sup{log h( z, w ), w extr }. Here extrf is the set of points of F that are not interior points of any segment laying in F.

18 8 FDRICO PIAZZON In order to be able to compare our approximate solution with the true extremal function, we build our test cases as particular instances of the Baran s Formula: test case, 4 := [, ] 2, test case 2, m the real m-agon centred at 0, test case 3, := B, the real unit dis. We measure the error with respect to the true solution V in terms of Card e ( f, V, Ω) := Ω 0 f (ζ i ) V (ζ i ), Card Ω 0 which should be intended as a quadrature approximation for the L norm of the error. Also we consider, as an approximation of the relative L error, the quantity Card Ω0 e rel ( f, V, Ω) := i= f (ζ i ) V (ζ i ) Card Ω0, i= V (ζ i ) and the following approximation of the error in the uniform norm i= e ( f, V, Ω) := max f (ζ i ) V (ζ i ). ζ i Ω 0 In all the tests we performed the rate of convergence is experimentally sublinear, i.e., s + / s as, where s := f + f for f being one of u, ũ, v, ṽ and being one of the pseudo-norms we used above for defining e and e. This slow convergence may be overcame by extrapolation at infinity with the vector rho algorithm (see [20]). We present below experiments regarding both the original and the accelerated algorithm Test case. In this case equation (33) reads as V [,] 2 (z) := max log h(ζ). ζ {,} 2 To build an admissible mesh for [, ] 2 we can use the Cartesian product of an admissible mesh X for one dimensional polynomials up to degree ; see [6]. We can pic X := cos(θ j ), θ j := jπ/ m, with j = 0,,..., m. First we compare the performance of our four approximations in terms of L error behaviour as grows large. Also, in order to better understand the rate of convergence, we compute the ratios s := e ( f +2, f +, Ω) e ( f +, f, Ω), for f in {u, v ũ, ṽ } and Ω = [ 2, 2] 2. We report the results in Figure and 2 respectively. The profile of convergence is slow but monotone, indeed the asymptotic constants s become rather close to. This suggest a sub-linear convergence rate. It is worth to say that in all the tests we made the point-wise error is much smaller far from than for points that lie near to : in the experiment reported in Figure and Figure 2 Ω is an equispaced real grid in [0, 2] 2, but the results are

19 PLURIPOTNTIAL NUMRICS 9 Figure. Comparison of the e (, Ω) error of u, ũ, v and ṽ for = [, ] 2 and Ω the 0000 points equispaced coordinate grid in [ 2, 2] Bergman Kernel w-bergman w-kernel Figure 2. Comparison of the s behaviour for f being u (Bergman), ũ (w-bergman), v (Kernel) or ṽ (w-kernel) for = [, ] 2 and Ω the 0000 points equispaced coordinate grid in [ 2, 2] Bergman Kernel w-bergman w-kernel the same for larger bounds and becomes much better if Ω = or when Ω is a purely imaginary set, i.e., Ω R 2 =.

20 20 FDRICO PIAZZON Fortunately, even if the convergence rate shown by the experiment of Figure, the vector rho algorithm, see for instance [20], wors effectively on our approximation sequences and actually allows to produce much better approximations than the original sequences {u }, {ũ }, {v }, {ṽ }. We report in Figure 3 a comparison among the errors, on [ 2, 2] 2 and [0, 20] 2 respectively, of the original sequence {u } and the accelerated sequence produced by the vector rho algorithm, together with a linear (with respect to log-log scale) fitting of the original errors. In contrast with the sub-linear convergence enlightened above, the accelerated sequence exhibits (in the considered interval for ) a super-quadratic behaviour Test case 2. To construct a wealy admissible mesh on the real regular m- agon {( ) ( ) ( cos 2π m := conv, m cos 2(m )π )} 0 sin 2π,......, m m sin 2(m )π, m we use the algorithm proposed in [7]. First the convex polygon is subdivided in non overlapping triangles, then an admissible mesh {Â,l } for each triangle l, l =, 2,..., m is created by the Duffy transformation; finally, the union {à } := { m l=â,l} of all meshes (with no point repeated) is an admissible mesh for the polygon by definition. The resulting mesh à has good approximation properties: for instance it can be constructed in such a way to have a very small constant, however à is not tailored to our problem. Points of à cluster, as one could expect, at any corner of the regular polygon, but also they cluster near the edges of each triangle in the subdivision. This spurious clustering is not coming from the geometry of the problem, instead it is an effect of the method we used to solve it. More importantly, this issue tends to deteriorate the convergence of SZF algorithm. Let us briefly give an insight on why this phenomena occurs in the following remar. Remar 3.4. In Section 5 we will prove (see Proposition 5.) that the sequence of measures B µ {ˆµ } := µ N µ, where denotes the convergence in the wea topology on Borel measures. Assume for simplicity that µ is absolutely continuous with respect to the Lebesgue measure m restricted to and consider an admissible mesh that tends to cluster points on a ball B(z 0, r) for which µ dm (z), z B(z 0, r), is very small. The asymptotic above implies in particular that B µ (z) will be very small for z supp µ B(z 0, r). On the other hand we have N µ B dµ = for any measure and thus there will be some points ẑ supp µ \ B(z 0, r) for which B µ (ẑ) is very large. Recall that the uniform convergence of Theorem 3. relies on the fact that B µ /2, thus we should aim to get an admissible mesh {A } having small constant, whose cardinality is slowly increasing, and whose Bergman function is not too large,

21 PLURIPOTNTIAL NUMRICS 2 Figure 3. Log-log plot of e (ũ, V, Ω), erel (ũ, Ω) and e (ũ, V, Ω) and the same quantities for the accelerated sequence of approximations for = [, ] 2 and Ω the 0000 points equispaced coordinate grid in [ 2, 2] 2 (above) and in [0, 20] 2 (below) absolute L error relative L- error absolute max error Acc. absolute L- error Acc. relative L error Acc. absolute max error absolute L error relative L- error absolute max error Acc. absolute L- error Acc. relative L error Acc. absolute max error e.g., is not too oscillating. From the observation above a good heuristic to achieve such an aim is to mimic the density of the equilibrium measure. In order to overcome this issue we can apply two strategies. The first one is to get rid of these extra points. To this aim we can use the AFP algorithm [9] to extract a set of discrete Feete points A of order 2 from à 2.

22 22 FDRICO PIAZZON Let us motivate this choice. First A has been shown (numerically) to be a wealy admissible mesh in many test cases, see [9] and reference therein. Also, even more importantly, the resulting Bergman function is (experimentally) much less oscillating than the one that can be computed starting by à and this turns in a smaller uniform norm on m of such a function. Lastly, heuristically speaing in view of Remar 3.4, we would lie to use a mesh which is mimicing the distribution of the equilibrium measure. We invite the reader to compare this last discussion with [0] where the definition of optimal measures is introduced. We can also perform a different choice which rests upon Theorem 3.2. Indeed the SZF-BW algorithm uses a Bergman weighting of µ to prevent the phenomena we discussed in Remar 3.4. We tested our algorithm SZF-BW for different values of m and several choices of Ω, here we consider the case m = 6. Again, the convergence is quite slow but (numerically) monotone and, apart from the region of Ω 0 close to R n, it is not affected by the particular choice of Ω; that is, we can appreciate numerically the global uniform convergence proved in Theorem 3.2. Moreover, extrapolation at infinity is successful even in this test case. We report profiles of convergence relative to two possible choices of Ω in Figure Test case 3. Lastly we consider = := {(z, z 2 ) : I(z i ) = 0, R(z ) 2 + R(z 2 ) 2 }; in such a case the true solution is computed by the Lundin Formula (32). We can easily construct an admissible mesh of degree for the real unit dis following [6], for, it is sufficient to tae a set of Chebyshev-Lobatto points {η 0, η,..., η s }, s > and set { ( π j ) ( π j )) } A := η h (cos, sin, j = 0,,..., s, h = 0,,..., s. 2s 2s We report the behaviour of the error and the convergence profile in Figure 5, again the sub-linear rate of convergence is rather evident. Also, we compare the profile of the error function e (z) := ṽ (z) V (z) on different two real dimensional squares in Figure 6. The global uniform convergence of ṽ to V theoretically proven in Theorem 3.2 reflects on our experiments: e is small (approximately 0 2 for 30) and very flat away from while it attains its maximum on with a fast oscillation near R 2. Again, the extrapolation at infinity improves the quality of our approximation. 4. Approximating the transfinite diameter In this section we present a method for approximating the transfinite diameter of a real compact set. 4.. Theoretical result. Given any basis B = {b,..., b N } of the space P and a measure µ inducing a norm on P, we denote by G (µ, B ) its Gram matrix, namely we set G (µ, B ) := [ b i ; b j L 2 µ ] i, j=,...,n.

23 PLURIPOTNTIAL NUMRICS 23 Figure 4. Log-log plot of e (ũ, V 6, Ω), e rel (ũ, Ω) and e (ũ, V 6, Ω) and the same quantities for the accelerated sequence of approximations Ω the 0000 points equispaced coordinate grid in [ 2, 2] 2 (above), and in [0, 20] 2 (below) absolute L error relative L- error absolute max error Acc. absolute L- error Acc. relative L error Acc. absolute max error absolute L error relative L- error absolute max error Acc. absolute L- error Acc. relative L error Acc. absolute max error The hermitian matrix G (µ, B ) has square root, indeed introducing the generalized Vandermonde matrix V (µ, B ) := [ q i (, µ); b j L 2 µ ] i, j=,...,n, we have V (µ, B ) H V (µ, B ) = G (µ, B ). Note that, for µ being the uniform probability measure supported on an array of unisolvent points of degree, V (µ, B ) is the standard Vandermonde matrix for the basis B and divided by N.

24 24 FDRICO PIAZZON Figure 5. Comparison of e (ũ, Ω), e rel (ũ, Ω) and e (ũ, Ω), their linear (in the log-log scale) fitting and the same quantities computed on the accelerated sequences of approximations for = 6, 8..., 38,, the real unit dis, and Ω the 0000 points equispaced coordinate grid in [ 2, 2] 2 (above) and in [0, 20] 2 (below) absolute L error relative L- error absolute max error Acc. absolute L- error Acc. relative L error Acc. absolute max error absolute L error relative L- error absolute max error Acc. absolute L- error Acc. relative L error Acc. absolute max error We recall, see [3], the following relation between Gram determinants and L 2 norms of Vandermonde determinants. (34) det G (µ, M ) =... det Vdm (ζ,..., ζ N ) 2 dµ(ζ )... dµ(ζ N ), N!

25 PLURIPOTNTIAL NUMRICS 25 Figure 6. Profile of the error ṽ V B in logarithmic scale for = 40, and Ω the points equispaced coordinate grid in [0, 2] 2 (high-left), [00, 02] 2 (above-right), (0.i + [0, 2]) 2 (below left) and (i + [0, 2]) 2 (below right) Where M denotes the (graded lexicographically ordered) monomial basis. Here is the main result this section. Theorem 4.. Let C n be a compact L-regular set and {A } a (wealy) admissible mesh for then, denoting by µ the uniform probability measure on A, we have 2nN (35) lim (det G (µ, M )) n+ = δ(). Proof. We use equation (34). Since µ is a probability measure, it follows that hence (36) det G (µ, M ) /2 2nN lim sup (det G (µ, M )) n+ ( lim sup ( lim max det Vdm (ζ,..., ζ N ) ζ,...,ζ N max det Vdm (ζ,..., ζ N ) ζ,...,ζ N max det Vdm (ζ,..., ζ N ) ζ,...,ζ N ) n+ nn ) n+ nn = δ(). On the other hand, by the sampling property of admissible meshes it follows that, for any polynomial p of degree at most we have p C p A C Card A p L 2 µ.

26 26 FDRICO PIAZZON Thus, since det Vdm (ζ,..., ζ N ) is a polynomial in each variable ζ i C n of degree not larger than, we get ( (det G (µ, M )) n+ 2nN =... Vdm (ζ,..., ζ N ) L 2 µ... C... max Card A Vdm (z, ζ 2..., ζ N ) L 2 z µ... L 2 µ ( ( C Card A ) n+ n max det Vdm (z,..., z N ) z,...,z N ) n+ nn n+ nn Lµ 2 Since (Card A ) /, being {A } wealy admissible, it follows that 2nN lim inf (det G (µ, M )) n+ δ(). ) n+ nn... In principle Theorem 4. provides an approximation procedure for δ(), given {A }, however the straightforward computation of the left hand side of (35) leads to stability issues. In the next subsection we present our implementation of an algorithm based on Theorem 4. and we discuss a possible way to overcome such difficulties Implementation of the TD-GD algorithm. We recall that we denote by T j (z) the classical j-th Chebyshev polynomial and we set T := {φ,..., φ N }, where φ j (z) := T α ( j)(π z)t α2 ( j)(π 2 z), j N, j > 0, where α : N N 2 is the one defined in () and π h is the h-coordinate projection. We denote by V = V (A, T ) the Vandermonde matrix of degree with respect the mesh A := {(x, y ),..., (x M, y M )} and the basis T, that is V := [ φ j (z i ) ] i=,...,m, j=,...,n, similarly we define W := V (A, M ) where the chosen reference basis is the lexicographically ordered monomial one. Now we notice that, setting M := Card A, thus we have m α, m β L 2 µ = M M h= (W ) α,h (W ) h,β, det G (µ ) = det Wt W M. The direct application of this procedure leads to a unstable computation that actually does not converge. On the other hand, the computation of the Gram determinant in the Chebyshev basis, det G (µ, T ) := det Vt V M,

27 is more stable and we have (det G (µ )) n+ 2nN = PLURIPOTNTIAL NUMRICS 27 ( det Wt W ) n+ 2nN M ( = det Pt Vt V ) n+ P 2nN nn = (det(p )) n+ 2nN det G (µ, T ) n+. M Here the matrix P is the matrix of the change of basis. Again the numerical computation of det P becomes severely ill-conditioned as grows large. Instead, our approach is based on noticing that P does not depend on the particular choice of, thus we can compute the term (det(p )) n+ once we now nn 2nN (det G (ˆµ )) n+ 2nN and (det G (ˆµ )) n+ for a particular ˆµ which is a Bernstein Marov measure for Ê [, ] 2 as (37) (det(p )) n+ nn = ( det G (ˆµ ) det G (ˆµ ) ) n+ 2nN. 2nN Also we can introduce a further approximation, since det G (ˆµ ) n+ δ(ê), we 2nN replace in the above formula det G (ˆµ ) n+ by δ(ê). Finally, we pic Ê := [, ] 2 and ˆµ uniform probability measure on an admissible mesh for the square, for instance the Chebyshev Lobatto grid with (2 + ) 2 points, thus our approximation formula becomes (38) δ() ( ) n+ 2nN det G (µ ), 2 det G (ˆµ ) where we used δ([, ] 2 ) = /2; [9]. Finally, to compute the determinants of the Gram matrices on the right hand side of equation (38) we use the SVD factorization of the square root of the Gram matrices, note that for instance ( ) N det G (ˆµ ) = det V H M V = (det S ) 2 = σ 2 j, where V and M has been defined above and S = diag(σ,..., σ N ) is the diagonal matrix with the singular values the matrix V / M Numerical test of the TD-GD algorithm. In order to illustrate how our algorithm wors in practice, we perform two numerical tests for real compact sets whose transfinite diameters have been computed analytically in [9]. Namely, we consider the case of the unit dis B 2 = {x R 2 : x } and the unit simplex S 2 := {x R 2 + : x + x 2 } For such sets Bos and Levenberg computed formulas that in the specific case of dimension n = 2 read as δ(b 2 ) = 2e, δ(s 2 ) = 2e. j=