Finite Range Method of Approximation for Balance Laws in Measure Spaces

Home , Dirac measure, Lipschitz continuity

Kinetic and Related Models doi:10.3934/krm.2017027 c American Institute of Mathematical Sciences Volume 10, Number 3, September 2017 pp. 669–688

FINITE RANGE METHOD OF APPROXIMATION FOR BALANCE LAWS IN MEASURE SPACES

Piotr Gwiazda∗, Piotr Orlinski and Agnieszka Ulikowska Institute of Applied Mathematics and Mechanics, University of Warsaw ul. Banacha 2, 02-097 Warsaw, Poland

(Communicated by Jos´eAntonio Carrillo)

Abstract. In the following paper we reconsider a numerical scheme recently introduced in [10]. The method was designed for a wide class of size structured population models with a nonlocal term describing the birth process. Despite its numerous advantages it features the exponential growth in time of the number of particles constituting the numerical solution. We introduce a new algorithm free from this inconvenience. The improvement is based on the application the Finite Range Approximation to the nonlocal term. We prove the convergence of the derived method and provide the rate of its convergence. Moreover, the results are illustrated by numerical simulations applied to various test cases.

1. Introduction. The aim of this paper is to present an improvement of the numerical scheme which was introduced in [10] and which possesses an unfavourable feature: in many cases the number of particles constituting the numerical solution can increase exponentially in time. To deal with the problem we apply a Finite Range Approximation method to the nonlocal term appearing on the right hand side of the model equation (1), see Subsection 2.1 for details. As shown in [10], the convergence of the scheme follows from the stability estimate [9, Theorem 2.11 (ii)]; however, its assumptions are not fulﬁlled after the application of the approximation procedure. For that reason we need to establish a relaxed version of the stability estimate and apply a new strategy in the proof of convergence of the scheme. The scheme under consideration follows a current trend which is based on a kinetic approach to population dynamics problems [2,3, 17, 21, 22, 23, 26]. Within this approach a population of individuals is divided into groups, so called cohorts. Each cohort is represented by the number of individuals and by their average state. Thus it seems natural to approximate the distribution of the population by a linear combination of Dirac measures. Such a method of approximation is very suitable for numerical studies, especially when it comes to compatibility of a model with experimental data. Indeed, a result of a measurement of a population usually boils down to providing the number of individuals with their average state within the underlying cohort. A good example of such measurements are demographical studies which provide data about a size of the age-cohorts.

2010 Mathematics Subject Classiﬁcation. P92D25, 65M12, 65M75. Key words and phrases. Structured population models, particle method, measure valued solutions, Radon measures, ﬂat metric. ∗ Corresponding author: Piotr Gwiazda.

669 670 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA

A broad group of methods originating from the kinetic theory are particle methods, which are designed to model the behavior of large groups of interacting particles or individuals. Over the last decades they have been successfully applied to solve numerically many problems originating from physics as the Euler equation in fluid mechanics [19, 32] and the Vlasov equation in plasma physics [5, 12, 18]. Recently, the particle methods have been used in problems related to crowd dynamics and flow of pedestrians [17, 26, 27], models of a collective motion of large groups of agents [8, 16,4] and population dynamics [9]. For more applications see [24, 25, 28, 29] and references therein. In this paper we focus on the population dynamics and the following size structured population model ∂ ∂ Z µ + (b(t, µ)µ) + c(t, µ)µ = (η(t, µ))(y) dµ (y), (1) ∂t ∂x R+ where t ∈ [0,T ] and x ≥ 0 denote, respectively, time and the size of an individual. Solutions of (1) are considered in a weak sense, see [9, Definition 2.2] for the exact definition. In general, the x variable can describe other physiological states (e.g. length or weight) but for sake of simplicity we stick to the size variable. The measure µ is a distribution of individuals with respect to x. We assume that an individual changes its size according to the following ODE x˙ = b(t, µ)(x), where b describes the dynamics of the transformation, that is, the speed of the individual’s growth. Function c(t, µ)(x) stands for the death rate, and the integral term describes the birth process. To explain briefly the meaning of the nonlocal term in (1), we assume for a moment that function η does not depend neither on time t nor the population state µ. Then, for a fixed y ≥ 0, function η(y) describes a distribution (with respect to x) of the offspring of individuals at size y. Additionally, in the particular case where all new born individuals have the same size xb we set

η(y) = β(y)δx=xb , (2) where β(y) is related to the probability that individuals at size y procreate. If (2) holds, then the integral in (1) transforms into a boundary condition and, as a consequence, (1) can be reduced to the following classical renewal equation with the nonlocal boundary condition ∂ ∂ µ + (bµ) + cµ = 0, for x ≥ x , ∂t ∂x b Z (3) b b b(x )Dλµ(x ) = β(y) dµ (y), R+ b where Dλµ(x ) is the Radon-Nikodym derivative, if it exists, of µ with respect to the Lebesgue measure at xb. The model (1) describes a population which undergoes processes of birth, death and development. The number of individuals in the population and its total biomass change in time, which is clearly indicated by the nonconservative character of the problem. We need to underline that the lack of mass conservation is the main challenge associated with the application of the particle methods in population dynamics. Let us briefly recall that the most common mathematical framework for the kinetic theory is a space of probability measures equipped with a Wasserstein FINITE RANGE APPROXIMATION METHOD 671 distance. Unfortunately, the 1-Wasserstein distance W1 between two Radon measures µ and ν such that R dµ =6 R dν is infinite, which is the reason why natural distances for measures, like the Wasserstein distances, cannot be exploited in the case of the nonconservative problems. Indeed, let µ, ν be finite Radon measures on R such that µ(R) =6 ν(R). Then, according to [30, Section 7.1] Z W1(µ, ν) = sup ϕ(x) d( µ − ν)(x): Lip(ϕ) ≤ 1 R Z ≥ a d( µ − ν)(x) = a(µ(R) − ν(R)), R yields W1(µ, ν) = +∞, as a ∈ R can be arbitrarily large. Therefore, to consider the well–posedness of the models of population dynamics in the space of measures, the suitable framework is needed in the first place. It has been recently established by replacing the Wasserstein distance by the flat metric (see Section3 for definitions and technical details). One of the first steps in that field has been made in [22, 23], where existence, uniqueness and stability of solutions to (3) in the space of finite, nonnegative Radon measures equipped with the flat metric were proved. Within this framework the first formal proof of convergence of a corresponding particle method for (3) has been conducted in [6]. The method is called the Escalator Boxcar Train (EBT), and although it was described for the first time in the 80’s in [13], the proof of its convergence and the convergence rate [20] is very recent. Well–posedness of a general size-structured population model (1) in the space of measures was established in [9], and a numerical scheme based on the particle methods was developed in [10]. In the latter paper an essential assumption is the particular form of the η function, namely r X η(y) = βp(y)δx=fp(y), (4) p=1 r which means that the size of a child belongs to a set {fp(y)}p=1, where y stands for b the size of its parent. For instance, letting r = 1, f1(y) = x , and β1(y) = β > 0 corresponds to the special case of (2), and leads to the equation (3). Another common example is a simple symmetric cell division model, which arises due to the 1 following settings; r = 1, f1(y) = 2 y, and β1(y) = β > 0. The asymmetric case may be obtained with r = 2, f1(y) = σy, f2(y) = (1 − σ)y, where 0 < σ < 1, and β1(y) = β1 > 0, β2(y) = β2 > 0. In both cases the cell division process is understood as the birth of two new cells and the death of a mother cell, which should be incorporated in the death rate. As has been already stated above, in the kinetic approach a solution is approximated by a linear combination of Dirac measures at each discrete time moment. In the case of the algorithm developed in [10] Dirac deltas represent cohorts, that is groups of individuals of a similar size. Since a population undergoes a process of births, at least one additional Dirac measure is created at each time step of the algorithm. In case of the particular choice of (2) it is exactly one Dirac measure, since all new born individuals have the same size. However, in the case of the symmetric cell division model the number of Dirac measures is doubled at each time step, which results in the exponential growth of particles. In order to prevent this phenomenon authors of [10] developed a reconstruction procedure, which is in fact an approximation procedure. More precisely, if too many Dirac deltas are created at a particular time step, they are simply approximated by a measure composed of 672 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA a smaller number of them. Nevertheless, the reconstruction has to be performed once every several time steps, which influences an accuracy of a numerical solution. Our improvement of the scheme presented in [10] is based on a new way of approximation. We postulate to approximate properly the η function given by (4) before performing any numerical simulation. Namely, we approximate the functions ε fp in (4) by piecewise constant functions fp and, as a consequence, new Dirac measures appear only at some fixed points of the ambient space. Therefore, we run the scheme with a slightly inaccurate coefficients, but as a consequence, we do not need to perform neither approximations nor reconstructions during its execution. The main problem of such approximation is that the new η, that is r X ε η˜(y) = βp(y)δx=fp (y), p=1 does not fulfill the assumptions [10, Assumptions (3.1) - (3.4)] providing well– posedness of (1) in the space of measures. Indeed, the piecewise constant functions are not a subclass of the Lipschitz functions. However, in the present paper we have overcome this obstacle by developing a relaxed version of the stability estimate, see Subsection 4.2 (Remark2 and Theorem 4.2). To accomplish the task we assumed that f(x) ≤ x, which is not a restrictive limitation since the sublinearity of f is biologically justified. Let us mention just the basic examples, i.e. the age of a new born individual is always equal to zero, which is not greater that its parent’s age, the sizes of daughter cells are smaller than the size of a mother cell before the mitosis process, a polymer chain is shorter after the division process. This paper is organized as follows. In Section2 we briefly describe the numerical scheme and a Finite Range Approximation method. Section3 consists of some basic facts about the space of finite, nonnegative Radon measures equipped with the flat metric. For the sake of completeness we also justify the choice of the latter space instead of the Banach space (W1,∞)∗. In Section4 we provide convergence results. In Section5 we show results of numerical simulations for several test cases.

2. Splitting-particle method with ﬁnite range approximation.

2.1. Finite range approximation. In the following subsection we show how to construct a Finite Range Approximation of a Lipschitz continuous function. The Lipschitz continuity of fp in (4) is the assumption required for well–posedness of (1), see [10, Assumption (3.4)]. Before we proceed let us take a closer look at the following example.

1 Example 1. Setting r = 1, f1(y) = 2 y, and β1(y) = β > 0 in (4) yields the symmetric cell division model. Application of the particle-based scheme developed in [10] to that model results in the exponential growth of Dirac measures approximating a solution. It is a consequence of the fact that a child’s size is exactly a half of its parent size. If we substitute the function f1 by a suitable piecewise constant approximation, then a set of all possible sizes of the children becomes ﬁnite. An example of such approximation is shown on Figure1.

Deﬁnition 2.1. Let f : R → R be a Lipschitz function. We say that f ε : R → R is a Finite Range Approximation of f on the interval [0,M), if ε ε #Im(f ) < ∞, and k(f − f )|[0,M)kL∞ < ε, FINITE RANGE APPROXIMATION METHOD 673

y x - size of a parent f(x) - size of an oﬀspring function f FRA of f

Figure 1. Function f and its Finite Range Approximation. where ε, M > 0 are arbitrary constants, Im(fε) is the image of the function f ε, and #A denotes the number of elements in a set A. The next lemma shows that a Finite Range Approximation exists, while its proof demonstrates how to construct one.

Lemma 2.2. Let ε, M > 0, and f : R → R be a Lipschitz continuous function such that f(x) ≤ x, for all 0 ≤ x < M. Then, there exists a function f ε that is a Finite Range Approximation of f and satisfies fε(x) ≤ x. M Proof. Since f(x) ≤ x, it holds that f|[0,M) < M. Let J = d ε e, and define Aj = −1 J (f|[0,M)) ( [(j − 1)ε, jε) ), for j = 1,...,J. Then, obviously, ∪j=1Aj = [0,M). ε Since f is defined for all x ∈ R, we redefine the sets A1 := (−∞, 0) ∪ A1, and ε AJ := AJ ∪ [M, +∞). The approximation f is thus given by the formula J ε X f (x) = ajχAj (x), (5) j=1 where aj = (j − 1)ε and χAj is the characteristic function of the set Aj. It follows directly from the construction of f ε that it is a Finite Range Approximation of f and f ε(x) ≤ x, for all 0 ≤ x < M. Henceforth, the Finite Range Approximation constructed in Lemma 2.2 is considered to be the canonical one, which will be referred to simply as the FRA. Remark 1. For our purposes, it is sufficient to consider a finite interval [0,M). Note that the equation ∂ ∂ µ + (b(t, µ)µ) + c(t, µ)µ = 0, (6) ∂t ∂x possesses the finite propagation speed property. Assume that the support of the initial data µo is contained in the interval [0,Mo), for some Mo > 0. Then, the support of a solution µ(t) at time t ∈ [0,T ] is a subset of the interval [0,M), for some M > 0. The constant M depends on Mo, a suitable norm of b, and the length of the time interval [0,T ]. Substituting the right hand side of (6) by R η dµ, where 674 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA

η is given by (4) and assuming that fp(x) ≤ x, for p = 1, . . . , r, does not inﬂuence the bound on the support of µ(t), for any t ∈ [0,T ]. In other words, the integral does not change the support of solution since for each y ≥ 0

supp(η(y)) = {x ∈ R+ : x < y} . 2.2. Description of the algorithm. For ﬁxed T > 0 and N ∈ N we deﬁne the length of a time step ∆t = T/N and a set of discrete time points tk = k∆t, where k = 0,...,N. Assume that the numerical solution µk at time tk is a linear combination of Nk Dirac deltas;

Nk X i µ = m δ i , k k xk i=1 i i where mk is a mass of the i-th Dirac delta and xk denotes its location. In particular, we assume that the initial data is a linear combination of Dirac measures. If this is not the case, the initial data can be approximated by such a sum, see Lemma 4.5 for details. Let M > 0 be a constant, such that the support of a solution to (1) is contained in the interval [0,M), for all t ∈ [0,T ]. Such a constant exists according to Remark1. Since the algorithm is based on the splitting technique [10], it is divided into two steps described below. Step 1. The ﬁrst step is to calculate the new locations of “existing” Dirac measures. These are obtained by solving the following system of ODEs d xi(t) = b (xi(t)), xi(t ) = xi , i = 1,...,N , where b = b(t , µ ), (7) dt k k k k k k k on a time interval (tk, tk+1]. A result of this step is a measure

Nk X i i i µ¯ = m δ i , where x = x (t +1). k k xk+1 k+1 k i=1 Step 2. The second step is to determine locations of the new Dirac measures, which correspond to the newly born individuals. We also need to recalculate masses of all Dirac measures. In order to determine the locations of the new particles we use the FRA of functions fp in (4) provided by Lemma 2.2. According to (5), new J particles appear only at points {aj}j=1, and these locations are ﬁxed in time. As a consequence, we obtain a set of Nk+1 := Nk + J Dirac measures. In order to recalculate the masses we solve the following system of ODEs  d i i i  m (t) = −ck(x +1)m (t),  dt k  i i  m (tk) = mk, 1 ≤ i ≤ Nk, (8) d  i i i PNk+1 Pr j j i  m (t) = −ck(xk+1)m (t) + j=1 p=1 m (t)αpk(xk+1, xk+1),  dt  i  m (tk) = 0,Nk + 1 ≤ i ≤ Nk+1, where ck = c(tk, µ¯k), βpk = βp(tk, µ¯k) and ( j j β (x ), if f ε(x ) = xi , j i pk k+1 p k+1 k+1 αpk(xk+1, xk+1) = 0, otherwise. After reassignment of indices (e.g., arranging all Dirac measures in the ascending PNk+1 i order with respect to the location) we obtain a measure µ +1 = m δ i , k i=1 k+1 xk+1 FINITE RANGE APPROXIMATION METHOD 675 which is the output of the algorithm at time tk+1. The essential feature of this algorithm is that after k time steps the number of Dirac deltas approximating the solution is equal to No + Jk, where No is the initial number of Dirac measures, as k opposed to e.g. No · 2 (for symmetric cell splitting).

+ + 3. The space of measures (M (R+), ρF ). Henceforth, M (R+) denotes the space of nonnegative Radon measures with bounded total variation on R+ := {x ∈ + R : x ≥ 0}. We equip M (R+) with the flat metric (Z ) 1 ρF (µ1, µ2) = sup ϕ d( µ1 − µ2): ϕ ∈ C (R+) and kϕkW1,∞ ≤ 1 , R+ where kϕkW1,∞ = max {kϕkL∞ , k∂xϕkL∞ }. Using the standard mollification procedure, ρF can be equivalently rewritten as (Z ) 1,∞ ρF (µ1, µ2) = sup ϕ d( µ1 − µ2): ϕ ∈ W (R+) and kϕkW1,∞ ≤ 1 , R+ + and therefore, for all µ1, µ2 ∈ M (R+) it holds that ρF (µ1, µ2) = kµ1 −µ2k(W1,∞)∗ . + Note that in this paper, the space M (R+) is equipped with the metric ρF and this + shall remain until said differently (see (33) in Section5). The space ( M (R+), ρF ) is complete and separable. This equality gives a rise to a question about the pos- 1,∞ ∗ sibility of setting the model in the Banach space (W ) , k · k(W1,∞)∗ . It is a natural question, considering the previous papers in which the problem of continuity of solutions with respect to time was addressed. In [31] the authors established the continuity of solutions to the age structured population model (3) in the L1 topology. Then, it was proved in [15] that solutions to balance laws in the space of measures are continuous in the weak-* topology of the Radon measures space. Un- fortunately, it turns out that the answer to our initial question is negative, because 1,∞ ∗ 1 shift operators are not continuous in (W ) , k · k(W1,∞)∗ in contrast to the L topology, see Lemma 3.1 for details. This is the obstacle one cannot overcome, since the continuity of the shift operators is essential for obtaining the continuity of solutions to the transport equation.

Lemma 3.1. One-parameter semigroup {T }y≥0 of the shift operators is not a strongly continuous semigroup on the space (W1,∞(R))∗. Moreover, for all y > 0 it holds that

kTy − IkL((W1,∞)∗) ≥ 1. where k · kL((W1,∞)∗) is the operator norm. Deﬁnition of the shift operator on (W1,∞(R))∗ and proof of Lemma 3.1 can be found in Appendix.

4. Convergence of the algorithm. 4.1. Theoretical results concerning well–posedness of (1). In this subsection we recall theoretical results from [9] concerning well–posedness of (1). Assume that α,1 + 1,∞ b, c, βp ∈ BC [0,T ] × M (R+); W (R+) , for p = 1, . . . , r,

fp ∈ Lip(R+; R+), fp(x) ≤ x, for p = 1, . . . , r, (9) + b(t, µ)(0) ≥ 0, for (t, µ) ∈ [0,T ] × M (R+). 676 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA

α,1 + 1,∞ 1,∞ Here, BC ([0,T ] × M (R+); W (R+)) is the space of W (R+) valued functions which are uniformly bounded in the k · kW1,∞ norm, Höldercontinuous with exponent 0 < α ≤ 1 with respect to time and Lipschitz continuous in the flat metric ρF with respect to the measure variable. This space is equipped with the k · kBCα,1 norm defined by

kfkBCα,1 = kfkBC + sup Lip (f(t, ·)) + sup Hα (f(·, µ)) , + t∈[0,T ] µ∈M (R+) where kfkBC = sup kf(t, µ)kW1,∞ , + t∈[0,T ],µ∈M (R+) Lip(f) is the Lipschitz constant of a function f and

kf(s1, µ) − f(s2, µ)kW1,∞ Hα(f(·, µ)) := sup α . s1,s2∈[0,T ] |s1 − s2| Since in [10] the specific form (4) of the η function has been assumed, we rewrite the original well–posedness theorem [9, Theorem 2.11] in the terms of βp and fp instead of η, see Theorem 4.1 below. Regularity of βp and fp imposed in (9) guarantees that η defined by (4) fulfills the assumptions of [9, Theorem 2.11] and thus, (1) is well posed. + Theorem 4.1. [9, Theorem 2.11] Let (9) hold and µo ∈ M (R+). Then, there exists a unique solution

+ µ ∈ (BC ∩ Lip) [0,T ]; (M (R+), ρF ) to (1). Moreover, the following properties are satisﬁed:

1. For all 0 ≤ t1 ≤ t2 ≤ T there exist constants K1 and K2, such that

K2(t2−t1) ρF (µ(t1), µ(t2)) ≤ K1 e µo(R+) · (t2 − t1). + i i i i 2. Let µ1(0), µ2(0) ∈ M (R+) and bi, ci, βi = (β1, . . . , βr), fi = (f1, . . . , fr) satisfy assumptions (9) for i = 1, 2, p = 1, . . . , r. Let µi solve (1) with initial datum µi(0) and coeﬃcients (bi, ci, βi, fi). Then, there exist constants C1, C2 and C3 such that

C1t ρF (µ1(t), µ2(t)) ≤ e ρF (µ1(0), µ2(0)) (10) C3t + C2t e (k(b1 − b2, c1 − c2, β1 − β2)kBC + kf1 − f2kL∞ ) , for all t ∈ [0,T ]. All constants in Theorem 4.1 depend on suitable norms of the model coefficients. Unfortunately, the constants C2,C3 in the second claim depend on the Lipschitz i i constants of f1, ··· , fr, for i = 1, 2. This is the main obstacle we need to overcome, since the FRA of a Lipschitz function is not a Lipschitz function (in fact it is not even continuous). In the next subsection we show how to deal with this problem. 4.2. Theoretical results concerning well–posedness of (1) with a relaxed version of the stability estimate. The first problem with the FRA is that it produces non-continuous functions, which implies that assumptions (9) are not ful- filled and, as a consequence, there are no results concerning well–posedness of (1). In this subsection we first show how to substitute these noncontinuous functions by suitable Lipschitz continuous functions in a way that does not affect the solutions of (7)–(8). This is a subject of the following remark. FINITE RANGE APPROXIMATION METHOD 677

Remark 2. Define a set D consisting of locations of all Dirac measures at all discrete timeslices N [ i Nk D := {xk}i=1, k=0 i where xk is the location of the i-th Dirac delta at time tk = k∆t. This is clearly a finite set, D ⊂ [0, Λ), where Λ ≥ 0 does not depend on N, which follows from the construction of numerical scheme, see Section 2.2 for details. More precisely, the transport problem (Subsection 2.2, Step 2) admits the finite propagation speed, and the ODE problem (Subsection 2.2, Step 3) does not widen the support (compare with Remark1). ε Let fp be the FRA of fp constructed in Lemma 2.2, for p = 1, . . . , r. Without ε ε loss of generality we assume that fp, fp : R+ → R+. For a fixed p, fp is a piecewise constant function, which can be rewritten in the following form

Jp ε X f (x) = a χ p (x), p i Di i=1 ε where ai ≥ 0, Jp is greater or equal to the number of elements in the image of fp , p p p { p}Jp p ∞ Di = [di , di+1), di i=1 is a finite, strictly increasing sequence, and dJp+1 = + Jp p such that ∪i=1Di = R+. Define p,max p di = max{x : x ∈ D ∩ Di }, for i = 1,...,Jp − 1, ¯ε and a piecewise linear function fp  a , for x ∈ [d p, d p,max), i < J ,  i i i p  a d p − d p,maxa f¯ε(x) = ai+1 − ai i i+1 i i+1 p,max p p p p,max x + p p,max , for x ∈ [di , di+1), i < Jp,  di+1 − di di+1 − di  aJp , for x ∈ DJp . (11) It follows directly from the construction that for all x ∈ D it holds that ¯ε ε ε ¯ε fp (x) = fp (x), and kfp − fp kL∞ ≤ ε. ε ¯ε An example of a piecewise constant function fp and its Lipschitz counterpart fp is shown of Figure2. Henceforth, we use the modification described above instead of the correspond- ¯ε ing FRA. Note that the functions fp are Lipschitz continuous, but their Lipschitz constants may increase due to the increase of a number of Dirac measures approximating a solution. Therefore, there is still some work to be done. Namely, we need to obtain an estimate analogous to (10), but with all constants independent ¯ε of the Lipschitz constants of the functions fp . This is the subject of the following theorem.

+ i i Theorem 4.2. Let µ1(0), µ2(0) ∈ M (R+) and bi, ci, βi = (β1, . . . , βr), fi = i i (f1, . . . , fr) satisfy assumptions (9) for i = 1, 2, p = 1, . . . , r. Let µi be the solution to (1) with initial datum µi(0) and the coeﬃcients (bi, ci, βi, fi). Then, there exists a constant C depending only on suitable norms of (b1, c1, β1, f1), such that Ct ρF (µ1(t), µ2(t)) ≤ e ρF (µ1(0), µ2(0)) (12) Ct + Ct e k(b1 − b2, c1 − c2, β1 − β2)kBC + kf1 − f2kL∞ , 678 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA

y function f ε function f¯ε elements from the set D

Figure 2. Function f ε and its approximation f¯ε.

where kfk = sup + kf(t, µ)k ∞ . BC t∈[0,T ],µ∈M (R+) L The significance of the estimate (12) lies in the fact that the constant C depends only on coefficients (b1, c1, β1, f1). Therefore, we can plug into (12) the original ¯ε functions fp and their approximations fp described in Remark2. Note that the ¯ε only place where the functions fp appear in (12) is the term kf1 − f2kL∞ , which can be estimated by ε, according to Lemma 2.2. The proof of Theorem 4.2 is based on formula (13) (see [7, Theorem 2.9]), which allows to consider equations locally in time. Before we proceed, we introduce some preliminary notions. Definition 4.3. Let (X, ρ) be a metric space. A map S : [0,T ] × X → X is called a Lipschitz semiflow, if

1. S0x = x for all x ∈ X, 2. St+sx = StSsx for all t, s, t + s ∈ [0,T ] and x ∈ X. 3. ρ(Stx, Ssy) ≤ L1ρ(x, y) + L2|t − s|. Proposition 1. [7, Theorem 2.9] Let S : [0,T ] × X → X be a Lipschitz semiﬂow. Then, for every Lipschitz continuous map ν : [0,T ] → X the following estimate holds Z ρ ντ+h,Shντ ρ νt,Stνo ≤ L1 lim inf dτ, (13) [0,t] h↓0 h where ρ is a corresponding metric.

Proof. Note that C1 in (10) depends only on (b1, c1, β1, f1). Indeed, let ν(t) be a solution to (1) with coeﬃcients (b1, c1, β1, f1) and initial condition µ2(0). Then,

ρF (µ1(t), µ2(t)) ≤ ρF (µ1(t), ν(t)) + ρF (ν(t), µ2(t)). Using estimate (10) from Theorem 4.1 yields C1t ρF (µ1(t), µ2(t)) ≤ e 1 ρF (µ1(0), µ2(0)) 1,2 1,2 C3 t + C2 t e (k(b1 − b2, c1 − c2, β1 − β2)kBC + kf1 − f2kL∞ ) , FINITE RANGE APPROXIMATION METHOD 679

1 1,2 where C1 = C1(b1, c1, β1, f1), Ci = Ci(b1, c1, β1, f1, b2, c2, β2, f2), i = 2, 3. There- fore, we may assume that µ1(t) and µ2(t) are solutions to (1) with the same initial data µo. As the ﬁrst step, we deﬁne the time dependent functions

j bj(t, x) = bj(t, µj(t))(x), cj(t, x) = cj(t, µj(t))(x), βj,p(t, x) = βp(t, µj(t))(x),

n i n for j = 1, 2, p = 1, . . . , r. Fix n ∈ N, deﬁne ∆t = T/2 , tn = i∆t for i = 0, 1,..., 2 , and approximate bj, cj and βj,p as follows:

2n−1 j X i bn(t, x) = bj(tn, x) χ i i+1 (t), [tn,tn ) i=0 2n−1 j X i cn(t, x) = cj(tn, x) χ i i+1 (t), [tn,tn ) i=0 2n−1 j,p X i βn (t, x) = βj,p(tn, x) χ i i+1 (t) . [tn,tn ) i=0

i i+1 Note, that on each interval [tn, tn ) functions defined above do not depend on t. j j Therefore, according to [9, Theorem 2.8], solving (1) with coefficients bn, cn, and j,p i i+1 j,i,n βn on the time interval [tn, tn ) yields a Lipschitz semigroup. Call S the j,n + + corresponding semigroup and define the map F : [0,T ] × M (R+) → M (R+) by

j,n j,i,n i−1 j,q,n i i+1 F µ = S i ◦ S n µ, for t ∈ [t , t ). t t−tn q=0 T/2 n n

k We assume that the empty concatenation, that is q=0 with k < 0, is the identity operator. It follows from the construction that F j,n is a Lipschitz semiﬂow. Without k 1,n 2,n loss of generality we assume that t = tn, for some k. Substituting F and F to (13) yields

1,n 2,n ρF Ft µo,Ft µo k−1 Z 1,i,n 1,n 2,i,n 1,n X ρF S F µo,S F µo ≤ Lip(F 1,n) lim inf h τ h τ dτ. i i+1 h↓0 h i=0 [tn,tn )

According to estimates [9, proof of Theorem 2.8] for the linear autonomous problem, it holds that

1,i,n 1,n 2,i,n 1,n Z ρF S F µo,S F µo lim inf h τ h τ dτ i i+1 h↓0 h [tn,tn ) Z 1,n 1 i 2 i 1 i 2 i ≤ (Fτ µo)(R+)dτ kbn(tn, ·) − bn(tn, ·)kL∞ + kcn(tn, ·) − cn(tn, ·)kL∞ i i+1 [tn,tn ) r r X 1,p i 2,p i X 1 2 + kβn (tn, ·) − βn (tn, ·)kL∞ + kfp − fp kL∞ p=1 p=1 Z 1,n ≤ k(b1 − b2, c1 − c2, β1 − β2)kBC + kf1 − f2kL∞ (Fτ µo)(R+)dτ. i i+1 [tn,tn ) (14) 680 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA

Combining (14) with (14), and summing over i = 0, . . . , k − 1 yields

1,n 2,n 1,n ρF Ft µo,Ft µo ≤ Lip(F ) k(b1 − b2, c1 − c2, β1 − β2)kBC + kf1 − f2kL∞ Z 1,n · (Fτ µo)(R+)dτ. [0,t] According to the estimate from [9, proof of Theorem 2.10, claim iii)] it holds that 1,n Cτ¯ ¯ (Fτ µo)(R+) ≤ e µo(R+), where C depends only on (b1, c1, β1, f1). Therefore, there exists a constant C, which depends on this set of coeﬃcients and µo, such that 1,n 2,n Ct ρF Ft µo,Ft µo ≤ Ct e k(b1 − b2, c1 − c2, β1 − β2)kBC + kf1 − f2kL∞ . (15)

j,n According to [9] (see the proof of Theorem 2.10), the map Ft µo converges uniformly with respect to time to µj(t), as n → +∞. Therefore, passing to the limit in (15) completes the proof.

4.3. Error estimates in ρF . The following theorem provides the estimate on the rate of convergence of the numerical method described in Subsection 2.2. Theorem 4.4. Let µ be a solution to (1) with coefficients (b, c, β, f) and initial data µo. Let µk be a numerical solution at time tk = k∆t obtained by systems (7) ¯ε δ ¯ε - (8) with coefficients (b, c, β, f ) and initial data µo, where f is the modified FRA δ of f defined by (11) and µo is a linear combination of Dirac deltas. Then, there exists a constant C depending on (b, c, β, f), µo, and T , such that α ρF (µk, µ(tk)) ≤ C (∆t + (∆t) + ε + I(µo)) , (16) where ∆t is the length of the time step, ε is the error of the modified FRA (11), ¯ε δ that is kf − f kL∞ < ε, and I(µo) := ρF (µo, µo) is the error of the initial data approximation. Remark 3. Theorem 4.4 proves the convergence of the numerical approximation with f ε, since the approximations with f ε and f¯ε are the same (see Subsection 4.2). Remark 4. The error estimate (16) accounts for different error sources. More specifically, the error of the order O(∆t) is a consequence of the splitting algorithm. The term of order O((∆t)α) follows from the fact that we solve ODEs with parameter functions independent of time, while b, c and η are in fact Höldercontinuous with exponent α with respect to time. The error of the initial data approximation I(µo) is inversely proportional to the number of Dirac deltas approximating µo, see Lemma 4.5. Hence, I(µo) can be arbitrarily small. Remark 5. It follows from Theorem 4.4 that limits ∆t → 0 and ε → 0 com- mute. However, passing to the limit with ε when ∆t is fixed causes that the Finite Range Approximation property is lost, and as a consequence, equation (1) cannot be numerically solved by (7)–(8). Proof. The proof is divided into several steps. For simplicity, in all estimates below, we will use a generic constant C, without specifying its exact form that may change from line to line. Step 1. The auxiliary scheme. In this step we exploit the splitting technique [10]. Let us define the auxiliary semi-continuous scheme, which consists in solving FINITE RANGE APPROXIMATION METHOD 681 subsequently the problems ( ∂ ∂ ¯ µ + (bk(x)µ) = 0, on [tk, tk+1] × R+ ∂t ∂x (17) µ(tk) = µk, and ( ∂ R µ + c¯k(x)µ = η¯k(y)dµt(y), on [tk, tk+1] × R+ ∂t R+ (18) µ(tk) =µ ¯k, + ¯ where µk ∈ M (R+),µ ¯k is the solution to (17) at time tk+1 and bk, c¯k, and η¯k are defined as

¯bk(x) =b (tk, µk)(x), (19)

r X c¯ (x) =c (t , µ¯ )(x), η¯ (y) = β (t , µ¯ )(y) δ ¯ε . (20) k k k k p k k x=fp (y) p=1

A solution to the second equation at time tk+1 is denoted by µk+1. Denote by νk+1 ¯ Pr a solution to (18) with η¯k deﬁned as p=1 βp(tk, µ¯k)(y) δx=fp(y). Step 2. Error of the FRA. According to (12), it holds that

C∆t ε C∆t ρF (µk+1, νk+1) ≤ C∆t e kf − f¯ kL∞ ≤ C∆t e ε ≤ Cε˜ ∆t, (21) where C˜ is such that C eCh ≤ C˜ for all h ∈ [0,T ]. Step 3. Error of splitting. Let ν(t) be a solution to (1) on a time interval [tk, tk+1] with initial datum µk and parameter functions ¯bk,c ¯k,η ¯k, where ¯bk is deﬁned by (19),

c¯k(x) =c (tk, µk)(x), (22) r r X X ¯ η¯k(y) = βp(tk, µk)(y) δx=fp(y) =: βp,k(y) δx=fp(y). (23) p=1 p=1 According to [11, Proposition 2.7] and [9, Proposition 2.7], the distance between νk+1 and ν(tk+1), that is, the error coming from the application of the splitting algorithm can be estimated as 2 ρF (νk+1, ν(tk+1)) ≤ C(∆t) , (24) where C depends on the k·kW1,∞ norm of b, c, β and the Lipschitz constant Lip(fp), p = 1, . . . , r. To estimate a distance between ν(tk+1) and µ(tk+1) consider ζ(t), which is a solution to (1) on a time interval [tk, tk+1] with initial data µ(tk) and coeﬃcients ¯bk,c ¯k,η ¯k. By triangle inequality

ρF (ν(tk+1), µ(tk+1)) ≤ ρF (ν(tk+1), ζ(tk+1)) + ρF (ζ(tk+1), µ(tk+1)). The first term of the inequality above is a distance between solutions to (1) with different initial data, that is, µk and µ(tk) respectively. The second term is equal to a distance between solutions to (1) with coefficients (¯bk, c¯k, η¯k) defined by (19), (22), (23) and (b(t, µ(t)), c(t, µ(t)), η(t, µ(t))), respectively. By the continuity of solutions to (1) with respect to the initial datum and coefficients in Theorem 4.2, we obtain

C∆t ρF (ν(tk+1), ζ(tk+1)) ≤ e ρF (µk, µ(tk)), (25) 682 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA and r ! C∆t ¯ X ¯ ρF (ζ(tk+1), µ(tk+1)) ≤ C∆t e k(bk − b, c¯k − c)kBC + kβp,k − βpkBC , p=1 (26) k ¯ − k k ¯ − k ∞ ∈ { } where fk f BC = supt∈[tk,tk+1] fk f(t, µ(t)) L , and f b, c, βp . By the assumptions (9) and deﬁnition (19) of ¯bk

k¯bk − b(t, µ(t))kL∞ ≤ kb(tk, µk) − b(tk, µ(t))kL∞ + kb(tk, µ(t)) − b(t, µ(t))kL∞ α ≤ Lip(b(tk, ·)) ρF (µk, µ(t)) + kbkBC|t − tk| . (27) Using the Lipschitz continuity of the solution µ(t) (Theorem 4.2) yields C∆t ρF (µk, µ(t)) ≤ ρF (µk, µ(tk)) + ρF (µ(tk), µ(t)) ≤ ρF (µk, µ(tk)) + C∆t e . Substituting the latter expression into (27) yields C∆t α k¯bk − b(t, µ(t))kL∞ ≤ Lip(b) ρF (µk, µ(tk)) + C∆t e + kbkBC(∆t) , ¯ where Lip(b) = supt∈[0,T ] Lip(b(t, ·)). Bounds for kc¯k − ckBC and kβp,k − βpkBC can be proved analogously. From the assumptions (9) it holds that r X k(b, c, β)kBCα,1 := Lip(b) + Lip(c) + k(b, c)kBC + Lip(βp) + kβpkBC < +∞, p=1 and as a consequence, we obtain r ¯ X ¯ kbk − bkBC + kc¯k − ckBC + kβp,k − βpkBC p=1 C∆t α ≤ k(b, c, β)kBCα,1 ρF (µk, µ(tk)) + C∆t e + (∆t) . Using this inequality in (26) yields

C∆t α ρF (ζ(tk+1), µ(tk+1)) ≤C∆t e ρF (µk, µ(tk)) + ∆t + (∆t) C∆t CT 2 CT 1+α ≤C∆t e ρF (µk, µ(tk)) + C e (∆t) + C e (∆t) . Combining the inequality above with (25) and redeﬁning C leads to C∆t 2 1+α ρF (ν(tk+1), µ(tk+1)) ≤ e (1 + C∆t)ρF (µk, µ(tk)) + C(∆t) + C(∆t) (28) 2C∆t 2 1+α ≤ e ρF (µk, µ(tk)) + C(∆t) + C(∆t) . Finally, putting together (21), (24), and (28) we conclude that 2C∆t 2 1+α ρF (µk+1, µ(tk+1)) ≤ e ρF (µk, µ(tk)) + C(∆t) + C(∆t) + Cε∆t. (29)

Step 4. Adding the errors. Application of the discrete Gronwall’s inequality to (29) yields eCk∆t − 1 ρ (µ , µ(t )) ≤ eCk∆t ρ (µδ, µ ) + C (∆t)2 + (∆t)1+α + ε∆t . F k k F o o eC∆t − 1 There exists a constant C∗ depending only on T such that eCk∆t − 1 < C∗k∆t, for each k∆t ∈ [0,T ]. Therefore, we deduce eCk∆t − 1 C∗k∆t C∗ ≤ = k, eC∆t − 1 C∆t C FINITE RANGE APPROXIMATION METHOD 683 and thus, Ck∆t α ρF (µk, µ(tk)) ≤ e I(µo) + Ck∆t (∆t + (∆t) + ε) .

Since k∆t = tk ≤ T , the assertion is proved. + In the following lemma we show that any measure ν ∈ M (R+) can be approximated in ρF with an arbitrarily small error by a sum of Dirac deltas. + R Lemma 4.5. Let ν ∈ M ( +) be such that Mν = dν =6 0. Then, for each R R+ σ > 0 there exists K ∈ N which is inversely proportional to σ, and a measure PK ν˜ = i=1 miδxi , such that ρF (ν, ν˜) ≤ σ. (30) + Proof. A measure ν ∈ M (R+) is tight. Therefore, for each ε > 0 there exists ε Kε > 0 such that ν(R+\[0,Kε]) ≤ ε. Define ν as a restriction of ν to [0,Kε]. Let 1,∞ ϕ ∈ W (R+). Then, Z Z Z ϕ(x) d( ν − νε)(x) = ϕ(x) d( ν − νε)(x) + ϕ(x) d( ν − νε)(x) R+ [0,Kε] (Kε,+∞) Z = ϕ(x) dν (x) ≤ kϕkL∞ ε. (Kε,+∞) 1,∞ Taking supremum over all ϕ ∈ W (R+) such that kϕkL∞ ≤ 1 yields ε ρF (ν, ν ) ≤ ε. (31) Let M ε = R dνε = R dν. Then, according to [10, Lemma 2.1 and (2.17)], ν R+ [0,Kε] the error of the fixed-location approximation of νε by a measureν ˜ consisting of Q Dirac deltas is equal to K ρ (ν , ν˜) ≤ M ε ε . (32) F ε v 2Q Let us recall that the fixed-location approximation is a procedure which divides a support of a measure into equal intervals and put a Dirac Delta with a proper mass in the middle of the interval. The mass of each Dirac Delta is equal to the mass contained in the corresponding interval. σ 4 ≥ Taking ε = in (31) and Q ε in (32) finishes the proof. 2 Mν Kεσ 5. Numerical results. In this section we present results of numerical simulations for the symmetric cell division model. We assume that a cell is not able to divide, unless it reaches size xo, and, with probability 1, it divides before reaching its maximal size xmax. From these assumptions the existence of the minimal cell size 1 follows, which is equal to 2 xo. Following [1], we set the model coefficients to

1 1 3 x0 = , x = 1, µ0(x) = (1 − x)(x − x0) , 4 max 2 b(x) = 0.1(1 − x), c(x) = β(x), η(t, µ)(y) = 2β(y)δ 1 , x= 2 y where ( 0, for y ∈ [0, x0) ∪ (xmax, +∞), β(y) = b(y)g(y) y ∈ 1−R g(x)dx , for y [x0, xmax], x0 and ( 160 (− 2 + 8 y)3, for y ∈ [x , x0+1 ], g(y) = 117 3 3 0 2 640 16 5 2 5120 5 3 8 11 x0+1 117 −1 + 2y + 3 (y − 8 ) + 9 (y − 8 ) ( 3 y − 3 ), for y ∈ ( 2 , xmax]. 684 PIOTR GWIAZDA, PIOTR ORLINSKI AND AGNIESZKA ULIKOWSKA

Note that b, c, and η do not depend on time. Therefore, the term of order α in (16) does not play a role in the order of convergence estimate (34). Since the main purpose of this section is to examine performance of the numerical scheme, our choice of the parameters is not dictated by any particular biological phenomenon. We consider the following concept of error

Err(T, ∆t, ε) := ρ(µref (T ), µk), where T is the final time, ∆t stands for the length of the time step, ε denotes the accuracy of FRA (see Lemma 2.2), µref (T ) is a reference solution, µk is the solution to (1) constructed in Section 2, and k is such that k∆t = T . The reference solution µref (T ) is the solution to (17)–(18) calculated by finite difference methods with the small parameters ∆t = ∆x = ε = 3.90625 · 10−05, where ∆x is proportional to I(µo), see Lemma4 for more details. Function ρ is defined by µ ν ρ(µ, ν) = min{Mµ,Mν }W1 , + |Mµ − Mν |, (33) Mµ Mν R where Mµ = dµ, and W1 is the 1-Wasserstein metric on the space of probability R+ measures on R+. W1 may be equivalently expressed as Z

W1(µ1, µ2) = |Fµ1 (x) − Fµ2 (x)|dx, R+ where Fµi denotes the cumulative distribution function of the measure µi. Function ρ is equivalent to the ﬂat metric ρF in the sense that there exists a constant C such that

Cρ(µ1, µ2) ≤ ρF (µ1, µ2) ≤ ρ(µ1, µ2), see [10, Lemma 2.1] for details. The order of the method is deﬁned by

Err(T,2∆t,ε) log Err(T,∆t,ε) q := lim . (34) ∆t→0 log 2 In Tables1-4 we present results of numerical simulations with different values of ∆t and ε. Parameter ∆x is always equal to the corresponding value of ∆t. For ∆t = ∆x << ε and a fixed ε, term log (Err(T, 2∆t, ε)/ Err(T, ∆t, ε)) in (34) tends to zero as ∆t → 0, see Table1. It is intuitively clear, since ε stands for the accuracy of FRA. Therefore, decreasing ∆t and ∆x does not decrease the error, since η is not sufficiently accurately approximated, compare with estimate (16). In the case of ∆t = ∆x = ε the order of convergence is close to 1, see Table4, which is consistent with the theoretical estimate from Theorem 4.4. The results concerning the order of convergence are shown on Figure3.

Acknowledgments. PG and AU were supported by the International PhD Pro- jects Programme of Foundation for Polish Science operated within the Innovative E- conomy Operational Programme 2007-2013 (PhD Programme: Mathematical Meth- ods in Natural Sciences). PG was also supported by the grant IdP2011/000661. FINITE RANGE APPROXIMATION METHOD 685

Figure 3. Order of convergence. Results for ε ∈ {0.1, 0.0125, 0.0015625, ∆t} presented in Tables 1–4.

Err(T,2∆t,ε) ∆t Err(1, ∆t, 0.1) log Err(T,∆t,ε) / log 2 1.0000 · 10−1 0.0008865973 - 5.0000 · 10−2 0.0005886514 0.59086543 2.5000 · 10−2 0.0003434678 0.77723882 1.2500 · 10−2 0.0002668756 0.36400752 6.2500 · 10−3 0.0002400045 0.15310595 3.1250 · 10−3 0.0002258416 0.08775007 1.5625 · 10−3 0.0002187803 0.04582869 7.8125 · 10−4 0.0002154824 0.02191237 Table 1. The error of the FRA method for ε = 0.1.

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Err(T,2∆t,ε) ∆t Err(1, ∆t, 0.0125) log Err(T,∆t,ε) / log 2 1.0000 · 10−1 6.702793 · 10−4 - 5.0000 · 10−2 4.255311 · 10−4 0.6554979 2.5000 · 10−2 1.757573 · 10−4 1.2756799 1.2500 · 10−2 9.330613 · 10−5 0.9135408 6.2500 · 10−3 5.658711 · 10−5 0.7214983 3.1250 · 10−3 4.055238 · 10−5 0.4806869 1.5625 · 10−3 3.332588 · 10−5 0.2831438 7.8125 · 10−4 2.973577 · 10−5 0.1644434 Table 2. The error of the FRA method for ε = 0.0125.

Err(T,2∆t,ε) ∆t Err(1, ∆t, 0.0015625) log Err(T,∆t,ε) / log 2 1.0000 · 10−1 6.616854 · 10−4 - 5.0000 · 10−2 4.139056 · 10−4 0.6768439 2.5000 · 10−2 1.570582 · 10−4 1.3980025 1.2500 · 10−2 7.418803 · 10−5 1.0820407 6.2500 · 10−3 3.649949 · 10−5 1.0820407 3.1250 · 10−3 1.883518 · 10−5 0.9544464 1.5625 · 10−3 1.032763 · 10−5 0.8669200 7.8125 · 10−4 6.277910 · 10−6 0.7181537 Table 3. The error of the FRA method for ε = 0.0015625.

Err(T,2∆t,2ε) ∆t Err(1, ∆t, ∆t) log Err(T,∆t,ε) / log 2 1.0000 · 10−1 8.865973 · 10−4 - 5.0000 · 10−2 4.797352 · 10−4 0.8860407 2.5000 · 10−2 1.991736 · 10−4 1.2682122 1.2500 · 10−2 9.330613 · 10−5 1.0939823 6.2500 · 10−3 4.477800 · 10−5 1.0591816 3.1250 · 10−3 2.162411 · 10−5 1.0501496 1.5625 · 10−3 1.032763 · 10−5 1.0661308 7.8125 · 10−4 4.763338 · 10−6 1.1164650 Table 4. The error of the FRA method for ε = ∆t.

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