Continuous piecewise linear representation of the approximate potential 𝑉 ℎ

One of the most successful applications of continuous piecewise-linear functions has been in nonlinear circuit theory, in particular, in the field of nonlinear resistive networks. In 1965, Katzenelson [45]

presented an effective approach for the search of the operating point of a piecewise-linear resistive network. Since then, the same idea has been extended, improved and generalized [11,13,18]. Among these extensions and generalizations are the resolution of general nonlinear equations of the form 𝑓(𝑥) = 0 where 𝑓 : ℝ^𝑑 → ℝ^𝑚 is a continuous mapping [12], and the introduction of a canonical piecewise-linear function [43, 66] which allows for the compact and closed representation of any function using the minimal number of parameters by taking advantage of a simplicial partition of the domain [39].

Our aim in this section is to describe a unique, systematic and efficient representation of the continuous piecewise-linear approximation of a function𝑓 :ℝ^𝑑 →ℝ, in general, and the potential energy𝑉 :ℝ^𝑑 →ℝ, in particular. With this goal in mind, we first consider a region 𝒮 ∈ℝ^𝑑 and its regular tessellation in hyperrectangles or orthotopes with characteristic length ℎ𝑗 ∈ℝ⁺ in the 𝑗-th direction, with𝑗= 1,2, . . . , 𝑑. We then subdivide each hyperrectangle into proper simplices by introducing a unique, regular triangulation𝒯ℎin accordance with Chien and Kuh [11], see Figure 3.2.

For later reference, we define the homeomorphism Λ :𝒮 → 𝒮𝐶, represented by a diagonal matrix𝑇, i.e.,𝑧=𝑇 𝑞, which maps hyperrectangles in𝒮 into hypercubes [0,1]^𝑑 in a new region𝒮_𝐶∈ℝ^𝑑. For

later reference, we name𝒱 and𝒱_𝐶 the sets of vertices contained in regions𝒮 and𝒮_𝐶, respectively.

Figure 3.2: Simplicial partition of hypercubes [0,1]² and [0,1]³.

Before we discuss further details of the continuous piecewise-linear representation of the approx- imate potential energy, we may first recall some properties of a simplex and its boundaries.

Definition 3.3.1 Let𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑be(𝑑+1)points in general position in the𝑑-dimensional space. A simplexΔ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑)is defined as the convex hull of𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑—the vertices of the simplex—

Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑) = {

𝑞:𝑞=

𝑑

∑

𝑖=0

𝜇𝑖𝑞^𝑖,1≥𝜇𝑖 ≥0, 𝑖= 0,1,2, . . . , 𝑑 and

𝑑

∑

𝑖=0

𝜇𝑖= 1 }

In addition, corresponding to the (𝑑+ 1)vertices, there are (𝑑+ 1) boundaries. The boundary 𝐵_𝑠 which corresponds to the vertex 𝑞^𝑠is defined as

𝐵𝑠={

𝑞:𝑞∈Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑)with 𝜇𝑠= 0}

Since 𝐵𝑠 contains all the vertices except 𝑞^𝑠, there is a one-to-one correspondence between vertices and boundaries.

We now present the key steps for achieving a unique and systematic representation of force- stepping trajectories. To this end, we first subdivide each hypercube which belongs to 𝒮𝐶 into

non-overlapping simplices by properly arranging vertices of 𝒱_𝐶 in a fixed order [11]. This ordering relation allows for a unique representation of every point 𝑧 ∈ 𝒮_𝐶 and 𝑞 ∈ 𝒮 (see Lemma 3.3.1).

Therefore, there is a unique representation of the affine function𝑉ℎ(𝑞) :ℝ^𝑑→ℝwith𝑉ℎ(𝑞^𝑖) =𝑉(𝑞^𝑖) for all vertices𝑞^𝑖 ∈ 𝒱 (see Lemma 3.3.2) upon which force-stepping trajectories are systematically built.

Lemma 3.3.1 Every𝑧∈ 𝒮𝐶 has a unique representation𝑧=∑𝑚

𝑖=0𝜇𝑖𝑧^𝑖 where𝜇𝑖>0,𝑧^𝑖 ∈ 𝒱𝐶, for 𝑖= 0,1, . . . , 𝑚(≤𝑑),∑𝑚

𝑖=0𝜇_𝑖 = 1, and the following ordering relation is attained 𝑧⁰ ⩽𝑧¹ ⩽. . .⩽ 𝑧^𝑚≦𝑧⁰+1. Likewise, every𝑞∈ 𝒮has a unique representation𝑞=∑𝑚

𝑖=0𝜇_𝑖𝑞^𝑖where𝑞^𝑖 =𝑇⁻¹𝑧^𝑖∈ 𝒱, and the following ordering relation is attained 𝑇 𝑞⁰⩽𝑇 𝑞¹⩽. . .⩽𝑇 𝑞^𝑚≦𝑇 𝑞⁰+ 1.

Proof. The proof in 𝒮𝐶 is given by Kuhn [47]. The proof in 𝒮 follows from the fact that the homeomorphism Λ(⋅) is defined by a positive-definite diagonal matrix𝑇 and therefore preserves the ordering of vertices.

Corollary 3.3.1 If 𝑚=𝑑,𝑧 is an interior point of the simplexΔ(𝑧⁰, 𝑧¹, . . . , 𝑧^𝑑). Otherwise,𝑧 lies on the boundary of a simplex. In addition, every 𝑑-dimensional simplex defined by(𝑑+ 1) vertices contains𝑧⁰ and𝑧⁰+ 1which define the hypercube 𝐶(𝑧⁰)containing the simplex.

Lemma 3.3.2 A nonlinear function𝑉(𝑞) :ℝ^𝑑 →ℝ is approximated by an affine function𝑉ℎ(𝑞) : ℝ^𝑑→ℝas follows

𝑉ℎ(𝑞) =∇𝑉Δ⋅𝑞+𝑉0

for 𝑞 ∈ Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑), ∇𝑉Δ = ¯𝐽ℎ,[1...𝑑]𝑇 ∈ ℝ^𝑑—the first 𝑑 elements of the vector—and 𝑉0 = 𝐽¯_ℎ,[𝑑+1] ∈ ℝ. The Jacobian matrix of the piecewise-linear transformation is unique for a given simplex Δand has the form

𝐽¯_ℎ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑) = (

𝑉(𝑞⁰) ⋅ ⋅ ⋅ 𝑉(𝑞^𝑑) )

⎛

⎜

⎜

⎝

𝑇 𝑞⁰ ⋅ ⋅ ⋅ 𝑇 𝑞^𝑑 1 ⋅ ⋅ ⋅ 1

⎞

⎟

⎟

⎠

−1

=:𝑉_Δ𝑍_Δ⁻¹

with𝑉Δ∈ℝ^𝑑+1 and𝑍_Δ⁻¹∈ℕ(𝑑+1)×(𝑑+1).

Proof. It follows from Lemma 3.3.1 that there is a unique representation of points𝑞∈Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑) and𝑧=𝑇 𝑞 given by

⎛

⎜

⎜

⎝ 𝑧 1

⎞

⎟

⎟

⎠

=

⎛

⎜

⎜

⎝ 𝑇 𝑞

1

⎞

⎟

⎟

⎠

=

⎛

⎜

⎜

⎝

𝑇 𝑞⁰ ⋅ ⋅ ⋅ 𝑇 𝑞^𝑑 1 ⋅ ⋅ ⋅ 1

⎞

⎟

⎟

⎠

𝜇=𝑍Δ𝜇 (3.22)

We now define the affine function𝑉ℎ(𝑞), with𝑉ℎ(𝑞^𝑖) =𝑉(𝑞^𝑖) for𝑖= 0,1, . . . , 𝑑, as

𝑉ℎ(𝑞) = (

𝑉(𝑞⁰) ⋅ ⋅ ⋅ 𝑉(𝑞^𝑑) )

⋅𝜇=𝑉Δ𝑍_Δ⁻¹⋅

⎛

⎜

⎜

⎝ 𝑧 1

⎞

⎟

⎟

⎠

= ¯𝐽ℎ⋅

⎛

⎜

⎜

⎝ 𝑇 𝑞

1

⎞

⎟

⎟

⎠

(3.23)

Then, the claim follows from writing the affine function as 𝑉_ℎ(𝑞) =∇𝑉Δ⋅𝑞+𝑉₀. We note that both matrices𝑍_Δ and𝑍_Δ⁻¹ belong toℕ(𝑑+1)×(𝑑+1)due to the fact that hypercubes in𝒮_𝐶 have unit volume, i.e., det(𝑍Δ) = 1.

We then show that force-stepping trajectories {𝑡𝑘, 𝑞𝑘,𝑞˙𝑘}, as described in Definition 3.2.1, can be systematically tracked in phase space. In particular, we show that successive times of boundary crossings𝑡0< 𝑡1< 𝑡2< . . .and the corresponding sequence of positions𝑞0, 𝑞1, 𝑞2, . . .can be obtained from the simplicial partition presented above. The procedure applied for solving this problem, namelythe time-step problem(see Proposition 3.3.1), is indeed the methodTime-Steprequired by the implementation of the force-stepping integrator presented in Algorithm 3,

{𝑡𝑘+1, 𝐵𝑠} ←Time-Step(

𝑞𝑘,𝑞˙𝑘,∇𝑉𝑘, 𝑡𝑘;𝑍_Δ⁻¹

𝑘

)

Proposition 3.3.1 (The time-step problem) A general trajectory of the form𝑞(𝑡) =𝑞𝑘+ (𝑡− 𝑡𝑘) ˙𝑞𝑘−¹₂(𝑡−𝑡𝑘)²𝑀⁻¹∇𝑉𝑘, with𝑞(𝑡⁺_𝑘)∈Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑)and𝑡 > 𝑡𝑘, intersects the boundary𝐵𝑠 of the simplexΔ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑) at𝑞(𝑡𝑘+1) =𝑞𝑘+1 with

𝑡_𝑘+1= min

𝑟∈[0,𝑑]inf{

𝑡 > 𝑡_𝑘 :𝜇_1,𝑟+ (𝑡−𝑡_𝑘)𝜇_2,𝑟+ (𝑡−𝑡_𝑘)²𝜇_3,𝑟 = 0} 𝑠= arg min

𝑟∈[0,𝑑]

{𝜇1,𝑟+ (𝑡𝑘+1−𝑡𝑘)𝜇2,𝑟+ (𝑡𝑘+1−𝑡𝑘)²𝜇3,𝑟 = 0}

where

(

𝜇1 𝜇2 𝜇3

)

=𝑍_Δ⁻¹

⎛

⎜

⎜

⎝

𝑇 𝑞𝑘 𝑇𝑞˙𝑘 −𝑀⁻¹𝑇∇𝑉𝑘/2

1 0 0

⎞

⎟

⎟

⎠

Proof. It follows from Definition 3.3.1 and Lemma 3.3.1 that a trajectory of the form 𝑞(𝑡) =

∑𝑑

𝑟=0𝜇𝑟(𝑡)𝑞^𝑟, with𝑞(𝑡⁺_𝑘)∈Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑) and𝑡 > 𝑡𝑘, intersects the boundary𝐵𝑟^∗ of the simplex Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑑) at 𝑞(𝑡^∗) if and only if the 𝑟^∗-th component of𝜇(𝑡^∗) is equal to zero with all other components positive. Then, the following equality holds

𝜇(𝑡) =𝑍_Δ⁻¹

⎛

⎜

⎜

⎝ 𝑇 𝑞(𝑡)

1

⎞

⎟

⎟

⎠

=𝑍_Δ⁻¹

⎛

⎜

⎜

⎝ 𝑇 𝑞_𝑘

1

⎞

⎟

⎟

⎠

| {z }

𝜇₁∈ℝ^𝑑+1

+(𝑡−𝑡𝑘)𝑍_Δ⁻¹

⎛

⎜

⎜

⎝ 𝑇𝑞˙_𝑘

0

⎞

⎟

⎟

⎠

| {z }

𝜇₂∈ℝ^𝑑+1

+(𝑡−𝑡𝑘)²𝑍_Δ⁻¹

⎛

⎜

⎜

⎝

−𝑀⁻¹𝑇∇𝑉_𝑘/2 0

⎞

⎟

⎟

⎠

| {z }

𝜇₃∈ℝ^𝑑+1

and the claim follows from the fact that𝑡_𝑘+1 is the earliest𝑡^∗> 𝑡_𝑘 for which𝜇_𝑟^∗_=𝑠(𝑡^∗) = 0 among all possible pairs (𝑡^∗, 𝑟^∗).

It is worth noting that the representation of the approximate potential energy𝑉ℎcan be restricted solely to the current simplex Δ𝑘 and the relevant simplex-related matrices are⟨𝑉Δ_𝑘, 𝑍Δ_𝑘, 𝑍_Δ⁻¹

𝑘⟩, with

𝑉Δ_𝑘 = (

𝑉(𝑞⁰) ⋅ ⋅ ⋅ 𝑉(𝑞^𝑠) ⋅ ⋅ ⋅ 𝑉(𝑞^𝑑) )

(3.24a)

𝑍_Δ𝑘 =

⎛

⎜

⎜

⎝

𝑧⁰ ⋅ ⋅ ⋅ 𝑧^𝑠 ⋅ ⋅ ⋅ 𝑧^𝑑 1 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ 1

⎞

⎟

⎟

⎠

(3.24b)

We then claim thatthe replacement rule (see Chien and Kuh [11] and Lemma 3.3.3) provides for a very efficient scheme to construct the adjacent simplex Δ𝑘+1 knowing only the boundary𝐵𝑠of the current simplex Δ𝑘 intersected by the trajectory. Moreover, we describe an efficient procedure for updating all simplex-related matrices required to compute the force-stepping trajectory in Δ𝑘+1. In particular, given all Δ𝑘-related matrices and the intersected boundary𝐵𝑠, matrices are updated as

follows

𝑉Δ_𝑘+1= (

𝑉(𝑞⁰) ⋅ ⋅ ⋅ 𝑉(𝑞^′𝑠) ⋅ ⋅ ⋅ 𝑉(𝑞^𝑑) )

(3.25a)

𝑍Δ_𝑘+1=

⎛

⎜

⎜

⎝

𝑧⁰ ⋅ ⋅ ⋅ 𝑧^′𝑠 ⋅ ⋅ ⋅ 𝑧^𝑑 1 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ 1

⎞

⎟

⎟

⎠

(3.25b)

𝑍_Δ⁻¹

𝑘+1=𝑍_Δ⁻¹

𝑘− (𝑍_Δ⁻¹

𝑘𝑢^𝑠)

⊗( 𝑣^𝑠𝑍_Δ⁻¹

𝑘

) 1 +𝑣^𝑠𝑍_Δ⁻¹

𝑘𝑢^𝑠 (3.25c)

where 𝑧^′𝑠 = 𝑧^𝑠+1+𝑧^𝑠−1−𝑧^𝑠, 𝑞^′𝑠 = 𝑇⁻¹𝑧^′𝑠, 𝑢^𝑠 = 𝑧^′𝑠−𝑧^𝑠, and 𝑣_𝑖^𝑠 = 𝛿𝑖,𝑠. Matrices 𝑍Δ_𝑘+1 and 𝑉Δ_𝑘+1 are obtained by replacing𝑧^𝑠with𝑧^′𝑠 and𝑉(𝑞^𝑠) with𝑉(𝑞^′𝑠), respectively. The matrix𝑍_Δ⁻¹

𝑘+1

is 1-rank updated by means of the Sherman-Morrison formula [28]—with algorithm complexity 𝑂(𝑑²)—instead of computing the inverse of 𝑍_Δ𝑘+1—with algorithm complexity𝑂(𝑑³). The overall gain in efficiency is remarkable.

Lemma 3.3.3 (The replacement rule) Let𝐵𝑠be the boundary shared by two adjacent simplices.

Given the simplex Δ(𝑞⁰, 𝑞¹, . . . , 𝑞^𝑠−1, 𝑞^𝑠, 𝑞^𝑠+1, . . . , 𝑞^𝑑), with 𝑇 𝑞⁰⩽𝑇 𝑞¹⩽. . .⩽𝑇 𝑞^𝑑, its neighbor is simply defined by replacing𝑞^𝑠with 𝑞^′𝑠. The new vertex is defined as

𝑞^′𝑠=𝑞^𝑠+1+𝑞^𝑠−1−𝑞^𝑠

where𝑞^𝑑+1≡𝑞⁰ and𝑞⁻¹≡𝑞^𝑑.

Corollary 3.3.2 The replacement rule preserves the order in the set of vertices𝒱.

Proof. For𝑠= 1, . . . , 𝑑−1, the following relation holds𝑇 𝑞^𝑠−1⩽𝑇 𝑞^′𝑠⩽𝑇 𝑞^𝑠+1. For𝑠= 0 (𝑠=𝑑) a simple backward (forward) shift of indices has to be introduced in the updated set of vertices. After the shift is performed, the new set of vertices verifies the ordering relation.

We conclude this section with the second method required by the implementation of the force- stepping integrator presented in Algorithm 3,

{∇𝑉𝑘+1,⟨𝑉Δ_𝑘+1, 𝑍Δ_𝑘+1, 𝑍_Δ⁻¹

𝑘+1⟩}

←Update(

∇𝑉𝑘,⟨𝑉Δ_𝑘, 𝑍Δ_𝑘, 𝑍_Δ⁻¹

𝑘⟩;𝐵𝑠

)

The method is defined in Algorithm 4 and is responsible for applyingthe replacement ruleas well as updating∇𝑉_𝑘 and all simplex-related matrices⟨𝑉_Δ𝑘, 𝑍_Δ𝑘, 𝑍_Δ⁻¹

𝑘⟩. We remark that, for all practical purposes, there is no need to shift indices after vertices𝑞⁰ or 𝑞^𝑑 are updated by the algorithm, as suggested by Corollary 3.3.2.

Algorithm 4Update(

∇𝑉𝑘,⟨𝑉Δ_𝑘, 𝑍Δ_𝑘, 𝑍_Δ⁻¹

𝑘⟩;𝐵𝑠)

Require: 𝑧^′𝑠=𝑧^𝑠+1+𝑧^𝑠−1−𝑧^𝑠,𝑢^𝑠=𝑧^′𝑠−𝑧^𝑠,𝑣^𝑠_𝑖 =𝛿𝑖,𝑠 and𝑞^′𝑠=𝑇⁻¹𝑧^′𝑠

1: 𝑍_Δ⁻¹

𝑘+1 ←𝑍_Δ⁻¹

𝑘−^(𝑍

−1

Δ𝑘𝑢^𝑠)⊗(𝑣^𝑠𝑍_Δ⁻¹

𝑘) 1+𝑣^𝑠𝑍⁻¹_Δ

𝑘𝑢^𝑠 2: 𝑍_Δ𝑘+1 ←

( 𝑧⁰ ⋅ ⋅ ⋅ 𝑧^′𝑠 ⋅ ⋅ ⋅ 𝑧^𝑑 1 ⋅ ⋅ ⋅ 1 ⋅ ⋅ ⋅ 1

)

3: 𝑉Δ_𝑘+1←(

𝑉(𝑞⁰) ⋅ ⋅ ⋅ 𝑉(𝑞^′𝑠) ⋅ ⋅ ⋅ 𝑉(𝑞^𝑑) )

4: 𝐽¯ℎ←𝑉Δ_𝑘+1𝑍_Δ⁻¹

𝑘+1

5: ∇𝑉𝑘+1←𝐽¯_{ℎ,[1...𝑑]}𝑇

6: return ∇𝑉𝑘+1,⟨𝑉Δ_𝑘+1, 𝑍Δ_𝑘+1, 𝑍_Δ⁻¹

𝑘+1⟩