Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation

The main target of this work is to construct and implement a stochastic algorithm which approximates the solution of a multidimensional porous media type equation with monotone possibly irregular coefficient.

Note that, when \(\beta (u)=u \cdot |u|^{m-1}, m>1\), the partial differential equation (PDE) in (1.1) is nothing else but the classical porous media equation. In this case \(\Phi (u) = |u|^\frac{m-1}{2}\).

Equation (1.3) constitutes a model intervening in some self-organized criticality (often called SOC) phenomena, see [4] for a significant monography on the subject and [10, 18] for recent related references.

Sand piles are typical related models, which were first introduced in the discrete setting: for instance the BTW (Bak–Tang–Wiesenfeld) model, see [5] and a refined version, the so-called Zhang model. Inspired from the latter model, Bantay and Janosi [6] introduced continuous sand pile models, in which appear a porous media equation of the type (1.1) with \(\beta \) defined in (1.3). Two different effects appear: the avalanche and the regular arrival of sand. A natural description of the global phenomenon is a stochastic perturbation by noise of the mentioned equation, i.e., a generalized stochastic porous media equation. The two effects appearing in very different scales, there is sense to analyze them separately. The deterministic PDE (1.1) is a natural description of the avalanche effect and in this paper we concentrate on that one. Recent work related to self-organized criticality and SPDEs was done by Barbu et al. [9, 10].

In the paper will also often appear the following hypothesis.

In the one-dimensional case, under Assumption B(\(1\)), [18, Proposition 3.4] proved existence and uniqueness of solutions (in the sense of distributions) for (1.1) when the initial condition \(u_0\) is a bounded integrable function. Indeed, that result was essentially a clarification of older and celebrated results quoted in [17]. In Proposition 3.1, we extend that result when the dimension \(d\) is greater than \(1\), under the validity of Assumptions A and B(\(\ell \)) for some \(\ell \ge 1\).

To the best of our knowledge the first author who considered a probabilistic representation for the solutions of non-linear deterministic partial differential equations was McKean [27]. However, in his case, the coefficients were smooth. In the one-dimensional case, a probabilistic interpretation of (1.1) when \(\beta (u)=u.|u|^{m-1}\), \(m>1\), was provided in [16]. Since the original article of McKean, many papers were produced and it is impossible to list them all: Belaribi et al. [14] provides a reasonable list. If \(\beta (u)=u.|u|^{m-1},~m\in ]0,1[\), the partial differential equation in (1.1) is in fact the so-called fast diffusion equation. In the case when \(d=1\), [15] provides a probabilistic representation for the Barenblatt type solutions of (1.1).

Under Assumptions A and B(\(1\)), supposing that \(u_0\) has a bounded density, [18] (resp. [12]) proves existence and uniqueness of the probabilistic representation (in law) when \(\beta \) is non-degenerate (resp. degenerate). Besides, a theoretical probabilistic representation of the PDE, perturbed by a multiplicative noise, is given in [11].

Earlier, in the multi-dimensional case, Jourdain and Méléard [24] concentrated on the case when \(\beta \) is non-degenerate and \(\Phi \) is Lipschitz, continuously differentiable at least up to order \(3\), and with some further regularity assumptions on \(u_{\scriptscriptstyle {0}}\). The authors established existence and uniqueness of the probabilistic representation and the so-called propagation of chaos, see [35] for a rigorous formulation of this concept. When \(\beta \) is not smooth, the probabilistic representation remains an open problem in the multidimensional case.

The connection between the solutions of (1.4) and (1.1) is given by the following result.

The proof of previous result is well-known, but we recall here the basic argument.

In the present paper, we do not solve the general problem of the probabilistic representation for (1.1), however, the solutions to (1.5) are natural candidates and we establish some theoretical related results. A mollified version of (1.4) will be stated in (4.1). It consists in replacing in the first line of (1.4), \(\Phi _d(u)\) with \(\Phi _d(K_H * v^H)\), where \(v^H\) are the marginal laws of the solution \(Y\) of (4.1) and \(K_H\) is a mollifying kernel. In Proposition 4.3, at least when \(\Phi \) is bounded, non-degenerate and continuous, we establish the equivalence between the existence for (4.1) and the existence for the corresponding partial integro-differential deterministic Eq. (4.11). In Proposition 4.2, under the same assumptions on \(\Phi \), we prove existence in law for (4.1). This also provides existence for (4.11).

Since a basic obstacle arises in dimension \(d > 1\), a natural simplified problem to study is the probabilistic representation of (1.1) when the initial condition \(u_0\) only depends on the radius (radially symmetric). In fact, in that case, the problem can be reduced to dimension one, studying the stochastic differential equation fulfilled by the norm of the process at power \(d\): this is the object of Section 5. The reduction to dimension \(1\) has also the interest to build a new class of non-linear diffusions with singular coefficients. Unfortunately also that type of equation is difficult to handle at the theoretical level since, even when \(\Phi = 1\) (and so the unique solution \(Y\) to (1.4) is a classical \(d\)-dimensional Brownian motion), some non-Lipschitz functions at zero naturally appear. In that case, the norm of \(Y\) is a \(d\)-dimensional Bessel process. When \(\beta \) is non-degenerate, the mentioned norm behaves then similarly to a \(d\)-dimensional Bessel process. Since that Bessel type process is recurrent for \(d \le 2\) and transient for \(d > 2\), it is expected that the zero (which is a singularity) point is much less visited when \(d > 2\).

As mentioned in the beginning of this introduction, one purpose of the present paper is to exploit the probabilistic representation in order to simulate the solutions of (1.1). For this we will implement an Euler scheme for stochastic differential equations and a non-parametric density estimation method using Gaussian kernel estimators, see [38]. Since we expect our methods to be robust when the coefficient \(\Phi \) is irregular, it is not reasonable to make use of higher order discretization schemes involving derivatives of \(\Phi \). Concerning the choice of the smoothing parameter \(\varepsilon \) for the density estimate, we extend to the multidimensional case the techniques used in [14].

Besides, we have carried out a deterministic numerical method in dimension \(d=2\) of space, based on a sophisticated procedure developed in [20], which is one of the most recent references in the subject. In fact, Cavalli et al. [20] coupled WENO (weighted essentially non-oscillatory) interpolation methods for space discretization, see [32], with IMEX (implicit explicit) Runge–Kutta schemes for time advancement, see [28], to obtain a high order method. We emphasize that WENO techniques help to prevent the onset of spurious oscillations.

The general stochastic particle algorithm is empirically investigated in dimension \(d = 2\). This is done in the following two cases. First, when \(\beta (u) = u^3\), comparing with the Barenblatt exact solutions. Second, when \(\beta \) is defined by (1.3), comparing with the deterministic numerical technique; indeed, in that case, exact explicit solutions for (1.1) are unknown.

In the radially symmetric case, we apply the stochastic particle algorithm to the one-dimensional reduced non-linear stochastic differential equation. As mentioned earlier, when \(d =2\) (resp. \(d>2\)), in most cases, the solutions are expected to be recurrent (resp. transient); so we suspected the simulations to be more performing when \(d>2\). However, our experiments show that the error of the approximations with respect to the exact solutions (derived by the classical porous media equation) remain stably low for all the values of \(d\) including \(d = 2\). In Section 8.2 we also compare the radial reduction method with the deterministic approach when \(d=2\) and \(\beta \) is of type (1.3).

The paper is organized as follows. After this introduction and some preliminaries, in Section 3 we discuss the well-posedness of the deterministic problem (1.1). In Section 4 we discuss the existence of a mollified version of the non-linear stochastic differential equation and its equivalence to the existence of a partial integro-differential equation which is a regularized version of (1.1). Section 5 handles the case of a radially symmetric initial condition. In Section 6 we describe the general particle algorithm for \(d = 2\). Section 7 summarizes the deterministic technique developed in [20] and finally Section 8 is devoted to numerical experiments.

From now on, Assumption A is supposed to be in force and it will not be recalled anymore.

From now on, without restriction of generality, \(d\) will be greater or equal to \(2\).

In our numerical simulations, we will consider the case where \(d=2\). The operational aspects of that method, in the one-dimensional case, were explained in details in [14, Section 5].

The probabilistic and deterministic algorithms were both carried out using Matlab. In order to speed up our probabilistic procedure, we have implemented, using the Matlab Parallel Computing Toolbox (PCT), a GPU version of the kernel density estimator in dimension \(1\) and \(2\) of space. Using \(10^5\) particles, this has \(500\) times reduced the CPU time on our reference computer. As mentioned in Section 7, the deterministic numerical solutions are performed via the method provided in [20]. In fact, we use the WENO spatial reconstruction of order 5 and a third order explicit Runge–Kutta IMEX scheme for time stepping. From now on, we will denote the related time step by \(\Delta t_{det}\) and the deterministic numerical solution by \(\hat{u}_{det}\).

The general stochastic particle algorithm for \(d=2\)

We have proceeded to the validation of our algorithm in three main situations: the classical porous media equation, the fast diffusion equation and the Heaviside case.

The porous media equation case

In the case where \(\beta (u)=u|u|^{m-1}\), \(m>1\), we recall that the PDE in (1.1) is nothing else but the classical porous media equation (PME). If \(u_{\scriptscriptstyle {0}}=\delta _{\scriptscriptstyle {0}}\), an exact solution is provided by [13] (see also [37, Section 17.5]) known as the density of Barenblatt-Pattle:

$$\begin{aligned} E(t,x)=t^{-\alpha }\left(D-\kappa \Vert x\Vert ^2t^{-2\beta }\right)^{\frac{1}{m-1}}_+, \ x\in \mathbb R ^d,\ \ t>0, \end{aligned}$$

(8.1)

where \(\ \alpha =\frac{\displaystyle {d}}{\displaystyle {(m-1)d+2}},\ \ \beta =\frac{\displaystyle {\alpha }}{\displaystyle {d}}, \ \ \kappa =\frac{\displaystyle {m-1}}{\displaystyle {m}}\beta , \ \ D=\left[\kappa ^{-\frac{d}{2}}I\mathfrak{C }\right]^{\frac{2(1-m)}{2+d(m-1)}} \ \text{ and} \)

$$\begin{aligned} I=\frac{{\Gamma \left(\frac{d}{2}\right)\Gamma \left(\frac{m}{m-1}\right)}}{{\Gamma \left(\frac{d}{2}+\frac{m}{m-1}\right)}}, \ \Gamma \ \text{ being} \text{ the} \text{ usual} \text{ Gamma} \text{ function.} \end{aligned}$$

We would now compare the exact solution (8.1) to an approximated probabilistic solution. Up to now, we are not able to perform an efficient bandwidth selection procedure in the case when the initial condition of PME is a Dirac probability measure, i.e., the law of a deterministic random variable. Since we are nevertheless interested in exploiting (8.1), we have considered a time translation of the exact solution \(E\) defined as

$$\begin{aligned} \mathcal U (t,x)=E(t+2,x) ,\ \ \ x\in \mathbb{R }^d,\ \ t\in [0,T]. \end{aligned}$$

(8.2)

Note that one can immediately deduce from (8.2) that \(\mathcal U \) still solves the PME but now with a smooth initial condition given by

$$\begin{aligned} u_0(x)=E(2,x), \ \ \ x\in \mathbb{R }^d. \end{aligned}$$

(8.3)

Simulation experiments. We set \(d=2\) and \(m=3\). We compute both the deterministic and probabilistic numerical solutions over the time-space grid \([0,3]\times [-2.5,2.5]\times [-2.5,2.5]\), with a uniform space step \(\Delta x= 0.0167\). We set \(\Delta t_{{det}}=7.5 \times 10^{-4}\), while, we use \(n=200,000\) particles and a time step \(\Delta t=10^{-2}\) for the probabilistic simulation. Figure 1a–d (resp. Fig. 2a–d) displays the numerical probabilistic (resp. deterministic) solutions at times \(t=0\), \(t=1\), \(t=2\) and \(t=T=3\), respectively. Besides, Fig. 3 describes the time evolution of the \(L^1\) probabilistic and deterministic errors on the time interval \([0,3]\).

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The fast diffusion equation (FDE) case

Now, we suppose that \(\beta (u)=u|u|^{m-1}\), \(m\in ]0,1[\). In that case the PDE in (1.1) corresponds to the so-called fast diffusion equation. Similarly as for the porous media equation, there also exists a Barenblatt type solution for the mentioned \(\beta \) when the initial condition \(u_{\scriptscriptstyle {0}}\) is a delta Dirac measure at zero. Indeed, it is given by the following expression:

$$\begin{aligned} E(t,x)=t^{-\alpha }\left(\widetilde{D}+\widetilde{\kappa }\Vert x\Vert ^2t^{-2\beta }\right)^{-\frac{1}{1-m}},\ x\in \mathbb{R }^d,\ t>0, \end{aligned}$$

(8.4)

where \(\ \alpha =\frac{\displaystyle {d}}{\displaystyle {(m-1)d+2}},\ \ \beta =\frac{\displaystyle {\alpha }}{\displaystyle {d}}, \ \ \widetilde{\kappa }=\frac{\displaystyle {1-m}}{\displaystyle {m}}\beta , \ \ \widetilde{D}=\left[\widetilde{\kappa }^{-\frac{d}{2}}I\mathfrak{C }\right]^{\frac{2(m-1)}{d(1-m)-2}}\) \( \text{ and} \)

$$\begin{aligned} I=\frac{{\Gamma \left(\frac{d}{2}\right)\Gamma \left(\frac{1}{1-m} -\frac{d}{2}\right)}}{{\Gamma \left(\frac{1}{1-m}\right)}}. \end{aligned}$$

Again, we consider a time shifted version of the explicit solution (8.4) for the numerical experiments. Indeed, we define

$$\begin{aligned} \mathcal U (t,x)=E(t+1,x),\ \ x\in \mathbb{R }^d,\ \ t\in [0,T]. \end{aligned}$$

Obviously, \(\mathcal U \) solves the FDE with \(u_{\scriptscriptstyle {0}}=E(1,\cdot )\) as initial condition.

Simulation experiments. We set \(d=2\) and \(m=\frac{1}{2}\). We consider the time-space grid \([0,1.5]\times [-15,15]\times [-15,15]\) over which the probabilistic, the deterministic and the exact solutions are computed. We fix \(\Delta t_{det}=1.5\times 10^{-3}\). We use a uniform space step \(\Delta x= 0.4\). For the probabilistic simulation we set \(n=200,000\) and \(\Delta t=10^{-2}\).

Figure 4a–d displays the numerical probabilistic solutions at times \(t=0\), \(t=0.5\), \(t=1\) and \(t=T=1.5\), respectively. Figure 5a–d shows the deterministic solution values at \(t=0\), \(t=0.5\), \(t=1\) and \(t=T=1.5\), respectively. Furthermore, Fig. 6 describes the time evolution of the \(L^1\) errors on the time interval \([0,1.5]\), related to both the probabilistic and deterministic algorithms.

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Using the analytical expression (8.4) of \(E(t,\cdot )\), we get

$$\begin{aligned} \sup _{x\in \mathbb{R }^d}E(t,x)\le \widetilde{D}^{-\frac{1}{1-m}}t^{-\alpha }, \end{aligned}$$

where the right hand side of previous expression goes to zero as \(t\rightarrow +\infty \). This convergence clearly appears in Figs. 4 and 5 when \(d=2\). However, the convergence in \(L^1\) does not hold since for every \(t>0\), \(\int _{\mathbb{R }^d}E(t,x)dx=1\) if \(m>m_c\), where \(m_c=0\) for \(d=1,2\) and \(m_c=\frac{d-2}{d}\) if \(d\ge 3\), see [36, Section 5.6].

Remark 8.1

In previous cases, we had some exact expressions of the solution of (1.1) at our disposal that we could compare with the approximations issued from the deterministic and probabilistic algorithms. The committed error using the deterministic approach is definitely lower than using the probabilistic one. Below we treat the Heaviside case: by default of exact expressions, the deterministic solutions will be used for evaluating the error related to the probabilistic method.

The Heaviside case

In this part, we will discuss the numerical experiments for a coefficient \(\beta \) given by (1.3). We recall that in this case we do not know an exact solution for the problem (1.1). Consequently, we will compare our probabilistic solution to the numerical deterministic solution obtained using the method developed in [20], see also Section 7. In fact, we will simulate both numerical solutions according to several initial data \(u_0\) and with different values of the critical threshold \(u_c\).

Empirically, after various experiments, similarly to the one-dimensional case investigated in [14, Section 6], it appears that for a fixed threshold \(u_c\), the numerical solution approaches some limit function which seems to belong to the “attracting” set

$$\begin{aligned} \mathcal J =\{f \in L^1(\mathbb R ^2)| \int f(x)dx=1,\ \ 0\le f \le u_c\}; \end{aligned}$$

(8.5)

in fact \(\mathcal J \) is the closure in \(L^1\) of \(\mathcal J _0=\{f : \mathbb R ^2 \rightarrow \mathbb R _+ | \ \beta (f)=0\}\). Again, the following theoretical questions arise.

1. (1)

   Does indeed \(u(t,\cdot )\) have a limit \(u_{\infty }\) when \(t \rightarrow \infty \)?

    
2. (2)

   If yes, does \(u_{\infty }\) belong to \(\mathcal J \)?

    
3. (3)

   If (2) holds, do we have \(u(t,\cdot )=u_{\infty }\) for \(t\) larger than a finite time \(\tau \)?

    

(a) Gaussian initial condition

For the mentioned \(\beta \) we consider an initial condition \(u_0\) being a Gaussian density with mean \(\mu \) and invertible covariance matrix \(\Sigma \), i.e.,

$$\begin{aligned} u_0(x)=p(x,\mu ,\Sigma ), \end{aligned}$$

where,

$$\begin{aligned} p(x,\mu ,\Sigma )=\frac{1}{(2\pi )^{\frac{d}{2}}|\Sigma |^{\frac{1}{2}}} \exp \left(-\frac{1}{2}(x-\mu )^t\Sigma ^{-1}(x-\mu )\right), \quad \ x\in \mathbb{R }^d. \end{aligned}$$

(8.6)

Simulation experiments. Test case 1. We set \(d=2\), \(u_c=0.07\), \(\mu =(0,0)\) and \(\Sigma =I_2\), where \(I_2\) is the unit matrix on \(\mathbb{R }^2\). We compute both deterministic and probabilistic solutions over the time-space grid \([0,0.9]\times [-4 , 4]\times [-4, 4]\) with a uniform space step \(\Delta x=0.05\). For the deterministic approximation we set \(\Delta t_{det}=2\times 10^{-4}\) while for the probabilistic one we use \(n=200,000\) particles and a time step \(\Delta t=2\times 10^{-4}\).

Figures 7, 8, and 9, show the deterministic and probabilistic numerical solutions at times \(t=0\), \(t=0.3\) and \(t=T=0.9\), respectively. Furthermore, Fig. 10 describes the time evolution of the \(L^1\)-norm of the difference of the two solutions over the time interval \([0,0.9]\).

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(b) Bimodal initial condition

Now, we suppose that the initial condition is a mixture of two Gaussian densities with separated modes, i.e.,

$$\begin{aligned} u_0(x)=\frac{1}{2}\left(p(x,\mu _1,\Sigma _1)+p(x,\mu _2,\Sigma _2)\right),\quad x\in \mathbb{R }^d, \end{aligned}$$

where \(p\) is defined in (8.6).

Simulation experiments. Test case 2. We set \(d=2\), \(u_c=0.1\). We fix \(\mu _1=(1,0)\), \(\mu _2=(-1,0)\), \(\Sigma _1=(0.1) I_2\) and \(\Sigma _2=(0.2) I_2\). The deterministic and probabilistic solutions are simulated over the time-space grid \([0,0.8]\times [-3.5 , 3.5]\times [-3.5 , 3.5]\) with a uniform space step \(\Delta x=0.05\). We set \(\Delta t_{det}=2\times 10^{-4}\), while we use \(n=200,000\) particles and a time step \(\Delta t=2\times 10^{-4}\), for the probabilistic approximation. Figures 11, 12, and 13, show the deterministic and probabilistic numerical solutions at times \(t=0\), \(t=0.27\) and \(t=T=0.8\), respectively. Furthermore, Fig. 14 displays the time evolution of the \(L^1\)-norm of the difference over the time interval \([0,0.8]\).

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(c) Trimodal initial condition

For the \(\beta \) given by (1.3), we consider an initial condition being a mixture of three Gaussian densities with three modes at some distance from each other, i.e.,

$$\begin{aligned} u_0(x)=\frac{\displaystyle {1}}{\displaystyle {3}}\left(p(x,\mu _1,\Sigma _1)+p(x,\mu _2,\Sigma _2)+p(x,\mu _3,\Sigma _3)\right),x\in \mathbb{R }^d, \end{aligned}$$

(8.7)

where \(p\) is defined in (1.3).

Simulation experiments. We fix again \(d=2\). For this specific type of initial condition \(u_0\), we consider two test cases depending on the value taken by the critical threshold \(u_c\). We set, for instance, \(\mu _1=(-2,2)\), \(\mu _2=(2,-2)\), \(\mu _3=(0,0)\), \(\Sigma _1=(0.1)^2I_2\), \(\Sigma _2=(0.2)^2I_2\) and \(\Sigma _3=\Sigma _2\).

Test case 3. We start with \(u_c=0.15\). We consider a time-space grid \([0,0.4]\times [-5,5]\times [-5,5]\), with a uniform space step \(\Delta x=0.05\). For the deterministic approximation, we set \(\Delta t_{det}=2\times 10^{-4}\). The probabilistic simulation uses \(n=200,000\) particles and a time step \(\Delta t=2\times 10^{-4}\). Figures 15, 16, and 17 display both the deterministic and probabilistic numerical solutions at times \(t=0\), \(t=0.14\) and \(t=T=0.4\), respectively. Besides, the time evolution of the \(L^1\)-norm of the difference between the two numerical solutions is depicted in Fig. 18.

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Test case 4. We choose now as critical value \(u_c=0.035\) and a time-space grid \([0,2]\times [-5,5]\times [-5,5]\), with a uniform space step \(\Delta x=0.05\). We set \(\Delta t_{det}=3\times 10^{-4}\) and the probabilistic approximation is performed using \(n=200,000\) particles and a time step \(\Delta t=4\times 10^{-4}\). Figures 19, 20, and 21 show the numerical (probabilistic and deterministic) solutions at times \(t=0\), \(t=0.66\) and \(t=T=2\). In addition, Fig. 22 describes the \(L^1\)-norm of the difference between the two.

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(d) Uniform and normal densities mixture initial condition

We proceed again with \(\beta \) defined in (1.3). We are now interested in an initial condition \(u_0\), being a mixture of a normal and an uniform density, i.e.,

$$\begin{aligned} u_0(x)=\frac{\displaystyle {1}}{\displaystyle {2}}\left(p(x,\mu ,\Sigma )+{\small 1}\!\!1_{[0,1]\times [-1,0]}(x)\right),\ x\in \mathbb{R }^2, \end{aligned}$$

where \(p\) is defined in (8.6).

Simulation experiments. Test case 5. We fix \(u_c=0.15\), \(\mu =(0,-1)\) and \(\Sigma =(0.076)^2I_2\). We perform both the approximated deterministic and probabilistic solutions on the time-space grid \([0,0.5]\times [-3,3]\times [-3,3]\), with a space step \(\Delta x=0.05\). We use \(n=200,000\) particles and a time step \(\Delta t=2\times 10^{-4}\) for the probabilistic simulation. Moreover, we set \(\Delta t_{det}=2 \times 10^{-4}\). Figures 23, 24, and 25 illustrate those approximated solutions at times \(t=0\), \(t=0.2\) and \(t=T=0.6\). Furthermore, we compute the \(L^1\)-norm of the difference between the numerical deterministic solution and the probabilistic one. That error is displayed in Fig. 26.

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Long-time behavior of the solutions

As it was mentioned previously, we are interested in the empirical behavior of solutions to (1.1) in the Heaviside case. For this, we first provide Fig. 27, which displays the time evolution of the \(L^1\)-norm of the difference between two successive time evaluations of the numerical solutions. That quantity was computed for both deterministic and probabilistic numerical solutions and in the different test cases 1 to 5. In fact, Fig. 27, shows that the numerical solutions approach some limit function \({\hat{u}}_{\infty }\). Indeed, they seem to reach \({\hat{u}}_{\infty }\) after a finite time \(\hat{\tau }\).

In addition, according to Fig. 28, this suggests the existence of a limit function \(u_{\infty }\), which of course depends on the initial condition \(u_0\) such that \(u(t,\cdot )=u_{\infty }\), for \(t\ge \tau \). Moreover, \(u_{\infty }\) is expected to belong to the ”attracting set” \(\mathcal J \), since \(\Vert \beta (\hat{u}(t,\cdot ))\Vert _{L^1(\mathbb R ^2)}\) equals zero when \(t\) is larger than \(\hat{\tau }\), at least when \(\hat{u}\) is the deterministic numerical solution.

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Long-time stability behavior of the general probabilistic algorithm \(\mathbf{d=2}\)

Now, we inquire about the long time behavior of the probabilistic particle algorithm. In fact, we are interested in the dependence of the error over the time. For this, we have simulated the solution of the PME with \(n=200,000\), \(T=50\), \(\Delta t=0.02\) and \(m=3\). Figure 29 displays the time evolution of the \(L^1\)-norm of the error. In particular, Fig. 29 shows (in the PME case) that the probabilistic algorithm seems to remain stable for a large time \(T\).

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The radially symmetric case

Validation on exact solutions in hyperspherical coordinates

We make the same conventions as in Section 5. Suppose that \(u_0\) is radially symmetric. \(u\) is the solution of (1.1) and \(\tilde{u}\) is given in Remark 5.5. \(\nu \) constitutes the marginal laws of \(S\) which is defined in (5.3) which is a solution of (5.11). \(\nu \) equals \(\tilde{u}\) up to a constant, see (5.4).

The Fokker–Planck equation for Bessel processes

When \(\beta (u)=u\), the PDE in (1.1) corresponds to the heat equation. The linked radial Eq. (5.8) with \(\Phi = 1\) (Bessel equation) and with initial condition \(\ell _{\scriptscriptstyle {0}}^d\), admits (5.16) as exact solution. We would like to compare the probabilistic approximation \(\widetilde{\nu }^{\varepsilon ,n}\) defined as the right-hand side of (5.21) to (5.16). In order to avoid computation difficulties for the bandwidth \(\varepsilon \) in the case when the initial condition \(\nu _{\scriptscriptstyle {0}}\) is the law of a deterministic random variable, we will consider a time shifted version of (5.16), given by

$$\begin{aligned} v(t,\rho )&= \frac{\displaystyle {\rho ^{\frac{2-d}{2d}}}}{\displaystyle {(t+t_{\scriptscriptstyle {0}})\ell _{\scriptscriptstyle {0}}^{\frac{d-2}{2}}d}} \exp \left(-\frac{\displaystyle {\ell _{\scriptscriptstyle {0}}^2+\rho ^{\frac{2}{d}}}}{\displaystyle {2(t+t_{\scriptscriptstyle {0}})}}\right)I_{\frac{d}{2}-1}\left(\frac{\displaystyle {\ell _{\scriptscriptstyle {0}}\rho ^{\frac{1}{d}}}}{\displaystyle {t+t_{\scriptscriptstyle {0}}}}\right),\nonumber \\&\quad t\in [0,T], \rho >0 , t_{\scriptscriptstyle {0}}> 0. \end{aligned}$$

(8.8)

In fact, with this reposition, \(v(t,\cdot )\) still solves (5.8) but now with a smooth initial data \(v(0,\rho )=\nu _{\ell _{\scriptscriptstyle {0}}^d}(t_{\scriptscriptstyle {0}},\rho )\).

Simulation experiments. We set \(\ell _{\scriptscriptstyle {0}}=1\) and \(t_{\scriptscriptstyle {0}}=10^{-3}\). We compute the probabilistic numerical solutions of (5.8) over a time-space grid \([0, 0.01]\times ]0,L]\), \(L>0\), for different values of the dimension \(d=2,5,10\) and with a space step \(\Delta x= 0.01\). We use a time step \(\Delta t=10^{-4}\) and \(n=200,000\) particles. Figures 30, 31, and 32a–c, show the exact and the numerical solutions at times \(t=0\), \(t=0.005\) and \(t=T=0.01\), for \(d=2,5,10\), respectively. The exact solution, defined in (8.8), is depicted by solid lines. Besides, Figs. 30e, 31e, and 32e describe the time evolution of the \(L^1\) error on the interval \([0,0.01]\), for \(d=2,5,10\), respectively.

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We point out that the performance of our algorithm is satisfying for all values of \(d\ge 2\) even though when \(d=2\) the solution is recurrent. In that case the process often attains zero, which is a non regular point of the diffusion term in (5.12).

The radial transformation of the classical porous media equation

When \(\beta (u)=u.|u|^{m-1},m>1\), and \(u_{\scriptscriptstyle {0}}=\delta _{\scriptscriptstyle {0}}\), we consider again the explicit Barenblatt type solution of (1.1), denoted by \({E}\) and given in (8.1). Once more, we will shift in time this exact solution, in order to avoid simulation problems in the case of a delta Dirac measure as initial condition. In fact, we set \(\mathcal U (t,x)=E(t+1,x)\), \(t\in [0,T]\), \(x\in \mathbb R ^d\). Then, \(\mathcal U \) still solves (1.1) for the mentioned \(\beta \) and with \(u_{\scriptscriptstyle {0}}(x)=E(1,x)\) as initial condition. Since \(u_{\scriptscriptstyle {0}}\) is radially symmetric, we deduce

$$\begin{aligned} \mathcal U (t,x)=\widetilde{\mathcal{U }}(t,\Vert x\Vert ^d)\!=\!(t\!+\!1)^{-\alpha }\left(D\!-\!\kappa \left(\Vert x\Vert ^d\right)^{\frac{2}{d}}(t\!+\!1)^{-2\beta }\right)^{\frac{1}{m-1}}_+, \ x\!\in \! \mathbb R ^d,\ \ t\!\ge \! 0. \end{aligned}$$

Then, using (5.4), we get

$$\begin{aligned} \nu _{ex}(t,\rho ) = \left\{ \begin{array}{ll} \frac{\displaystyle {\mathfrak{C }}}{\displaystyle {d}}(t+1)^{-\alpha }{\left(D-\kappa \rho ^{\frac{2}{d}}(t+1)^{-2\beta }\right)}^{\frac{1}{m-1}}&\text{ if} \rho \in \left[0, \left(\frac{\displaystyle {D}}{\displaystyle {\kappa }}\right)^{\frac{d}{2}}(t+1)^{\alpha }\right],\\ \\ 0&\text{ otherwise}.\\ \end{array}\right. \nonumber \\ \end{aligned}$$

(8.9)

Simulation experiments. We compute the probabilistic numerical solutions of (5.8) when \(\beta (u)=u^3\) (radial PME), over the time-space grid \([0, 1]\times [0,2.5]\) for different values of the dimension \(d=2,5,10\) and with a space step \(\Delta x= 0.01\). We consider a time step \(\Delta t=10^{-2}\) and \(n=200,000\) particles. Figures 33, 34, and 35a–c display the exact and the numerical solutions at times \(t=0\), \(t=0.5\) and \(t=T=1\) respectively, for \(d=2,5,10\), respectively.

The exact solution, defined in (8.9), is depicted by solid lines. Besides, Figs. 33e, 34e, and 35e describe the evolution of the \(L^1\) norm of the error on the time interval \([0,1]\), for \(d=2,5,10\), respectively.

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Comparison between the radial stochastic algorithm and the 2-dimensional deterministic approach in the Heaviside case

In this part of the work we exploit, at least empirically, the existing relation between the solutions of the multidimensional problem (1.1) and the solutions of the one-dimensional PDE (5.8), in the case when the initial condition \(u_0\) of (1.1) is a radially symmetric function. For this we will simulate the \(d\)-power norm of the solution \(Y\) of (1.4) via the one-dimensional non-linear diffusion introduced in (5.12). Let \(u:[0,T]\times \mathbb{R }^d\rightarrow \mathbb{R }_+\) be a solution of (1.1) with initial condition \(u_0\) which is radially symmetric. According to Remark 5.5, there exists \(\tilde{u}:[0,T]\times ]0,+\infty [\rightarrow \mathbb{R }_+\) such that \(u(t,x)=\tilde{u}(t,\Vert x\Vert ^d)\), \(\forall (t,x)\in [0,T]\times \mathbb{R }^d\setminus \{0\}\). If \(Y\) solves (1.4), the second item of Proposition 5.10 says that \(S=\Vert Y\Vert ^d\) verifies the non-linear diffusion Eq. (5.12). If \(\nu :[0,T]\times \mathbb{R }_+\rightarrow \mathbb{R }_+\) such that \(\nu (t,\cdot )\) is the law of \(S_t\), \(t\in [0,T]\), then by Lemma 5.6, we have \(\tilde{u}=\frac{d}{\mathfrak{C }}\nu \) and therefore

$$\begin{aligned} u(t,x)=\frac{d}{\mathfrak{C }}\nu (t,\Vert x\Vert ^d), \ \forall (t,x)\in [0,T]\times \mathbb{R }^d\setminus \{0\}. \end{aligned}$$

(8.10)

Consequently, previous expression allows us to simulate the solution \(u\) of (1.1) using the solution \(\nu \) of (5.8). Indeed, it will be enough to replace \(\nu \) in (8.10) with its kernel density estimator \(\widetilde{\nu }^{\varepsilon ,n}\) given in (5.21). Those approximations will be compared to the ones obtained via the \(d\)-dimensional general stochastic algorithm.

From now on we will fix \(d=2\) and \(\beta \) given by (1.3).

Test case (A)

Let \(u_0\) be a Gaussian density, i.e.,

$$\begin{aligned} u_0(x)=p(x,\mu ,\Sigma ), \ x\in \mathbb{R }^2, \end{aligned}$$

where \(p\) is defined in (8.6), \(\mu =(0,0)\) and \(\Sigma =I_2\). Thus, the initial condition \(\nu _0\) of (5.8) is the probability density of an exponential distribution of parameter \(\lambda =\frac{1}{2}\), i.e.,

$$\begin{aligned} \nu _0(\rho )=\lambda \exp (-\lambda \rho ), \ \forall \rho \in \mathbb{R }_+. \end{aligned}$$

Simulation experiments. We fix \(u_c=0.07\). We compute the probabilistic solution obtained through the one-dimensional radial algorithm using \(n=200,000\) particles and a time step \(\Delta t=2\times 10^{-4}\), and we compare it to the 2-dimensional deterministic approximation presented in Section 7. We fix \(\Delta t_{det}=4\times 10^{-4}\). We represent both the 2-dimensional deterministic and probabilistic solutions, on the time-space grid \([0,0.9]\times [-4,4]\times [-4,4]\), with a uniform space step \(\Delta x=0.05\).

Figures 36, 37, and 38 illustrate those approximated solutions at times \(t=0\), \(t=0.3\) and \(t=T=0.9\). Furthermore, we compute the \(L^1\)-norm of the difference between the numerical deterministic solution and the probabilistic one. Values of that error are displayed in Fig. 39.

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Test case (B)

Proceeding with the same \(\beta \), we consider now an initial condition \(u_0\) given by

$$\begin{aligned} u_0(x)=\frac{1}{\pi }\sqrt{\frac{2}{\pi }}\exp (-\frac{\Vert x\Vert ^4}{2}),\ x\in \mathbb{R }^2. \end{aligned}$$

The corresponding \(\tilde{u}_0\) is related, via (5.4), to the law density of an absolute value of a standard Gaussian r.v. Indeed, the initial condition \(\nu _0\) of (5.8) is a probability density defined by

$$\begin{aligned} \nu _0(\rho )=\sqrt{\frac{2}{\pi }}\exp (-\frac{\rho ^2}{2}), \ \forall \rho \in \mathbb{R }_+. \end{aligned}$$

Simulation experiments. We set \(u_c=0.08\). Then, using \(n=200,000\) particles and a time step \(\Delta t=2.8\times 10^{-4}\), we simulate the probabilistic solution applying the one-dimensional radial approach. The 2-dimensional deterministic and probabilistic simulations are computed over the time-space grid \([0,1.4]\times [-4,4]\times [-4,4]\), with a uniform space step \(\Delta x=0.05\) and fixing \(\Delta t_{det}=2.8\times 10^{-4}\).

Figures 40, 41, and 42 illustrate those approximated solutions at times \(t=0\), \(t=0.46\) and \(t=T=1.4\). Besides, Fig. 43 displays the \(L^1\)-norm of the difference between the numerical deterministic solution and the probabilistic one.

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Test case (C)

We assume that the initial condition \(u_0\) is defined by

$$\begin{aligned} u_0(x)=\frac{1}{2\pi }\left(g(\Vert x\Vert ^2,m_1,\sigma _1)+g(\Vert x\Vert ^2,m_2,\sigma _2)\right),\ x\in \mathbb{R }^2, \end{aligned}$$

where \(g(\rho ,m,\sigma )=f(\rho ,m,\sigma )+f(-\rho ,m,\sigma )\), \(\rho \ge 0\), and \(f\) is the density function of a one-dimensional Gaussian distribution with mean \(m\) and standard deviation \(\sigma \). Therefore, \(\nu _0\) of (5.8) is given by

$$\begin{aligned} \nu _0(\rho )=\frac{1}{2}\left(g(\rho ,m_1,\sigma _1)+g(\rho ,m_2,\sigma _2)\right), \ \forall \rho \in \mathbb{R }_+. \end{aligned}$$

Simulation experiments. We fix for instance \(u_c=0.07\), \(m_1=0\), \(m_2=6\), \(\sigma _1=0.2\) and \(\sigma _2=0.3\). We consider \(n=200,000\) particles and a time step \(\Delta t=2\times 10^{-4}\), in order to perform the probabilistic numerical solution via the one-dimensional radial algorithm. We compare it to the 2-dimensional deterministic approximation. We set \(\Delta t_{det}=2\times 10^{-4}\). We represent both the 2-dimensional deterministic and probabilistic solutions, on the time-space grid \([0,1]\times [-4,4]\times [-4,4]\), with a uniform space step \(\Delta x=0.05\). Figures 44, 45, and 46 display those approximated solutions at times \(t=0\), \(t=0.33\) and \(t=T=1\). Besides, Fig. 47 shows the \(L^1\)-norm of the difference between the two solutions.

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Remark 8.2

1. (i)

   The probabilistic algorithm can be parallelized on a graphical processor unit (GPU) such that we can speed-up its time machine execution; on the other hand, for the deterministic algorithm, this operation is far from being obvious.

    

1. (ii)

   At this point, even though it provides reliable approximation of the solutions, the implementation of the deterministic algorithm in dimension 2 is not optimal. Indeed, it costs a huge amount of CPU time comparing to the deterministic one-dimensional procedure and to the probabilistic algorithm in dimension 1 and 2.

    
2. (iii)

   In general, empirically, the different errors committed by the probabilistic algorithm seem to be reasonable, even though not very good. (a) The general two-dimensional probabilistic algorithm behaves well in the case of an unimodal initial condition. Some difficulties arise in the multimodal case; on the other hand we obtain satisfying results in Fig. 26 which represents an evolution in the Heaviside case of a bimodal and irregular initial condition. (b) The probabilistic algorithm in the radial case behaves quite well for all \( d\ge 2\), if the initial condition is unimodal and the coefficient \(\beta \) is smooth. If \(\beta \) is of Heaviside type, the error becomes more important when \(d =2\). Unfortunately we have no mean to validate the algorithm for larger values than \(d = 2\), in which case we could expect a better performance.

    

In conclusion the use of probabilistic methods in higher dimension is justified. Contrarily to the one-dimensional case, treated in [14], the probabilistic techniques are much simpler to formulate than the analytical method.