2.1 Existence and uniqueness

We first establish a comparison principle.

Theorem 2.3. Assume $F$ is monotone and has width $N$ . Let $u,v:{\mathbb {Z}^n_h} \to {\mathbb {R}}$ satisfy

(2.9) \begin{equation} u- F(\nabla ^2_h u) \leq v - F(\nabla ^2_h v) \ \ \text {on } {\mathbb {Z}^n_h}, \end{equation}

and

(2.10) \begin{equation} \lim _{|x|\to \infty } \frac {u(x) - v(x)}{|x|^2 }=0. \end{equation}

Then $u\leq v$ on $\mathbb {Z}^n_h$ .

Proof. For ${\varepsilon }\gt 0$ we define

\begin{equation*}u_{\varepsilon }(x) = u(x) - \frac {{\varepsilon }}{2}|x|^2 - {\varepsilon } N^2 - {\varepsilon }.\end{equation*}

We claim that $u_{\varepsilon } \leq v$ for all ${\varepsilon }\gt 0$ , from which the result follows. Fix ${\varepsilon }\gt 0$ and assume by way of contradiction that $\sup _{{\mathbb {Z}^n_h}}(u_{\varepsilon } - v) \gt 0$ . By (2.10), there exists $R\gt 0$ , depending on ${\varepsilon }\gt 0$ , such that $u_{\varepsilon }(x) \lt v(x)$ for $|x| \gt R$ . Thus, $u_{\varepsilon } - v$ attains its maximum over $\mathbb {Z}^n_h$ at some $x_0\in {\mathbb {Z}^n_h}$ , and $u_{\varepsilon }(x_0) \gt v(x_0)$ . Since $u_{\varepsilon }(x) - v(x) \leq u_{\varepsilon }(x_0) - v(x_0)$ for all $x\in {\mathbb {Z}^n_h}$ we have

\begin{equation*}u_{\varepsilon }(x) - u_{\varepsilon }(x_0) \leq v(x) - v(x_0).\end{equation*}

It follows that

\begin{align*} \nabla ^2_h u_{\varepsilon }(x_0,{\mathbf {v}}) &= \frac {1}{h^2}(u_{\varepsilon }(x_0+h{\mathbf {v}}) - u(x_0) + u_{\varepsilon }(x_0-h{\mathbf {v}}) - u(x_0)) \\ &\leq \frac {1}{h^2}(v(x_0+h{\mathbf {v}}) - v(x_0) + v(x_0-h{\mathbf {v}}) - v(x_0)) = \nabla ^2_h v(x_0,{\mathbf {v}}) \end{align*}

for all directions ${\mathbf {v}}\in {\mathbb {Z}^n}$ . Since $F$ is monotone we have

(2.11) \begin{equation} F(\nabla ^2_h u_{\varepsilon }(x_0)) \leq F(\nabla ^2_h v(x_0)). \end{equation}

We now compute

\begin{equation*} \nabla ^2_h u_{\varepsilon } (x,{\mathbf {v}}) = \nabla ^2_h u (x,{\mathbf {v}}) - {\varepsilon } |{\mathbf {v}}|^2,\end{equation*}

from which it follows that

\begin{align*} u_{\varepsilon }(x_0) - F(\nabla ^2_h u_{\varepsilon }(x_0)) &= u(x_0) - \frac {{\varepsilon }}{2}|x_0|^2 -{\varepsilon } N^2 - {\varepsilon } - F(\nabla ^2_h u(x_0) - {\varepsilon }|{\mathbf {v}}|^2)\\ &\leq u(x_0) - F(\nabla ^2_h u(x_0)) - \frac {{\varepsilon }}{2}|x_0|^2- {\varepsilon }\\ &\lt v(x_0) - F(\nabla ^2_h v(x_0))\\ &\leq v(x_0) -F(\nabla ^2_h u_{\varepsilon }(x_0)), \end{align*}

where the last line follows from (2.11). Therefore $u_{\varepsilon } (x_0) \leq v(x_0)$ , which is a contradiction.

Existence of a solution follows from the comparison principle and the Perron method.

Theorem 2.4. Assume $F$ is monotone, has width $N$ , and $F(0)=0$ . Suppose there exists $C_g\gt 0$ so that

(2.12) \begin{equation} |g(x)| \leq C_g (1 + |x|) \ \ \text {for all } x\in {\mathbb {Z}^n_h}. \end{equation}

Then there exists a unique solution $u:{\mathbb {Z}^n_h} \to {\mathbb {R}}$ of (2.6) satisfying $\lim _{|x|\to \infty } \frac {u(x)}{|x|^2 }=0$ . Furthermore, we have

(2.13) \begin{equation} |u(x)| \leq C C_g(\textrm {Lip}_N(F)N^3 + 1 + |x|) \ \ \text {for all } x\in {\mathbb {Z}^n_h}, \end{equation}

where $C$ depends only on $n$ .

Proof. We define

\begin{equation*}\psi (x) = \sqrt {1 + |x|^2},\end{equation*}

and note that $\psi$ is a smooth function with linear growth satisfying

(2.14) \begin{equation} \frac {1}{ \sqrt {2}}(1+|x|) \leq \psi (x) \leq 1+|x| \ \ \text {for all } x\in {\mathbb {R}}^n. \end{equation}

For any ${\mathbf {v}}\in {\mathbb {Z}^n}$ and $x\in {\mathbb {Z}^n_h}$ , we have

\begin{align*} |\nabla ^2_h \psi (x,{\mathbf {v}})| &\leq |\nabla ^2_h \psi (x,{\mathbf {v}}) - {\mathbf {v}}^T \nabla ^2 \psi (x){\mathbf {v}} | + C|{\mathbf {v}}|^2\\ &\leq |\nabla ^2_h \psi (x,{\mathbf {v}}) - {\mathbf {v}}^T \nabla ^2 \psi (x){\mathbf {v}} | + C|{\mathbf {v}}|^2\\ &\leq C\left (|{\mathbf {v}}|^3h+ |{\mathbf {v}}|^2\right ) \leq C |{\mathbf {v}}|^3, \end{align*}

where we used a Taylor series expansion of $\nabla ^2_h \psi (x,{\mathbf {v}})$ in the last line, along with $h\leq 1$ and $|{\mathbf {v}}|\geq 1$ . Since $F(0)=0$ we have

(2.15) \begin{equation} |F(\nabla ^2_h\psi )| = |F(\nabla ^2_h\psi ) - F(0)|\leq \textrm {Lip}_N(F) \|\nabla ^2_h \psi \|_{N,\infty }\leq C\textrm {Lip}_N(F)N^3=:\xi . \end{equation}

We now define

\begin{equation*}w = C_g(\xi + \sqrt {2} \psi ).\end{equation*}

Then by (2.14) and (2.15), we have

\begin{equation*} w - F(\nabla ^2_h w) = C_g\xi + \sqrt {2}C_g \psi - F(C_g \nabla ^2_h \psi ) \geq \sqrt {2}C_g \psi \geq C_g(1+|x|) \geq g.\end{equation*}

A similar argument shows that $v=-w$ satisfies $v - F(\nabla ^2_h v) \leq g$ .

Define

(2.16) \begin{equation} {\mathcal F} = \left \{v:{\mathbb {Z}^n_h} \to {\mathbb {R}} \, : \, v - F(\nabla ^2_h v) \leq g \ \text {and} \ v \leq w\right \} \end{equation}

and

(2.17) \begin{equation} u(x) = \sup \{v(x)\, : \, v \in {\mathcal F}\}. \end{equation}

Since $v=-w$ belongs to $\mathcal F$ , the set $\mathcal F$ is nonempty and we have $-w \leq u \leq w$ . Hence, $u$ satisfies (2.13).

We now claim that

\begin{equation*}u - F(\nabla ^2_h u) \leq g \ \ \text {on } \mathbb {Z}_h.\end{equation*}

To see this, fix $x_0 \in {\mathbb {Z}^n_h}$ and let $v_k\in {\mathcal F}$ such that $\lim _{k\to \infty }v_k(x_0)=u(x_0)$ . By passing to a subsequence, if necessary, we can assume that $\lim _{k\to \infty }v_k(x)$ exists for all $x\in {\mathbb {Z}^n_h}$ . Let us denote $v(x)\,:\!=\,\lim _{k\to \infty }v_k(x)$ , noting that $v(x_0)=u(x_0)$ . By continuity we have $v - F(\nabla ^2_h v) \leq g$ and $v \leq w$ , thus $v\in {\mathcal F}$ and $v\leq u$ . Since $v(x_0)=u(x_0)$ , we have that $v-u$ attains its maximum at $x_0$ . As in the proof of Theorem2.3 we have $\nabla ^2_hv(x_0,{\mathbf {v}}) \leq \nabla ^2_h u(x_0,{\mathbf {v}})$ for all $\mathbf {v}$ , and so by the monotonicity of $F$ we have $F(\nabla ^2_h v(x_0)) \leq F(\nabla ^2_h u(x_0))$ , which when combined with $u(x_0)=v(x_0)$ and $v - F(\nabla ^2_h v)\leq g$ , establishes the claim.

To complete the proof, we show that

\begin{equation*}u - F(\nabla ^2_h u) \geq g \ \ \text {on } {\mathbb {Z}^n_h}.\end{equation*}

Assume to the contrary that there is some $x_0\in {\mathbb {Z}^n_h}$ such that

\begin{equation*}u(x_0) - F(\nabla ^2_h u(x_0)) \lt g(x_0).\end{equation*}

Define

\begin{equation*}v(x) = \begin{cases} u(x_0) + {\varepsilon },& \text {if } x=x_0\\ u(x),& \text {otherwise.} \end{cases}\end{equation*}

By continuity, we can choose ${\varepsilon }\gt 0$ small enough so that

\begin{equation*}v(x_0) - F(\nabla ^2_h v(x_0)) \leq g(x_0).\end{equation*}

For $x\neq x_0$ , we already have

\begin{equation*}v(x) - F(\nabla ^2_h v(x)) \leq g(x),\end{equation*}

and the definition of $v$ and the monotonicity of $F$ imply that $\nabla ^2 v(x,{\mathbf {v}}) \geq \nabla ^2_h u(x,{\mathbf {v}})$ for all $x\neq x_0$ and ${\mathbf {v}}\in {\mathbb {Z}^n}$ . This completes the proof.

2.2 Properties of solutions

We recall that a function $u:{\mathbb {Z}^n_h} \to {\mathbb {R}}$ is Lipschitz continuous if there exists $C\gt 0$ such that

\begin{equation*}|u(x) - u(y)| \leq C\|x-y\|\end{equation*}

holds for all $x,y \in {\mathbb {Z}^n_h}$ . The Lipschitz constant of $u$ , denoted $\textrm {Lip}(u)$ , is the smallest such constant, given by

\begin{equation*}\textrm {Lip}(u) = \sup _{\substack {x,y\in {\mathbb {Z}^n_h} \\ x\neq y}} \frac {|u(x)-u(y)|}{\|x-y\|}.\end{equation*}

Proof. (i) Let $z\in {\mathbb {Z}^n_h}$ and define $w(x) = u(x+z) - \textrm {Lip}(g) \|z\|$ . Then we have

\begin{equation*}w(x) - F(\nabla ^2_h w(x)) = u(x+z) - \textrm {Lip} (g) \|z\| - F(\nabla ^2_h u(x+z)) = g(x+z) - \textrm {Lip}(g) \leq g(x).\end{equation*}

By the comparison principle, Theorem2.3, we have $w \leq u$ , and so

\begin{equation*}u(x+z) - u(x) \leq \textrm {Lip}(g)\|z\|.\end{equation*}

Since this holds for all $x,z\in {\mathbb {Z}^n_h}$ , the proof of (i) is complete.

(ii) If $F\geq 0$ then $u(x) = g(x) + F(\nabla ^2_h u(x)) \geq g(x)$ .

(iii) Let ${\overline {g}}:{\mathbb {R}}^n\to {\mathbb {R}}$ be the piecewise constant extension of $g$ to a function on ${\mathbb {R}}^n$ , defined so that ${\overline {g}}(y)=g(x)$ for all $y\in x+[0,h)^n$ and any $x\in Zh$ . While the extension is discontinuous, it satisfies

\begin{equation*}|\nabla _h {\overline {g}}(y,{\mathbf {v}})| = \left | \frac {g(x+h{\mathbf {v}}) - g(x)}{h}\right | \leq \textrm {Lip}(g)|{\mathbf {v}}|\end{equation*}

provided ${\mathbf {v}}\in {\mathbb {Z}^n}$ , where $x\in {\mathbb {Z}^n_h}$ satisfies $y\in x+[0,h)^n$ . Let $\varepsilon \gt 0$ and define the standard mollification $g_\varepsilon \,:\!=\,\eta _\varepsilon * {\overline {g}}$ , where $\eta _{\varepsilon }(x) = \frac {1}{{\varepsilon }^d}\eta \left ( \frac {x}{{\varepsilon }}\right )$ , and $\eta _{\varepsilon }$ is compactly supported in $B(0,{\varepsilon })$ . Note that

\begin{equation*}\nabla _h g_{\varepsilon }(x,{\mathbf {v}}) = \int _{{\mathbb {R}}^n} \eta _\varepsilon (y) \nabla _h {\overline {g}}(x-y,{\mathbf {v}}) \, dy = \int _{{\mathbb {R}}^n }\eta _\varepsilon (x-z) \nabla _h {\overline {g}}(z,{\mathbf {v}}) \, dz,\end{equation*}

where we used the change of variables $z=x-y$ above. We also have

\begin{equation*}\nabla _h g_{\varepsilon }(x,-{\mathbf {v}}) = \int _{{\mathbb {R}}^n} \eta _\varepsilon (y) \nabla _h {\overline {g}}(x-y,-{\mathbf {v}}) \, dy = -\int _{{\mathbb {R}}^n }\eta _\varepsilon (x-z - h{\mathbf {v}}) \nabla _h {\overline {g}}(z,{\mathbf {v}}) \, dz,\end{equation*}

where we now use the change of variables $z=x-y - h{\mathbf {v}}$ . Therefore

\begin{align*} \nabla ^2_h g_{\varepsilon }(x,{\mathbf {v}}) &= \frac {1}{h}(\nabla _h {\overline {g}}_{\varepsilon }(x,{\mathbf {v}}) + \nabla _h {\overline {g}}_{\varepsilon }(x,-v)) \\ &= -\int _{{\mathbb {R}}^n}\nabla _h \eta _{\varepsilon }(x-z,-{\mathbf {v}}) \nabla _h {\overline {g}}(z,{\mathbf {v}})\, dz. \end{align*}

Therefore,

\begin{equation*} |\nabla ^2_h g_{\varepsilon }(x,{\mathbf {v}})| \leq \int _{B(x,{\varepsilon })\cup B(x-h{\mathbf {v}},{\varepsilon })}|\nabla _h \eta _{\varepsilon }(x-z,-{\mathbf {v}})| |\nabla _h {\overline {g}}(z,{\mathbf {v}})|\, dz \leq \frac {C}{{\varepsilon }}\textrm {Lip}(g)|{\mathbf {v}}|^2.\end{equation*}

Since $F$ has width $N$ , it follows that

(2.19) \begin{equation} |F(\nabla ^2_h g_{\varepsilon }(x))| = |F(\nabla ^2_h g_{\varepsilon }(x)) - F(0)| \leq \textrm {Lip}_N (F)\|\nabla ^2_h g_{\varepsilon }\|_{N,{\varepsilon }} \leq \frac {C}{{\varepsilon }}\textrm {Lip}_N(F)\textrm {Lip}(g)N^2. \end{equation}

We also have that $|{\overline {g}}(y) - g(x)| \leq |y-x| + \sqrt {d}h$ , and so

\begin{align*} |g_{\varepsilon }(x) - g(x)| &= \left |\int _{B(x,{\varepsilon })} \eta _{\varepsilon }(x-y)( {\overline {g}}(y) - g(x))\, dy \right |\\ &\leq \int _{B(x,{\varepsilon })} \eta _{\varepsilon }(x-y)|{\overline {g}}(y) - g(x)|\, dy\\ &\leq \textrm {Lip}(g)({\varepsilon } + \sqrt {d}h). \end{align*}

Combining this with (2.19) we have

\begin{equation*}|g_\varepsilon (x) + F(\nabla ^2 g_\varepsilon (x)) - g(x)| \leq \textrm {Lip}(g)({\varepsilon } + \sqrt {d}h) + \frac {C}{{\varepsilon }}\textrm {Lip}_N(F)\textrm {Lip}(g)N^2.\end{equation*}

Choosing ${\varepsilon }=N \sqrt {\textrm {Lip}_N(F)}$ yields

\begin{equation*}|g_\varepsilon (x) + F(\nabla ^2 g_\varepsilon (x)) - g(x)| \leq C\textrm {Lip}(g)\left ( N\sqrt {\textrm {Lip}_N(F)} + h\right ).\end{equation*}

By the comparison principle, Theorem2.3, we have

\begin{equation*}g_{\varepsilon } - C\textrm {Lip}(g)\left ( N\sqrt {\textrm {Lip}_N(F)} + h\right ) \leq u \leq g_{\varepsilon } + C\textrm {Lip}(g)\left ( N\sqrt {\textrm {Lip}_N(F)} + h\right ),\end{equation*}

which completes the proof.

To study further properties of $u$ , we need some additional definitions.

Definition 2.6. We say that $F:{\mathcal H}_n\to {\mathbb {R}}$ is convex if for all $X,Y\in {\mathcal H}_n$ and $\lambda \in [0,1]$ we have

\begin{equation*}F(\lambda X + (1-\lambda )Y) \leq \lambda F(X) + (1-\lambda )F(Y).\end{equation*}

We note that by (2.4), the convexity of $u$ is equivalent to the inequality

\begin{equation*}u(x) \leq \frac {1}{2}(u(x+h{\mathbf {v}}) + u(x-h{\mathbf {v}}))\end{equation*}

holding for all $x\in {\mathbb {Z}^n_h}$ and ${\mathbf {v}} \in {\mathbb {Z}^n}$ . We also note that the choice of $F$ given in (2.7) is convex.

We also consider a permutation invariance property.

We note that $F$ given in (2.7) is permutation invariant.

Finally, we also study translation properties.

Definition 2.10. For ${\mathbf {v}}\in {\mathbb {Z}^n}$ , we say $u:{\mathbb {Z}^n_h}\to {\mathbb {R}}$ satisfies the $\mathbf {v}$ -translation property if there exists a constant $c_{\mathbf {v}}$ such that

(2.20) \begin{equation} u(x+s{\mathbf {v}}) = u(x) + c_{\mathbf {v}} s \end{equation}

for all $x\in {\mathbb {Z}^n_h}$ and $s\in {\mathbb {R}}$ such that $s{\mathbf {v}} \in {\mathbb {Z}^n_h}$ .

The next lemma shows that these properties for $u$ are inherited from $F$ and $g$ .

Proof. (i) Let ${\mathbf {v}}\in {\mathbb {Z}^n}$ and define

\begin{equation*}w(x) = \frac {1}{2}(u(x+h{\mathbf {v}}) + u(x-h{\mathbf {v}})).\end{equation*}

Then for $x\in {\mathbb {Z}^n_h}$ we compute, using the convexity of $F$ , that

\begin{align*} w(x) - F(\nabla ^2_h w(x)) &= \frac {1}{2}u(x+h{\mathbf {v}}) + \frac {1}{2}u(x-h{\mathbf {v}}) - F\left ( \frac {1}{2}\nabla ^2_h u(x+h{\mathbf {v}}) + \frac {1}{2}\nabla ^2_h u(x-h{\mathbf {v}})\right )\\ &\geq \frac {1}{2}u(x+h{\mathbf {v}}) + \frac {1}{2}u(x-h{\mathbf {v}}) - \frac {1}{2}F(\nabla ^2_h u(x+h{\mathbf {v}})) - \frac {1}{2}F(\nabla ^2_h u(x-h{\mathbf {v}})) \\ &= \frac {1}{2}g(x+h{\mathbf {v}}) + \frac {1}{2}g(x-h{\mathbf {v}})\\ &= \frac {h^2}{2}\nabla ^2_h g(x,{\mathbf {v}}) + g(x)\\ &\geq g(x), \end{align*}

where the last line follows from the convexity of $g$ . By Theorem2.3 we have $w \geq u$ , and so

\begin{equation*}\nabla ^2_h u(x,{\mathbf {v}}) = \frac {2}{h^2}(w(x) - u(x)) \geq 0\end{equation*}

for all $x\in {\mathbb {Z}^n_h}$ . Since ${\mathbf {v}}\in {\mathbb {Z}^n}$ is arbitrary, we have $\nabla ^2_h u(x)\geq 0$ for all $x\in {\mathbb {Z}^n}$ , hence $u$ is convex.

(ii) Let $\sigma$ be a permutation and set $w = u\circ \sigma$ . Then we have

\begin{equation*}w(x) - F(\nabla ^2_h w(x)) = u(\sigma (x)) - F(\nabla ^2_h (u\circ \sigma )(x)).\end{equation*}

We note that for any ${\mathbf {v}}\in {\mathbb {Z}^n}$ we have $\nabla ^2_h (u\circ \sigma )(x,{\mathbf {v}}) = \nabla ^2_h u(\sigma (x),\sigma ({\mathbf {v}}))$ . Therefore $\nabla ^2_h (u\circ \sigma )(x) = \nabla ^2_h u(\sigma (x)) \circ \sigma$ . Since $F$ is permutation invariant we have

\begin{equation*}F(\nabla ^2_h (u\circ \sigma )(x)) = F(\nabla ^2_h u(\sigma (x))\circ \sigma ) = F(\nabla ^2_h u(\sigma (x))).\end{equation*}

Therefore,

\begin{equation*}w(x) - F(\nabla ^2_h w(x)) = u(\sigma (x)) - F(\nabla ^2_h u(\sigma (x))) = 0.\end{equation*}

By uniqueness of the solution $u$ of (2.6), we have $w=u$ , which completes the proof.

(iii) Let $s\in {\mathbb {R}}$ such that $s{\mathbf {v}} \in {\mathbb {Z}^n_h}$ , and define $w(x) = u(x + s{\mathbf {v}}) - c_{\mathbf {v}} s$ . Then,

\begin{equation*}w(x) - F(\nabla ^2_h w(x)) = u(x+s{\mathbf {v}}) - c_{\mathbf {v}} s - F(\nabla ^2_h u(x+s{\mathbf {v}})) = g(x+s{\mathbf {v}}) - c_{\mathbf {v}} s = g(x).\end{equation*}

Since the solution of (2.6) is unique, we have $w=u$ , which completes the proof.

2.3 Convergence rates

In this section, we prove convergence of the numerical scheme (2.6) towards the viscosity solution of the second-order degenerate elliptic equation

(2.21) \begin{equation} u - F(\nabla ^2 u) = g \ \ \text { on } {\mathbb {R}}^n. \end{equation}

As before, we assume $F:{\mathcal H}_n\to {\mathbb {R}}$ , and we interpret $F(X)$ for an $n\times n$ symmetric matrix $X$ as

\begin{equation*}F(X) \,:\!=\, F({\mathbf {v}} \mapsto {\mathbf {v}}^T X{\mathbf {v}}).\end{equation*}

We recall the definitions of viscosity solutions in Appendix A. Throughout this section, we will use the notation $\text {USC}({\mathcal O})$ (resp. $\text {LSC}({\mathcal O})$ ) for the set of functions that are upper (resp. lower) semicontinuous at all points in ${\mathcal O} \subset {\mathbb {R}}^n$ . For more details on the theory of viscosity solutions, we refer the reader to the user’s guide [Reference Crandall, Ishii and Lions19] and [Reference Calder10].

Existence and uniqueness of a linear growth viscosity solution to (2.21) is standard material for viscosity solutions, and proofs can be found in [Reference Calder10, Reference Crandall, Ishii and Lions19]. For use later on, we recall the comparison principle for (2.21) in Lemma 2.12 below. The proof of this result is standard in viscosity solution theory; a self-contained proof can be found in [Reference Calder10, Lemma 12.17].

Lemma 2.12. Assume $F$ is uniformly continuous, monotone, has width $N$ , and satisfies $F(0)=0$ . Let $u\in \text {USC}({\mathbb {R}}^n)$ be a viscosity subsolution of (2.21) and let $v \in \text {LSC}({\mathbb {R}}^n)$ be a viscosity supersolution of (2.21) . If

(2.22) \begin{equation} \lim _{|x|\to \infty }\frac {u(x)-v(x)}{|x|^2}=0 \end{equation}

then $u\leq v$ on ${\mathbb {R}}^n$ .

Using this comparison principle and the Perron method, we can prove existence of a linear growth solution (Theorem2.13 below). The application of the Perron method is standard and a self-contained proof can be found in [Reference Calder10, Theorem 12.18].Footnote ²

Theorem 2.13. Assume $F$ is monotone, has width $N$ , and satisfies $F(0)=0$ . Assume $g$ is Lipschitz continuous and there exists $C_g\gt 0$ such that

(2.23) \begin{equation} |g(x)| \leq C_g (1 + |x|). \end{equation}

Then there exists a unique viscosity solution $u\in C({\mathbb {R}}^n)$ of (2.21) satisfying

\begin{equation*}\lim _{|x|\to \infty }\frac {u(x)}{|x|^2} = 0.\end{equation*}

Furthermore, there exists $C\gt 0$ such that

(2.24) \begin{equation} |u(x)| \leq C(1 + |x|). \end{equation}

Convergence of the discrete scheme (2.6) to the PDE (2.21) is a standard result in viscosity solution theory. We state the theorem in the following result and briefly sketch the proof.

Theorem 2.14. Assume $F$ is monotone, has width $N$ , and satisfies $F(0)=0$ . Assume $g:{\mathbb {R}}^n\to {\mathbb {R}}$ is Lipschitz continuous and satisfies (2.23) . Let $u_h$ be the solution of (2.6) and let $u$ be the viscosity solution of (2.21) . Then $u_h$ converges to $u$ locally uniformly as $h\to 0$ , that is for all $R\gt 0$ we have

(2.25) \begin{equation} \lim _{h\to 0} \max _{\substack {x\in {\mathbb {Z}^n_h} \\ |x| \leq R}} |u_h(x) - u(x)| = 0. \end{equation}

Proof. By Lemma 2.5 (i), we have $\textrm {Lip}(u_h)\leq \textrm {Lip}(g)$ . By the Arzelá-Ascoli Theorem there exists a Lipschitz continuous function $u:{\mathbb {R}}^n\to {\mathbb {R}}$ such that, upon passing to a subsequence $u_{h_k}$ , we have

\begin{equation*}\lim _{k\to \infty } \max _{\substack {x\in \mathbb {Z}^n_{h_k} \\ |x| \leq R}} |u_{h_k}(x) - u(x)| = 0\end{equation*}

for all $R\gt 0$ . The proof will be completed by showing that $u$ is a viscosity solution of (2.21). By uniqueness of viscosity solutions of (2.21), the whole sequence $u_h$ converges locally uniformly to $u$ .

We verify the subsolution property for $u$ ; the supersolution property is similar. Let $x_0 \in {\mathbb {R}}^n$ and ${\varphi }\in C^\infty ({\mathbb {R}}^n)$ such that $u-{\varphi }$ has a local maximum at $x_0$ . Without loss of generality, we can assume $x_0$ is a strict global maximum of $u-{\varphi }$ . It follows that there exists $x_k \to x_0$ such that $u_{h_k} - {\varphi }$ attains its maximum over $\mathbb {Z}_{h_k}^n$ at $x_k$ . Therefore $\nabla ^2_h u_{h_k}(x_k,{\mathbf {v}}) \leq \nabla ^2_h {\varphi }(x_k)$ , and since $F$ is monotone we have

\begin{equation*}u(x_k) - F(\nabla ^2_{h_k} {\varphi }(x_k)) \leq {\varphi }(x_k) - F(\nabla ^2_h u_{h_k}(x_k)) =u(x_k) -u_{h_k}(x_k) - g(x_k).\end{equation*}

Sending $k\to \infty$ we obtain

\begin{equation*}u(x_0) - F(\nabla ^2 {\varphi }(x_0)) \leq g(x_0),\end{equation*}

which completes the proof.

If we have additional regularity for the solution $u$ of (2.21), we can prove convergence rates. We recall the $C^{k,1}$ seminorm of $u:{\mathbb {R}}^n\to {\mathbb {R}}$ is defined by

\begin{equation*}[u]_{C^{k,1}({\mathbb {R}}^n)} = \sum _{1\leq |\alpha | \leq k} \textrm {Lip}(D^\alpha u).\end{equation*}

Theorem 2.15. Assume $F$ is monotone, has width $N$ , and satisfies $F(0)=0$ . Assume $g:{\mathbb {R}}^n\to {\mathbb {R}}$ is Lipschitz continuous and satisfies (2.23) . Let $u_h$ be the solution of (2.6) and let $u$ be the viscosity solution of (2.6) . If $[u]_{C^{k,1}({\mathbb {R}}^n)} \lt \infty$ for $k=2$ or $k=3$ , then we have

(2.26) \begin{equation} \|u - u_h\|_\infty \leq C\textrm {Lip}_N(F)[u]_{C^{k,1}({\mathbb {R}}^n)}N^{k+1}h^{k-1} \end{equation}

Proof. By Taylor expansion, we have

\begin{equation*} |\nabla ^2_h u(x,{\mathbf {v}}) - {\mathbf {v}}^T \nabla ^2 u(x) {\mathbf {v}}| \leq C[u]_{C^{k,1}({\mathbb {R}}^n)}|{\mathbf {v}}|^{k+1}h^{k-1}.\end{equation*}

Therefore,

\begin{equation*}|F(\nabla ^2_h u(x)) - F(\nabla ^2 u(x))| \leq C\textrm {Lip}_N(F)[u]_{C^{k,1}({\mathbb {R}}^n)}N^{k+1}h^{k-1}=:{\varepsilon }.\end{equation*}

It follows that

\begin{equation*}u(x) - F(\nabla ^2_hu(x)) \leq u(x) - F(\nabla ^2 u(x)) + {\varepsilon } = g(x) + {\varepsilon } \ \ \text {for } x\in {\mathbb {Z}^n_h}.\end{equation*}

By Theorem2.3, we have $u \leq u_h + {\varepsilon }$ on $\mathbb {Z}^n_h$ . The opposite inequality is obtained similarly.

From the convergence results in Theorems2.14 and 2.15, we immediately obtain that all the properties of the discrete solutions proved in Section 2.2 extend to the viscosity solution $u$ .

We note that in (iv), the notion of convexity for $F$ is the same as in Definition 2.6, while for $u$ it is the usual one for functions on ${\mathbb {R}}^n$ , that is

\begin{equation*}u(\lambda x + (1-\lambda ) y) \leq \lambda u(x) + (1-\lambda ) u(y).\end{equation*}

In (v), the definition of permutation invariance for $u:{\mathbb {R}}^n\to {\mathbb {R}}$ is identical to Definition 2.8; namely $u=u\circ \sigma$ . The translation property is defined similarly, but is slightly different so we give the definition below for functions on ${\mathbb {R}}^n$ .

The proof of Proposition 2.16 follows from Lemmas 2.5 and 2.11, and the convergence result in Theorem2.14.

2.4 Restricting the domain

In order to compute the solution of the discrete scheme (2.6) on an unbounded domain $\mathbb {Z}^n_h$ , it is necessary to restrict the domain to a compact set. In this section, we study restrictions of (1.3) to computational domains of the form $\Omega _{T,h} \,:\!=\, [\!-T,T]^n\cap {\mathbb {Z}^n_h}$ . In this section, we always assume $T=mh$ for some integer $m\geq 1$ . We define the width $N$ boundary as

(2.28) \begin{equation} \partial _N \Omega _{T,h} = \{x\in {\mathbb {Z}^n_h}\setminus \Omega _{T,h}\, : \, \text {there exists }y\in \Omega _{T,h} \text { with } |x-y|_\infty \leq N\}. \end{equation}

We set a Dirichlet boundary condition on $\partial _N \Omega _{T,h}$ and show that this condition affects the solution only near the boundary, and the solution remains accurate in the interior of $\Omega _{T,h}$ .

In this section, we study the equation

(2.29) \begin{equation} u - F(\nabla ^2_h u) = g \ \ \text { on } \ \Omega _{T,h}, \end{equation}

which is the restriction of (2.6) to the computational domains $\Omega _{T,h}$ . We first recall the comparison principle for (2.29).

Lemma 2.18. Assume $F$ is monotone, has width $N$ , and satisfies $F(0)=0$ . Let $u,v:{\mathbb {Z}^n_h}\to {\mathbb {R}}$ satisfy

(2.30) \begin{equation} u - F(\nabla ^2_h u) \leq v - F(\nabla ^2_h v) \ \ \text { on } \ \Omega _{T,h} \end{equation}

and $u\leq v$ on $\partial _N \Omega _{T,h}$ . Then $u\leq v$ on $\Omega _{T,h}$ .

Proof. Let $x_0\in \Omega _{T,h}\cup \partial _N \Omega _{T,h}$ be a point where $u-v$ attains its maximum value over $\Omega _{T,h}\cup \partial _N \Omega _{T,h}$ . If $x_0 \in \partial _N \Omega _{T,h}$ then $u\leq v$ , so we may assume $x_0 \in \Omega _{T,h}$ . In this case we have $\nabla ^2_h u(x_0,{\mathbf {v}}) \leq \nabla ^2_h v(x_0,{\mathbf {v}})$ , and since $F$ is monotone, we obtain

\begin{equation*} u(x_0) - v(x_0) \leq F(\nabla ^2_h u(x_0)) - F(\nabla ^2_h v(x_0)) \leq 0.\end{equation*}

This completes the proof.

We now establish localization of solutions of (2.29).

Theorem 2.19. Assume $F$ is monotone, has width $N$ , and satisfies $F(0)=0$ . Let $u,v:{\mathbb {Z}^n_h}\to {\mathbb {R}}$ satisfy (2.30) . Then for any $\alpha \in (0,1)$ we have

(2.31) \begin{equation} \max _{\Omega _{\alpha T,h}}(u-v) \leq \frac {C \textrm {Lip}_N(F)N^2}{(1-\alpha )^2T^2}\left (\log \left ( \frac {T^2}{\textrm {Lip}_N(F) N^2}\right )^2+1 \right )\max _{\partial _N \Omega _{T,h}}(u-v). \end{equation}

Remark 2.20. The estimate (2.31) in Theorem 2.19 shows that the Dirichlet boundary conditions on $\partial _N \Omega _{T,h}$ have a limited domain of influence on the solution in $\Omega _{T,h}$ when $T\gt 0$ is large. Indeed, if we fix, say, $\alpha =\tfrac 12$ , and if $u,v:{\mathbb {Z}^n_h}\to {\mathbb {R}}$ are solutions of (2.29) , then $u$ and $v$ satisfy (2.30) with equality, and so it follows from two applications of Theorem 2.19 , applied to $u-v$ and $v-u$ , that

(2.32) \begin{equation} \max _{\Omega _{T/2,h}}|u-v|\leq \frac {C\log (T)^2}{T^2}\max _{\partial _N \Omega _{T,h}}|u-v|, \end{equation}

holds, where $C$ depends on $\textrm {Lip}_N(F)$ and $N$ , and $T\geq 3$ . This shows that errors in the Dirichlet condition on $\partial _N\Omega _{T,h}$ can be tolerated numerically, provided $T\gt 0$ is large enough. For example, by Lemma 2.5 (iii), we can set $u=g$ on $\partial _N \Omega _{T,h}$ and obtain an ${\mathcal O}(\log (T)^2/T^2)$ approximation of the solution of (2.21) on the interior domain $\Omega _{T/2,h}$ . We show how to obtain even more accurate solutions with better choices of boundary conditions in Section 3.1 .

Proof. Let $\mu = \max _{\partial _N \Omega _{T,h}}(u-v)$ and define

(2.33) \begin{equation} w(x) = v(x) + \gamma + \mu \sum _{i=1}^n \left (e^{\lambda \left ( \frac {x_i}{T}-1 \right )} + e^{-\lambda \left ( \frac {x_i}{T}+1 \right )}\right ) \end{equation}

for parameters $\lambda, \gamma \gt 0$ to be determined. For $x\in \partial _N \Omega _{T,h}$ , there exists $i$ such that $x_i\geq T$ or $x_i\leq -T$ , and so $w(x) \geq v(x) + \mu \geq u(x)$ . Note that we have

\begin{equation*}|F(\nabla ^2_h w) - F(\nabla ^2_h v)| \leq \textrm {Lip}_N(F) \|\nabla ^2_h w - \nabla ^2_h v\|_{N,\infty } \leq C\textrm {Lip}_N(F)\frac {\mu \lambda ^2N^2}{T^{2}}.\end{equation*}

Choosing $\gamma = C\textrm {Lip}_N(F)\mu \lambda ^2N^2T^{-2}$ we find that

\begin{equation*}w(x) - F(\nabla ^2_h w(x)) \geq v(x) + \gamma - F(\nabla ^2_h v(x)) - C\textrm {Lip}_N(F)\mu \lambda ^2N^2T^{-2} \geq 0.\end{equation*}

By Lemma 2.18 we have $w \geq u$ on $\Omega _{T,h}$ , and so

\begin{equation*}u(x) - v(x) \leq \gamma + \mu \sum _{i=1}^n \left (e^{\lambda \left ( \frac {x_i}{T}-1 \right )} + e^{-\lambda \left ( \frac {x_i}{T}+1 \right )}\right )\end{equation*}

for all $x\in \Omega _{T,h}$ . For $x\in \Omega _{\alpha T,h}$ with $\alpha \in (0,1)$ we have

\begin{equation*}u(x) - v(x) \leq \gamma + 2n\mu e^{-\lambda (1-\alpha )}. \end{equation*}

Choosing $\lambda$ so that $(1-\alpha )\lambda = \log \left ( \frac {T^2}{\textrm {Lip}_N(F) N^2}\right )$ completes the proof.

3.1 Reducing the dimension

We show here how to reduce the problem from $n$ dimensional to $n-1$ dimensional. This requires the translation property (2.20) and is based on the following lemma.

Lemma 3.1. Suppose $u$ satisfies the translation property (2.20) with ${\mathbf {v}} = \unicode {x1D7D9}$ and $c_{\mathbf {v}} = 1$ . Then for all ${\mathbf {v}} \in {\mathcal B}^n$ we have

(3.2) \begin{equation} \nabla ^2_h u(x,{\mathbf {v}}) = \nabla ^2_h u(x,\unicode {x1D7D9}-{\mathbf {v}}). \end{equation}

Proof. By assumption we have

(3.3) \begin{equation} u(x + s\unicode {x1D7D9}) = u(x) + s \end{equation}

for all $s\in \mathbb {Z}_h$ and $x\in {\mathbb {Z}^n_h}$ . We now compute, using (3.3), that

\begin{align*} h^2\nabla ^2_h u(x,{\mathbf {v}}) &= u(x+h{\mathbf {v}}) - 2u(x) + u(x-h{\mathbf {v}}) \\ &=u(x + h{\mathbf {v}} - h\unicode {x1D7D9}) + h - 2u(x) + u(x -h{\mathbf {v}}+h\unicode {x1D7D9}) - h\\ &=u(x - h(\unicode {x1D7D9}-{\mathbf {v}})) - 2u(x) + u(x + h(\unicode {x1D7D9} - {\mathbf {v}}))\\ &=h^2 \nabla ^2_h u(x,\unicode {x1D7D9}-{\mathbf {v}}), \end{align*}

which completes the proof.

For $x\in {\mathbb {R}}^{n-1}$ and $a\in {\mathbb {R}}$ we define $(x,a)\in {\mathbb {R}}^n$ by

\begin{equation*}(x,a) = (x_1,x_2,\ldots, x_{n-1},a).\end{equation*}

The next lemma shows how we can use the translation property (2.20) to reduce (3.1) to a similar equation in one less variable.

Proof. By Lemma 3.1, for any $x\in \mathbb {Z}^{n-1}_h$ we have

\begin{equation*}\max _{{\mathbf {v}} \in {\mathcal B}^n}\nabla ^2_h u((x,0),{\mathbf {v}}) = \max _{{\mathbf {v}} \in {\mathcal B}^{n-1}}\nabla ^2_h u((x,0),({\mathbf {v}},0)) = \max _{{\mathbf {v}} \in {\mathcal B}^{n-1}}\nabla ^2_h w(x,{\mathbf {v}}).\end{equation*}

Since $u$ satisfies (3.1) we have

\begin{equation*}w(x) - \frac {1}{2}\max _{{\mathbf {v}} \in {\mathcal B}^{n-1}}\nabla ^2_h w(x,{\mathbf {v}}) = g(x,0)=f(x),\end{equation*}

which completes the proof.

(ii) Since $g$ is permutation invariant, so is $u$ (by Lemma 2.11) and hence $w$ . Let $\sigma _{ij}$ denote the permutation swapping coordinates $i$ and $j$ . Let $x\in {\mathbb {R}}^{n-1}$ and let $y=(x,0)$ . For any $1 \leq i\leq n-1$ we use the translation property and permutation invariance to obtain

\begin{align*} w(x)=u( \left (x,0\right )) &= u\left ( (x,0) - x_i\unicode {x1D7D9}\right ) + x_i\\ &= u\left ( \left (x - x_i\unicode {x1D7D9},-x_i\right )\right ) + x_i\\ &=u\left (\sigma _{i,n}\left (x - x_i\unicode {x1D7D9},-x_i\right )\right ) + x_i\\ &=u\left (x - x_i\unicode {x1D7D9} - x_i{\mathbf {e}}_i,0\right ) + x_i\\ &=w\left (x - x_i\unicode {x1D7D9} - x_i{\mathbf {e}}_i\right ) + x_i, \end{align*}

which completes the proof.

By Lemma 3.2 and Remark 2.20, we may instead solve the equation

(3.6) \begin{equation} \left \{ \begin{aligned} w(x) - \frac {1}{2}\max _{{\mathbf {v}}\in {\mathcal B}^{n-1}}\nabla ^2_h w(x,{\mathbf {v}}) &= g(x,0),&& \text {if } x \in \Omega _{T,h}\\ w(x) &= g(x,0), && \text {if } x \in \partial _1 \Omega _{T,h} \end{aligned} \right . \end{equation}

in dimension $n-1$ , where we take $T=mh$ to be a multiple of the grid resolution. Provided we restrict our attention to the localized interior set $\Omega _{\alpha T,h}$ for some $\alpha \in (0,1)$ , then as per Remark 2.20 we incur an $O(1/T^2)$ error term, up to logarithmic factors.

3.2 Reducing the domain to a sector

It turns out that in addition to reducing the dimension of the equation from $n$ to $n-1$ , we can also drastically reduce the size of the computational grid by restricting our attention to the sector

(3.7) \begin{equation} {\mathbb {D}}_n = \{x\in {\mathbb {Z}^n_h}\, : \, x_1 \geq x_2 \geq \cdots \geq x_n\}, \end{equation}

and the positive sector

(3.8) \begin{equation} {\mathbb {D}}^+_n = \{x\in {\mathbb {D}}_n\, : \, x_n \geq 0\}. \end{equation}

Whenever $g$ is permutation invariant, e.g., the max-regret $g(x)=\max \{x_1,\ldots, x_n\}$ , the solution $u$ of (3.1) and the reduced solution $w(x)=u(x,0)$ are also permutation invariant. As we show in this section, this, combined with the translation property, or (3.5), allows us to reduce the domain of the discrete PDE (3.1) to ${\mathbb {D}}_n^+$ , which is drastically smaller than the full computational grid $\Omega _{T,h}$ .

In order to do this in a computational setting, for $x\in {\mathbb {D}}_{n-1}^+$ and $v\in {\mathcal B}^{n-1}$ , we need to be able to evaluate $w(x+{\mathbf {v}})$ and $w(x-{\mathbf {v}})$ in terms of only the values of $w$ within the positive sector ${\mathbb {D}}_{n-1}^+$ . This will allow us to evaluate the discrete second derivative $\nabla ^2_h w(x,{\mathbf {v}})$ without reference to the values of $u$ outside of ${\mathbb {D}}^{n-1}_+$ . To do this, we need to define two operations. First, let $\pi _n:{\mathbb {R}}^n\to {\mathbb {R}}^n$ be the sorting function that sorts the coordinates of an $n$ -dimensional vectors. Thus, $\pi _n({\mathbb {Z}^n_h}) = {\mathbb {D}}_n$ . We also define the function $\xi :{\mathbb {D}}^n\to {\mathbb {R}}^n$ by

(3.9) \begin{equation} \xi _n(x) = \begin{cases} x,& \text {if } x\in {\mathbb {D}}_n^+,\\ x +h(\unicode {x1D7D9} + {\mathbf {e}}_i),& \text {where } x_1,\ldots, x_{i-1} \geq 0 \gt x_i\geq \cdots \geq x_n. \end{cases} \end{equation}

To indicate the cases above, we write

\begin{equation*}\unicode {x1D7D9}_{\xi _n}(x) = \begin{cases} 0,& \text {if } \xi _n(x) = x\\ 1,& \text {otherwise.} \end{cases}\end{equation*}

The following lemma shows how to evaluate $w(x\pm {\mathbf {v}})$ within the positive sector ${\mathbb {D}}_{n-1}^+$ .

Proof. Part (i) follows directly from the permutation invariance of $w$ and that $\mathbf {v}$ is a binary vector.

For part (ii), let $y = \pi _{n-1}(x - h{\mathbf {v}}) \in {\mathbb {D}}_{n-1}$ . If $y\in {\mathbb {D}}_{n-1}^+$ , then $\xi _{n-1}(y)=y$ , $\unicode {x1D7D9}_{\xi _{n-1}}(y)=0$ , and the result follows from the permutation invariance of $w$ , as in part (i). If $y\not \in {\mathbb {D}}_{n-1}^+$ , then $\xi _{n-1}(y)\neq y$ and $\unicode {x1D7D9}_{\xi _{n-1}}(y)=1$ . Since $x\in {\mathbb {D}}_{n-1}^+$ we must have

\begin{equation*}y_1\geq \cdots \geq y_{i-1}\geq 0 \gt -h=y_i = \cdots = y_{n-1},\end{equation*}

for some $1 \leq i \leq n-1$ . Therefore

\begin{equation*}\xi _{n-1}(y) = y + h(\unicode {x1D7D9} + {\mathbf {e}}_i) = y - y_i(\unicode {x1D7D9} + {\mathbf {e}}_i),\end{equation*}

and so by (3.5) and the permutation invariance of $w$ we have

\begin{equation*}w(x - h{\mathbf {v}}) = w(y) = w(\xi _{n-1}(y)) + y_i = w(\xi _{n-1}(\pi _{n-1}(x-h{\mathbf {v}}))) - h,\end{equation*}

which completes the proof.

Lemma 3.3 allows us to restrict the reduced $n-1$ dimensional equation for $w$ to the sector ${\mathbb {D}}^+_{n-1}$ , yielding the equation

(3.10) \begin{equation} \left \{ \begin{aligned} w(x) - \frac {1}{2}\max _{{\mathbf {v}}\in {\mathcal B}^{n-1}}\nabla ^2_h w(x,{\mathbf {v}}) &= g(x,0),&& \text {if } x\in {\mathbb {D}}^+_{n-1}\cap \{x_1 \leq T - h\}\\ w(x) &= g(x,0), && \text {if } x \in {\mathbb {D}}^+_{n-1}\cap \{x_1=T\}, \end{aligned} \right . \end{equation}

where $T=mh$ is a multiple of the grid resolution. By Lemma 3.3 we can compute the derivative $\nabla ^2_h w(x,{\mathbf {v}})$ for $x\in {\mathbb {D}}^+_{n-1}\cap \{x_1 \leq T - h\}$ via

(3.11) \begin{equation} \nabla ^2_h w(x,{\mathbf {v}}) = \frac {w(y_+) + w(\xi _{n-1}(y_-)) - h\unicode {x1D7D9}_{\xi _{n-1}}(y_-) - 2w(x)}{h^2}, \end{equation}

where $y_+ = \pi _{n-1}(x + h{\mathbf {v}})$ and $y_- = \pi _{n-1}(x - h{\mathbf {v}})$ . By Lemma 3.3 the expression on the right-hand side of (3.11) involves evaluating $w$ only at points in the sector ${\mathbb {D}}^+_{n-1}\cap \{x_1\leq T\}$ . This is essentially equivalent to setting boundary conditions on the sector ${\mathbb {D}}^+_{n-1}$ , though the explicit identification of those boundary conditions is a complicated task that we do not undertake here.

Now, the number of grid points in the full computational grid $[\!-T,T]^n\cap \mathbb {Z}_h^n$ grows exponentially in the dimension $n$ as ${\mathcal O}((2Th^{-1})^n)$ , which is known as the curse of dimensionality. However, the number of grid points in the sector ${\mathbb {D}}^+_{n-1}\cap \{x_1\leq T\}$ grows much slower in $n$ , as the following lemma shows.

Proof. Let $T = mh$ and define

(3.12) \begin{equation} a_{m,n} = \sum _{i_1=0}^{m}\sum _{i_2=0}^{i_1}\cdots \sum _{i_{n-1}=0}^{i_{n-2}}\sum _{i_n=0}^{i_{n-1}}1. \end{equation}

Note that $G_{n,T}=a_{m,n}$ . Hence, the proof boils down to showing that $a_{m,n}\leq \frac {1}{n!}(m+n)^{n}$ . We will prove this with induction on $n$ , using the recursive identity

\begin{equation*}a_{m,n}=\sum _{i=0}^m a_{i,n-1},\end{equation*}

which follows directly from (3.12). For the base step of $n=1$ , we have $a_{m,1} = m+1$ by direct computation. For the inductive step, assume that for some $n\geq 1$ and all $m$ we have $a_{m,n}\leq \frac {1}{n!}(m+n)^{n}$ . Then we compute

\begin{equation*}a_{m,n+1} = \sum _{i=0}^m a_{i,n} \leq \sum _{i=0}^m \frac {1}{n!}(i+n)^n \leq \int _0^{m+1}\frac {1}{n!}(x + n)^n \, dx = \frac {1}{(n+1)!}(m+n+1)^{n+1}, \end{equation*}

which completes the inductive step and hence the proof.

If we choose $h$ small enough so that $n \leq Th^{-1}$ , then the bound in Lemma 3.4 implies

\begin{equation*}G_{n,T} \leq \frac {(2 T h^{-1})^n}{n!}.\end{equation*}

This is a factor of $n!$ smaller than the number of grid points on the full computational domain $[\!-T,T]^n\cap \mathbb {Z}_h^n$ , which reflects the exponentially small size of the sector ${\mathbb {D}}^+_{n}$ . In order to understand better how $G_{n,T}$ scales with $n$ in Lemma 3.4, we use a version of Stirling’s formula $n! \geq \sqrt {2\pi n} (n/e)^n$ [Reference Robbins41] to obtain

\begin{equation*}G_{n,T}\leq \frac {1}{n!}(Th^{-1} + n)^n \leq \frac {e^n}{\sqrt {2\pi n}}(1 + Th^{-1}n^{-1})^n \leq \frac {e^{n + Th^{-1}}}{\sqrt {2\pi n}}.\end{equation*}

While this complexity is still exponential in $n$ and $h^{-1}$ , these two quantities do not directly interact. For example, with the full grid $[\!-T,T]^n\cap \mathbb {Z}_h^n$ , increasing from $n$ to $n+1$ dimensions requires a factor of ${\mathcal O}(h^{-1})$ more grid points, while for the sector ${\mathbb {D}}^+_{n}\cap \{x_1\leq T\}$ we require at most $e$ times as many grid points, which is independent of the grid resolution $h$ . However, it is important to point out that the curse of dimensionality is not overcome. It is rather more accurate to say that we have postponed the curse to larger values of $n$ . For example, in Section 4, we conduct experiments with up to $n=7$ experts, using the reduced computational grid in dimension $n-1=6$ , with a grid resolution of $h=0.1$ . Using the full computational grid we are only able to compute solutions of the $n=4$ expert problem, on the reduced $n-1=3$ dimensional grid.

4.1 Full computational grid

Figure 1. Plots of the numerical solution $w$ versus the true solutions for $n=2$ and $n=3$ experts.

We first present results on the full computational grid, so we solve the equation (3.6). Due to the curse of dimensionality, we can only solve the equation for $n=2,3,4$ experts. The finest grid we used was $h=0.01$ for $n\leq 3$ and $h=0.025$ for $n=4$ . In Figure 1 we show plots of the numerical solutions versus the true solutions for $n=2,3$ experts. Since we solve the reduced $d=n-1$ dimensional equation, these are PDEs in $d=1$ and $d=2$ dimensions. The $n=2$ expert solution is accurate on the full domain $[\!-5,5]$ since the boundary condition $u(\pm 5) = \max \{5,0\}$ is exponentially accurate; see (1.6). For $n=3$ experts, the solution loses accuracy away from the restricted domain $[\!-1,1]^2$ .

Figure 2. Convergence rates and optimal strategies for $n=2,3,4$ experts, computed from the numerical solutions. The dashed red line indicates the COMB strategy, which is numerically observed to be optimal for $n=2,3,4$ experts, as the theory predicts.

In Figure 2 (a) we show a convergence analysis for varying grid resolution $h$ for $n=2,3,4$ experts, where the exact solutions of the PDEs are known. In all cases, we observe second-order ${\mathcal O}(h^2)$ convergence rates. Figure 2 also shows the optimality of each adversarial strategy. We define the optimality of strategy ${\mathbf {v}}\in {\mathcal B}^{n-1}$ at a grid point $x$ by the ratio

\begin{equation*}\textrm {opt}(x,{\mathbf {v}}) = \frac {\nabla ^2_h w(x,{\mathbf {v}})}{\max _{\mathbf {p}\in {\mathcal B}^{n-1}}\nabla ^2_h w(x,\mathbf {p})} = \frac {\nabla ^2_h w(x,{\mathbf {v}})}{2(u(x) - g(x,0))}.\end{equation*}

An optimality value of $\textrm {opt}(x,{\mathbf {v}})=1$ indicates that strategy $\mathbf {v}$ is optimal at grid point $x$ . In order to measure the optimality of all the competing strategies, we plot the minimum, maximum, and average optimality values over the intersection of the computational grid with the positive sector ${\mathbb {D}}^{+}_{n-1}$ . Since the solution is permutation invariant, the scores are the same in all other sectors. If the minimum score is $1$ , up to the numerical precision $O(h^2)$ , then that strategy is globally optimal over the unit box $[\!-1,1]^{n-1}$ . In Figure 2 we denote the strategies, which are binary vectors, by the decimal number that strategy corresponds to, and we indicate the COMB strategy with a dashed red line. Since $\mathbf {v}$ and $\unicode {x1D7D9}-{\mathbf {v}}$ are equivalent strategies, we only show the optimality scores for the first half of the strategies; the plot for the second half is a mirror image of the plots shown.

In Figure 2 (b), we see that strategies $1=(0,1)$ and $2=(1,0)$ are optimal, both of which correspond to the COMB strategy. In Figure 2 (c), we see that for $n=3$ experts, the COMB strategy $2=(0,1,0)$ is optimal, as well as the non-COMB strategy $3=(0,1,1)$ . In this case, we recall that the complement strategies $(1,0,1)$ and $(1,0,0)$ are equivalent, and hence also optimal, but are not depicted. For $n=4$ experts in Figure 2 (d), we see that the COMB strategy $5=(0,1,0,1)$ is optimal, as well as the non-COMB strategy $6=(0,1,1,0)$ . All of these results have already been established theoretically in previous work; see Section 1.1. We presented these results to verify that the numerical solvers are working properly and give results that agree with previous work.

We are unable to solve the PDE (3.6) for $w$ on a full computational grid for the $n=5$ expert problem, so for this we must resort to the sparse grid method that restricts attention to the positive sector.

4.2 Sparse grids

We now turn to computations on the positive sector ${\mathbb {D}}^+_{n-1}\cap [0,T]^{n-1}$ , which is far smaller than the full grid and allows us to run experiments for $n=5,6,7,8,9,10$ experts. In this case, we are solving equation (3.10) using the methods described in Section 3.2. To facilitate computations, we flatten the sector ${\mathbb {D}}^+_{n-1}\cap [0,T]^{n-1}$ to a one-dimensional array, and store the solution $w$ as a one-dimensional array with linear indexing. We pre-computed and stored the stencils for second derivatives in all directions ${\mathbf {v}}\in {\mathcal B}^{n-1}$ using the methods outlined in Section 3.2 prior to the running the iteration (4.1) to solve the equation. All code is written in Python using the Numpy package and fully vectorized operations.

Figure 3. Numerical computation of strategy optimality for the $n=5,6$ expert problems.

We used grid resolutions of $h=0.025$ for $n=5$ , $h=0.05$ for $n=6$ and $h=0.1$ for $n=7$ , $h=0.2$ for $n=8$ , $h=0.25$ for $n=9$ , and $h=0.35$ for $n=10$ experts. Table 1 shows the number of grid points used in each dimension, compared to the number of grid points that would be required on the full grid. The simulations required between 25 GB and 75 GB of memory and each took less than one day to run on a single processor. Figures 3, 4, and 5 show the numerically computed optimality of the adversary’s strategies for $n=5,6,7,8,9,10$ experts. In all cases, we have strong numerical evidence to indicate that the COMB strategy is not globally optimal. This corroborates numerical evidence from [Reference Chase17] for $n=5$ .

Table 1. Number of grid points in the sector computational domain ${\mathbb {D}}^+_d\cap [0,5]^d$ compared to the full grid $\mathbb {Z}_h^d\cap [\!-5,5]^d$ . We use fewer grid points as the dimension increases since evaluating the PDE involves computing derivatives in $2^d$ directions, so the computational time and memory storage increase exponentially with $d$

Figure 4. Numerical computation of strategy optimality for the $n=7,8$ expert problems.

Figure 5. Numerical computation of strategy optimality for the $n=9,10$ expert problems.

We have strong numerical evidence in Figure 3 that the strategy $11=(0,1,0,1,1)$ is optimal for the $n=5$ expert problem over the box $[\!-1,1]^5$ . Furthermore, the numerical evidence points to this being the only optimal strategy, over the unit box, for $n=5$ experts. The minimum optimality score for strategy $11$ is $0.9999999999991459$ , which is far more accurate than the second-order $O(h^2)$ accuracy for $h=0.05$ would suggest. The second best competing strategy is $13=(0,1,1,0,1)$ with a minimum optimality score of $0.966$ , which is well outside the range of numerical precision, suggesting that strategy $13$ is not globally optimal.

The remaining plots in Figures 3, 4, and 5 provide very strong numerical evidence that there are no globally optimal adversarial strategies for $n=6,7,8,9,10$ experts. The highest minimum optimality scores are well outside of numerical precision. The one exception is the $n=10$ expert problem, where there are strategies with minimum optimality scores above $0.98$ , while the grid resolution of $h=0.35$ yields $h^2 = 0.1225$ . However, we expect this is an artefact from using a very coarse grid. We observed a similar phenomenon with $n=9$ experts, where some strategies appeared more optimal on a coarse grid.