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  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    PDEs with nonlocal nonlinearities:Generation and solution

    Margaret Beck, Anastasia Doikou, Simon J.A. Malham,Ioannis Stylianidis and Anke Wiese

    Sheffield 2018: November 21st

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Outline

    1 Motivation.

    2 PDEs with nonlocal nonlinearities (examples).

    3 Infinite dimensional Riccati flows.

    4 PDEs with nonlocal nonlinearities (revisited in detail).

    5 Nonlinear PDEs and SPDEs.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Motivation: Integrable systems and Fredholm determinants

    Dyson; Miura; Ablowitz, Ramani & Segur; Pöppe; Sato; Segal &Wilson; Tracy & Widom...

    Quoting Pöppe:

    “For every soliton equation, there exists a linear PDE(called a base equation) such that a map can be definedmapping a solution f of the base equation to a solutionu of the soliton equation. The properties of the solitonequation may be deduced from the corresponding proper-ties of the base equation which in turn are quite simpledue to linearity. The map f → u essentially consists ofconstructing a set of linear integral operators using f andcomputing their Fredholm determinants.”

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Motivation: Marchenko equation

    Ablowitz, Ramani & Segur: Marchenko equation, y > x :

    p(x , y) = g(x , y) +

    ∫ ∞x

    g(x , z)q′(z , y ; x)dz ,

    1 Scattering data: p = p(x + y) and q′.

    2 KdV q′ = −p:

    ∂tp + ∂3p = 0 ⇒ u = −2(d/dx)g(x , x) satisfies KdV.

    3 Pöppe: operator level.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Example PDEs with nonlocal nonlinearities

    1 Fisher–Kolmogorov–Petrovskii–Piskunov (FKPP):

    ∂tg(x ; t) = d(∂x)g(x ; t)− g(x ; t)∫Rb(z , ∂z) g(z ; t) dz .

    2 Quadratic:

    ∂tg(x , y ; t) = d(∂x) g(x , y ; t)−∫Rg(x , z ; t) g(z , y ; t) dz .

    3 Odd degee:

    ∂tg = −ih(∂1) g − g ? f ?(g ? g †).

    4 Rational:

    ∂tg(x , y ; t) = d(√−∆x

    )g(x , y ; t)− g(x , y ; t)b(y)g(y , y ; t).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Smoluchowski’s coagulation equation

    ∂tg(x ; t) =12

    ∫ x0

    K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸coagulation gain

    − g(x ; t)∫ ∞

    0K (x , y) g(y ; t) dy︸ ︷︷ ︸

    coagulation loss

    .

    Applications: polymerisation, aerosols, clouds/smog, clusteringstars/galaxies, schooling/flocking.

    Special cases: (i) K = 2; (ii) K = x + y and (iii) K = xy .

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Riccati Flows: Quadratic degree

    1. ∂tQ = AQ + BP,

    2. ∂tP = CQ + DP,

    3. P = G Q.

    ⇒(∂tG

    )Q = ∂tP − G ∂tQ

    = CQ + DP − G (AQ + BP)= (C + DG )Q − G (A + BG )Q

    ⇒ ∂tG = C + DG − G (A + BG ).

    (Q0 = I , P0 = G0.)

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Riccati Flows: Higher odd degree

    1. ∂tQ = f (PP†)Q,

    2. ∂tP = DP,

    3. P = G Q.

    Require:

    QQ† = id,

    f (x) = i∑m>0

    αmxm.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Fredholm Grassmann Flows: Higher odd degree II

    f † = −f ⇒ ∂t(QQ†

    )= f (QQ†)− (QQ†) f

    ⇒ QQ† = id⇒ PP† = GG †

    (∂tG

    )Q = ∂tP − G ∂tQ

    = DG Q − G f (PP†)Q= DG Q − G f (GG †)Q.

    ⇒ ∂tG = D G − G f (GG †).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    “Big matrix” PDEs

    ∂tG = DG − G BG

    ⇔ ∂tg(x , y ; t) = d(∂x) g(x , y ; t)−∫Rg(x , z ; t) b(z) g(z , y ; t)dz .

    We set (big matrix product)(q ? q̃

    )(x , y ; t) :=

    ∫Rq(x , z ; t)q̃(z , y ; t) dz .

    ∂tG = D G − G f (GG †)⇔ ∂tg = −ih(∂1) g + g ? f ?(g ? g †).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    In practice: Quadratic “Big matrix” PDE

    Suppose we wish to solve the PDE

    ∂tg(x , y ; t) = d(∂x) g(x , y ; t)−∫Rg(x , z ; t) b(z) g(z , y ; t) dz .

    Then our prescription says set up:

    1. ∂tp(x , y ; t) = d(∂x) p(x , y ; t),

    2. ∂tq(x , y ; t) = b(x) p(x , y ; t).

    3. p(x , y ; t) = g(x , y ; t) +

    ∫Rg(x , z ; t) q′(z , y ; t)dz .

    Here we set q(x , y ; t) = δ(x − y) + q′(x , y ; t).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    In practice: What have we gained?

    ∂tp(x , y ; t) = d(∂x) p(x , y ; t),

    ∂tq(x , y ; t) = b(x) p(x , y ; t).

    p̂(k, y ; t) = ed(2πik) t p̂0(k , y),

    q̂′(k, y ; t) = e2πiky +

    ∫Rb̂(k − κ) Î (κ, t) p̂0(κ, y) dκ,

    Î (k , t) :=(ed(2πik) t − 1

    )/d(2πik).

    We can solve the linear equations for p and q explicitly andevaluate them for any given time t > 0. Then we can determinethe solution to the PDE at that time by solving the linearFredholm equation for g .

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    In practice: Higher odd degree

    Suppose we wish to solve the PDE

    ∂tg = −ih(∂1) g + g ? f ?(g ? g †).

    Our prescription says set up:

    1. ∂tp = −ih(∂1) p,2. ∂tq = f

    ?(p ? p†

    )? q,

    3. p = g ? q.

    1. ∂t p̂ = −ih(2πik) p̂,

    2. ∂t q̂ = f̂?(p̂ ? p̂†

    )? q̂,

    3. p̂(k , κ; t) =

    ∫Rĝ(k , λ; t)q̂(λ, κ; t)dλ.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    In practice: Higher odd degree II

    p̂(k , κ; t) = exp(−it h(2πik)

    )p̂0(k , κ),

    θ̂(k , κ; t) := exp(it h(−2πik)

    )q̂(k , κ; t).

    ∂t θ̂ =(f̂ ?(p̂0 ? p̂

    †0) + ih δ

    )? θ̂,

    θ̂(k, κ; t) = exp?(t(f̂ ?(p̂0 ? p̂

    †0) + ih δ

    )),

    Solve: p̂(k , κ; t) =

    ∫Rĝ(k, λ; t)q̂(λ, κ; t) dλ.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Generalized NLS

    h(x) = x4, f (x) = sin(x).

    ⇒ equation for g = g(x , y ; t):

    i∂tg = ∂41 g + g ? sin

    ?(g ? g †

    ),

    g0(x , y) := sech(x + y) sech(y).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Generalized NLS figures

    0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

    real(det)

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    ima

    g(d

    et)

    Determinant in complex plane

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Classical example

    1. ∂tp(y) = d(∂y ) p(y),

    2. ∂tq(y) = b(∂y ) p(y),

    3. p(y) =

    ∫Rg(z) q(z + y) dz .

    ⇒∫R∂tg(z) q(z + y)dz

    =∂tp(y)−∫Rg(z) ∂tq(z + y) dz

    = d(∂y ) p(y)−∫Rg(z) b(∂z) p(z + y) dz

    =

    ∫Rg(z) d(∂y ) q(z + y) dz −

    ∫Rg(z) b(∂z)

    ∫Rg(ζ) q(ζ + z + y)dζ dz

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Classical example cont’d

    ⇒∫R∂tg(z) q(z + y) dz

    =

    ∫R

    (d(−∂z) g(z)

    )q(z + y) dz

    −∫R

    (b(−∂z) g(z)

    ) ∫Rg(ζ) q(ζ + z + y) dζ dz

    =

    ∫R

    (d(−∂z) g(z ; t)

    )q(z + y ; t) dz

    −∫R

    (b(−∂z) g(z ; t)

    ) ∫Rg(ξ − z ; t) q(ξ + y ; t)dξ dz .

    ⇒ ∂tg(η) = d(−∂η) g(η)−∫R

    (b(−∂z) g(z)

    )g(η − z) dz .

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Fisher–Kolmogorov–Petrovskii–Piskunov equation (FKPP)

    Consider nonlocal FKPP (see Britton or Bian, Chen & Latos):

    ∂tg(x ; t) = d(∂x)g(x ; t)− g(x ; t)∫Rb(z , ∂z) g(z ; t) dz .

    1. ∂tp(x) = d(∂x) p(x),

    2. ∂tq(x) = b(x , ∂x) p(x),

    3. p(x) = g(x)

    ∫Rq(z) dz .

    (= g(x)q

    ).

    (∂tg(x ; t)

    )q(t) = ∂tp(x ; t)− g(x ; t) ∂tq(t)

    = d(∂x)p(x ; t)− g(x ; t)∫Rb(z , ∂z) p(z ; t) dz

    = d(∂x)g(x ; t) q(t)− g(x ; t)∫Rb(z , ∂z) g(z ; t)dz q(t).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    FKPP II

    Given data g0, set q(0) = 1 & p(x ; 0) = g0(x) ⇒

    g(x ; t) =p(x ; t)

    q(t)

    Consider the case b = 1:

    p̂(k; t) = exp(d(2πik) t

    )ĝ0(k),

    q(t) = 1 +

    (exp(t d(0))− 1

    d(0)

    )ĝ0(0).

    d(0) = 0 ⇒ q(t) = 1 + t ĝ0(0).

    ⇒ explicit solution for any diffusive or dispersive d = d(∂x).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Smoluchowski’s coagulation equation

    ∂tg(x ; t) =12

    ∫ x0

    K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸coagulation gain

    − g(x ; t)∫ ∞

    0K (x , y) g(y ; t) dy︸ ︷︷ ︸

    coagulation loss

    .

    g(x , t) = density of clusters of mass x ;

    Tag cluster mass x : rate it merges with cluster mass yproportional to the density of clusters;

    Constant of proportionality is K = K (x , y) or frequency;

    Rate coalesce (y , x−y)→ x is 12K (y , x−y) g(y ; t)g(x−y ; t);Loss rate is K (x , y) g(x ; t)g(y ; t).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Smoluchowski’s coagulation equation II

    ∂tg(x ; t) =12

    ∫ x0

    K (y , x − y) g(y ; t)g(x − y ; t) dy︸ ︷︷ ︸coagulation gain

    − g(x ; t)∫ ∞

    0K (x , y) g(y ; t) dy︸ ︷︷ ︸

    coagulation loss

    .

    Consider the case K = 1. With q(t) :=∫∞

    0 q(z) dz :

    1. ∂tp(x) = 0,

    2. ∂tq(x) = −12p(x) q2,

    3. p(x) =

    ∫ ∞0

    g(z) q(z + x)dz1

    q2.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Smoluchowski’s coagulation equation III

    ⇒ ∂tq = −12g q.

    ⇒∫ ∞

    0∂tg(z) q(z + x)dz

    1

    q2

    = −∫ ∞

    0g(z) ∂tq(z + x) dz

    1

    q2+ 2

    ∫ ∞0

    g(z) q(z + x) dz∂tq

    q3

    = 12

    ∫ ∞0

    g(z)

    ∫ ∞0

    g(ζ) q(ζ + z + x)dζ dz1

    q2−∫ ∞

    0g(z) q(z + x)dz

    g

    q2

    = 12

    ∫ ∞0

    g(z)

    ∫ ∞z

    g(ξ − z) q(ξ + x)dξ dz 1q2−∫ ∞

    0g(z) q(z + x)dz

    g

    q2.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Smoluchowski’s gain-only coagulation equation

    ∂tg(x ; t) =12

    ∫ x0

    K (y , x − y) g(y ; t)g(x − y ; t) dy

    1. ∂tp(x) = 0,

    2. ∂tq(x) = −12p(x),

    3. p(x) =

    ∫ ∞0

    H(x , z) g(z) q(z + x)dz ,

    Inversion:

    ∫ ∞0

    H(x , z) q(z + x)q∗(x , y) dx = δ(y − z).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Smoluchowski’s gain-only coagulation equation II

    Similar calculation ⇒∫ ∞0

    H(x , z) ∂tg(z) q(z + x) dz

    = 12

    ∫ ∞0

    ∫ z0

    H(x , ξ)H(x + ξ, z − ξ)︸ ︷︷ ︸=K(z,ξ)H(x ,z)

    g(ξ)g(z − ξ) dξ q(z + x) dz .

    Example cases:

    1 H(x , z) = eαxz ⇒ K (ξ, z − ξ) = eαξ(z−ξ);2 H(x , z) = eα(x

    2z+xz2) ⇒ . . .;3 Exponential of symmetric polynomials and further

    generalisations.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Smoluchowski’s coagulation equation: Add/Multive kernels

    Kernel cases K (x , y) = x + y and K (x , y) = xy intimately related.

    ∂tg(x ; t) =12x

    ∫ x0

    g(y ; t)g(x−y ; t) dy−g(x ; t)∫ ∞

    0(x+y) g(y ; t) dy .

    Desingularised LT: π(s, t) =

    ∫ ∞0

    (1− e−sx) g(x , t) dx .

    Menon & Pego (2003) ⇒

    K = 1 : ∂tπ +12π

    2 = 0;

    K = x + y : ∂tπ + π∂sπ = −π;K = xy : ∂t π̃ + π̃∂s π̃ = 0.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Optimal nonlinear control: Riccati PDEs

    Byrnes (1998) and Byrnes & Jhemi (1992) ⇒

    Nonlinear state evolution: q̇ = b(q) + σ(q)u;

    Nonlinear cost function:∫ T

    0 L(q, u) dt + Q(q(T )

    );

    Bolza problem ⇒ optimal u∗ = u∗(q, p) with

    q̇ = ∇pH∗, ṗ = −∇qH∗ and pT = −∇Q(q(T )

    );

    Goal: Find map π s.t. p = π(q, t) ⇒ u∗ = u∗(q, π(q, t)

    );

    Generates ∂tπ = ∇qH∗(q, π) + (∇qπ)(∇πH∗(q, π)

    ); (offline)

    L(q, u) = |u|2 ⇒ p = u and q̇ = p, ṗ = 0 so ∂tπ = (∇π)π.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Inviscid Burgers flow

    Characteristics: Assume π sat. ∂tπ + (∇π)π = 0; set q̇ = π.

    Now assume for q = q(a, t) and p = p(a, t) with q(a, 0) = a:

    q̇ = p, ṗ = 0, and p = π(q, t).

    Then 0 = ṗ = ∂tπ + (∇qπ)q̇ = ∂tπ + (∇π)π. To find π = π(x , t):

    π(q(a, t), t) = π0(a) and q(a, t) = a + tπ0(a).

    Hence if

    q(a, t) = x ⇔ x = a + tπ0(a) ⇔ a = (id + tπ0)−1 ◦ x ,

    then π(x , t) = π0 ◦ (id + tπ0)−1 ◦ x .

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Riccati flow is a Burgers subflow

    Riccati flow is Burgers subflow corres. to linear data π0(a) = π0a.

    In this case:

    x = a + tπ0a ⇔ a = (id + tπ0)−1x ,

    π(x , t) = π0a = π0(id + tπ0)−1x .

    Note π(x , t) is linear in x so set πR(t) := π0(id + tπ0)−1:

    ∂tπ(x , t) = −π0(id + tπ0)−1π0(id + tπ0)−1x

    = −(πR(t)

    )2x ,

    (∇xπ)π = πR(t)πR(t)x .

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Burgers flow/SPDEs

    1. Qt(a) = a +

    ∫ t0Ps(a)ds +

    √2νBt ,

    2. ∂tπ+(∇π)π + ν∆π = 0,3. Pt(a) = π

    (Qt(a), t

    ).

    Itô: π(Qt(a), t

    )= π0(a) +

    ∫ t0

    (∂sπ + (∇π)π + ν∆π

    )(Qs(a), s

    )ds

    +√

    ∫ t0

    (∇π)(Qs(a), s)dBs .

    ⇒ π(x , t) = E[π0(Q−1t (x)

    )].

    Constantin & Iyer (2008,. . . )Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Burgers flow/SPDEs II

    1. Qt(a) = a +

    ∫ t0Ps(a) ds +

    √2νBt ,

    2. Pt(a) = P0(a),

    3. Pt(a) = πt(Qt(a), t

    ),

    4. u(x , t) = E[π0(Q−1t (x)

    )]. (Observations—hope?)

    Generalised Itô:

    πt(Qt(a), t

    )= π0(a) +

    ∫ t0

    ((∇πs)πs − ν∆πs

    )(Qs(a), s

    )ds

    +√

    ∫ t0

    (∇πs)(Qs(a), s) dBs +∫ t

    0πs(Qs(a), ds

    ).

    ⇒ dπt+((∇πt)πt − ν∆πt

    )dt +

    √2ν(∇πt) dBt = 0.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Looking forward

    Coordinate patches:(qp

    )=

    (q

    πt ◦ q

    )=

    (idπt

    )◦ q.

    Eg. instead choose q = π′t(p), i.e. π′ = π−1.

    q̇ = p, ṗ = 0 ⇒ p0 = p(t) = q̇ = (∂tπ′t) ◦ p(t) = (∂tπ′t) ◦ p0.

    Setting y = p0 ⇒ π′t ◦ y = π′0 ◦ y + ty .

    Girsanov and Cole–Hopf transformations;

    Q, P and P = π(Q, t) all operators;

    SPDEs, Madelung, dispersion...

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Thank you

    Thank you for listening!

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Nonlocal reaction-diffusion system

    With d11 = ∂21 + 1, d12 = −1/2, b12 = 0 and b11 = N(x , σ):

    ∂tu = d11u + d12v − u ? (b11u)− u ? (b12v)− v ? (b12u)− v ? (b11v),∂tv = d11v + d12u − u ? (b11v)− u ? (b12v)− v ? (b12v)− v ? (b11u),

    u0(x , y) := sech(x+y) sech(y) and v0(x , y) := sech(x+y) sech(x).

    p =

    (p11 p12p12 p11

    ), q =

    (q11 q12q12 q11

    )and g =

    (g11 g12g12 g11

    ).

    Similar forms for d and b. Riccati equation: ∂tG = dG − G (bG )

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Nonlocal reaction-diffusion system II

    -0.1

    10

    0

    0.1

    5 10

    0.2

    u

    0.3

    Direct method: T=0.5

    5

    y

    0

    0.4

    x

    0.5

    0-5

    -5-10 -10

    -0.2

    10

    -0.1

    0

    5 10

    v

    0.1

    Direct method: T=0.5

    5

    y

    0.2

    0

    x

    0.3

    0-5

    -5-10 -10 0 0.1 0.2 0.3 0.4 0.5

    t

    0

    20

    40

    60

    80

    100

    120

    140

    de

    t, n

    orm

    Determinant and Hilbert--Schmidt norm

    Determinant

    HS-norm

    -0.1

    10

    0

    0.1

    5 10

    0.2

    g11

    0.3

    Riccati method: T=0.5

    5

    y

    0

    0.4

    x

    0.5

    0-5

    -5-10 -10

    -0.2

    10

    -0.1

    0

    5 10

    g12

    0.1

    Riccati method: T=0.5

    5

    y

    0.2

    0

    x

    0.3

    0-5

    -5-10 -10

    0

    10

    1

    5 10

    2

    ab

    s-r

    ea

    l

    10 -5

    Euclidean Difference: T=0.5

    5

    3

    y

    0

    x

    4

    0-5

    -5-10 -10

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Stochastic PDEs with nonlocal nonlinearities

    ∂tQ = AQ + BP,

    ∂tP = CQ + DP,

    P = G Q.

    ∂tG = C + DG − G (A + BG ).

    ∂tg = ∂21g + Ẇ ∗ g − g ? g .

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Stochastic PDEs with nonlocal nonlinearities II

    On T = [0, 2π]:

    Wt(x) :=1√π

    ∑n>1

    1

    nW nt cos(nx)

    Suppose pt = pt(x , y) satisfies

    ∂tpt = ∂21pt + Ẇ ∗ pt .

    pt(x , y) =1

    π

    ∑n>0,m>0

    (pssnm sin(nx) sin(my) + p

    csnm cos(nx) sin(my)

    + pscnm sin(nx) cos(my) + pccnm cos(nx) cos(my)

    ).

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Stochastic PDEs with nonlocal nonlinearities II

    s(x) =

    sin(x)sin(2x)...

    and c(x) =

    cocos(x)

    cos(2x)...

    pt(x , y) =(sT(x) cT(x)

    )(pss pscpcs pcc

    )(s(y)c(y)

    )

    =(sT(x) cT(x)

    )P

    (s(y)c(y)

    )

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

  • Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

    Stochastic PDEs with nonlocal nonlinearities III

    Ξ :=1√πdiag{W 1t ,

    1

    2W 2t , . . . , 0,W

    1t ,

    1

    2W 2t , . . .}

    D := −diag{1, 22, 32, . . . , 0, 1, 22, 32, . . .}

    ⇒ ∂tP = DP + Ξ̇P

    ⇒ ∂tpnm = −n2pnm +1

    nẆ nt pnm

    ⇒ pnm = exp(−n2t −

    √πn W

    nt − 12

    πn2t)pnm(0)

    ⇒ p0m = p0m(0)

    ⇒ q′nm =∫ t

    0pnm(τ) dτ.

    Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities

    Grassmann/Riccati flowsPDEs with nonlocal nonlinearities

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Grassmann/Riccati flows PDEs with nonlocal nonlinearities PDEs with nonlocal nonlinearities: Generation and solution Margaret Beck, Anastasia Doikou, Simon J.A. Malham, Ioannis Stylianidis and Anke Wiese Sheffield 2018: November 21st Beck, Doikou, Malham, Stylianidis, Wiese PDEs with nonlocal nonlinearities
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